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††thanks: These authors contributed equally††thanks: These authors contributed equally

Spin order and fluctuations in the EuAl4 and EuGa4 topological antiferromagnets: A μ\muSR study

X. Y. Zhu Key Laboratory of Polar Materials and Devices (MOE), School of Physics and Electronic Science, East China Normal University, Shanghai 200241, China    H. Zhang Key Laboratory of Polar Materials and Devices (MOE), School of Physics and Electronic Science, East China Normal University, Shanghai 200241, China    D. J. Gawryluk Laboratory for Multiscale Materials Experiments, Paul Scherrer Institut, Villigen CH-5232, Switzerland    Z. X. Zhen Key Laboratory of Polar Materials and Devices (MOE), School of Physics and Electronic Science, East China Normal University, Shanghai 200241, China    B. C. Yu Key Laboratory of Polar Materials and Devices (MOE), School of Physics and Electronic Science, East China Normal University, Shanghai 200241, China    S. L. Ju Swiss Light Source, Paul Scherrer Institut, Villigen CH-5232, Switzerland    W. Xie DESY, Notkestraβ\betae 85, D-22607 Hamburg, Germany    D. M. Jiang Key Laboratory of Polar Materials and Devices (MOE), School of Physics and Electronic Science, East China Normal University, Shanghai 200241, China    W. J. Cheng Key Laboratory of Polar Materials and Devices (MOE), School of Physics and Electronic Science, East China Normal University, Shanghai 200241, China    Y. Xu Key Laboratory of Polar Materials and Devices (MOE), School of Physics and Electronic Science, East China Normal University, Shanghai 200241, China    M. Shi Swiss Light Source, Paul Scherrer Institut, Villigen CH-5232, Switzerland    E. Pomjakushina Laboratory for Multiscale Materials Experiments, Paul Scherrer Institut, Villigen CH-5232, Switzerland    Q. F. Zhan Key Laboratory of Polar Materials and Devices (MOE), School of Physics and Electronic Science, East China Normal University, Shanghai 200241, China    T. Shiroka [email protected] Laboratory for Muon-Spin Spectroscopy, Paul Scherrer Institut, Villigen PSI, Switzerland Laboratorium für Festkörperphysik, ETH Zürich, CH-8093 Zürich, Switzerland    T. Shang [email protected] Key Laboratory of Polar Materials and Devices (MOE), School of Physics and Electronic Science, East China Normal University, Shanghai 200241, China
Abstract

EuAl4 and EuGa4 are two candidate materials for studying the interplay between correlated-electron phenomena, topological spin textures, and topologically nontrivial bands. Both compounds crystallize in a centrosymmetric tetragonal BaAl4-type structure (space group I​4/m​m​mI4/mmm) and show antiferromagnetic (AFM) order below TN=15.6T_{\mathrm{N}}=15.6 and 16.4 K, respectively. Here, we report on systematic muon-spin rotation and relaxation (μ\muSR) studies of the magnetic properties of EuAl4 and EuGa4 single crystals at a microscopic level. Transverse-field μ\muSR measurements, spanning a wide temperature range (from 1.5 to 50 K), show clear bulk AFM transitions, with an almost 100% magnetic volume fraction in both cases. Zero-field μ\muSR measurements, covering both the AFM and the paramagnetic (PM) states, reveal internal magnetic fields Bint​(0)=0.33B_{\mathrm{int}}(0)=0.33 T and 0.89 T in EuAl4 and EuGa4, respectively. The transverse muon-spin relaxation rate λT\lambda_{\mathrm{T}}, a measure of the internal field distribution at the muon-stopping site, shows a contrasting behavior. In EuGa4, it decreases with lowering the temperature, reaching its minimum at zero temperature, λT​(0)=0.71\lambda_{\mathrm{T}}(0)=0.71 μ\mus-1. In EuAl4, it increases significantly below TNT_{\mathrm{N}}, to reach 58 μ\mus-1 at 1.5 K, most likely reflecting the complex magnetic structure and the competing interactions in the AFM state of EuAl4. In both compounds, the temperature-dependent longitudinal muon-spin relaxation λL​(T)\lambda_{\mathrm{L}}(T), an indication of the rate of spin fluctuations, diverges near the onset of AFM order, followed by a significant drop at T<TNT<T_{\mathrm{N}}. In the AFM state, spin fluctuations are much stronger in EuAl4 than in EuGa4, while being comparable in the PM state. The evidence of robust spin fluctuations against the external magnetic fields provided by μ\muSR may offer new insights into the origin of the topological Hall effect and the possible magnetic skyrmions in the EuAl4 and EuGa4 compounds.

I Introduction

Topological materials are at the forefront of quantum matter and material science research due to their great potential for applications [1, 2]. Recently, the discovery of nontrivial band topology and extremely large magnetoresistance in the BaAl4 compound has stimulated considerable interest in this family of materials [3]. The tetragonal BaAl4-type structure with a space group of I​4/m​m​mI4/mmm (No. 139) represents the prototype for many binary- and ternary derivative compounds [4], as e.g., heavy-fermion compounds and iron-based high-TcT_{c} superconductors.

Upon replacing Ba with Sr or Eu, or when replacing Al with Ga, all AE(Al,Ga)4 (AE = Sr, Ba, and Eu) crystallize in the same tetragonal structure, while Ca(Al,Ga)4 adopts a monoclinic crystal structure with a space group C​2/mC2/m (No. 12) [3, 5, 5]. Among these materials, the Eu-4ff electrons bring new intriguing aspects to the topology. Both EuAl4 and EuGa4 are antiferromagnets below their critical temperatures TNT_{\mathrm{N}} = 15.6, and 16.4 K, respectively, with the former also undergoing a CDW transition at TCDW∼140T_{\mathrm{CDW}}\sim 140 K [6, 7, 8, 9, 10, 11, 12]. Further, while EuGa4 exhibits only one antiferromagnetic (AFM) transition, EuAl4 undergoes four subsequent AFM transitions below TNT_{\mathrm{N}}. More interestingly, by applying a magnetic field along the cc-axis, both EuAl4 and EuGa4 undergo a series of metamagnetic transitions in the AFM state [6, 7, 10]. Within a field range of ∼\sim1–2.5 T (EuAl4) or ∼\sim4–7 T (EuGa4), a clear hump-like anomaly is observed in the Hall resistivity, most likely a manifestation of the topological Hall effect (THE) [6, 7]. Very recently, a THE has been observed also in Al-doped EuGa4 [13], which exhibits comparable critical fields to EuAl4 [6].

The topological Hall effect is considered to be the hallmark of spin textures with a finite scalar spin chirality. Such topological spin textures usually exhibit a nonzero Berry phase, here acting as an effective magnetic field, giving rise to the topological Hall resistivity [14]. THE is frequently observed in magnetic materials with non-coplanar spin textures, such as magnetic skyrmions [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]. Skyrmions are one of the most intriguing topologically nontrivial spin textures that can be easily manipulated [26], hence holding a promise for diverse applications, such as high-density spintronics [27, 28]. THE has been observed mostly in magnetic compounds whose crystal structure lacks an inversion center, while centrosymmetric compounds that host magnetic skyrmions are rare [15, 16, 17, 18, 19, 29, 30]. Eu(Al,Ga)4 represent such rare cases where to look for the possible existence of magnetic skyrmions [6, 7, 13]. According to neutron diffraction studies, in the AFM state, the magnetic qq-vector of EuAl4 changes from 𝒒1\bm{q}_{1} = (0.085, 0.085, 0) at TNT_{\mathrm{N}} = 13.5 K to 𝒒2\bm{q}_{2} = (0.170, 0, 0) at 11.5 K and slightly to 𝒒3\bm{q}_{3} = (0.194, 0, 0) at 4.3 K [31]. Unlike the complex incommensurate transitions observed in EuAl4, the AFM structure of EuGa4 is described by a simple 𝒒\bm{q} = (0, 0, 0) magnetic vector, with the Eu moments lying in the basal a​bab-plane [32]. Noncollinear spins with incommensurate propagation vectors have been reported also in the isostructural EuGa2Al2 [13].

Refer to caption
Figure 1. : Time-domain wTF-μ\muSR spectra of (a) EuAl4 and (b) EuGa4 single crystals, collected in the AFM (1.5 K) and PM (18 K) states in a weak transverse field of 5 mT. The solid lines represent fits to Eq. (1). The inset in (a) depicts the EuAl4 crystals, aligned with their cc axis parallel to the muon momentum direction, i.e., 𝒑μ\bm{p}_{\mu} ∥\parallel cc.

As an extremely sensitive magnetic probe at a microscopic level, the muon-spin rotation and relaxation (μ\muSR) technique lends itself naturally to studying the temperature evolution of the magnetic properties of EuAl4 and EuGa4 single crystals. As shown in detail below, we report: i) the intrinsic fields at the muon implantation sites in EuAl4 and EuGa4 across the respective phase diagrams in the absence of external magnetic fields; ii) the magnetic volume fraction in the AFM state; iii) evidence of strong spin fluctuations.

II Experimental details

Single crystals of EuAl4 and EuGa4 were grown by a molten Al- and Ga flux method, respectively, the details of growth being reported elsewhere [6, 7]. The crystal orientation was checked by x-ray diffraction (XRD) measurements using a Bruker D8 diffractometer with Cu Kα radiation. The magnetic susceptibility measurements were performed on a Quantum Design magnetic properties measurement system (MPMS) with the applied magnetic field along the cc-axis.

μ\muSR experiments were carried out at the general-purpose surface-muon (GPS) instrument at the π\piM3 beam line of the Swiss muon source (Sμ\muS) at Paul Scherrer Institut (PSI) in Villigen, Switzerland. In this study, we performed three kinds of experiments: weak transverse-field (wTF)-μ\muSR, zero-field (ZF)-, and longitudinal-field (LF)-μ\muSR measurements. As to the former, we could determine the temperature evolution of the magnetic volume fraction. As to the latter two, we aimed at studying the temperature evolution of the magnetically ordered phase and the dynamics of spin fluctuations.

The aligned EuAl4 and EuGa4 crystals were positioned on a thin aluminum tape, with their cc-axes parallel to the muon-momentum direction, i.e., 𝒑μ\bm{p}_{\mu} ∥\parallel cc [see inset in Fig. I(a)]. For the wTF-μ\muSR measurements, the applied magnetic field BapplB_{\mathrm{appl}} was perpendicular to the muon-spin direction (i.e., Bappl⟂𝑺μB_{\mathrm{appl}}\perp\bm{S}_{\mu}), while it was parallel for the LF-μ\muSR measurements (i.e., Bappl∥𝑺μB_{\mathrm{appl}}\parallel\bm{S}_{\mu}). In both wTF- and LF-μ\muSR cases, the crystals were cooled in an applied magnetic field down to the base temperature (i.e., 1.5 K). For the ZF-μ\muSR measurements, to exclude the possibility of stray magnetic fields, the magnets were degaussed before the measurements. All the μ\muSR spectra were collected upon heating and were analyzed by means of the musrfit software package [33].

Refer to caption
Figure 2. : Temperature dependence of the ANMA_{\mathrm{NM}} asymmetry (left-axis) of wTF-μ\muSR spectra for (a) EuAl4 and (b) EuGa4 single crystals. We report also the magnetic susceptibilities χZFC\chi_{\mathrm{ZFC}} (right axes), measured in a field of μ0​H=0.1\mu_{0}H=0.1 T after zero-field cooling (ZFC). In both cases, the insets show the magnetic volume fraction vs temperature. Here, lines are fits to a phenomenological function [see Eq. (2)]. The vertical arrows mark the AFM transitions. The magnetic susceptibility data were taken from Refs. 6, 7.

III Results and discussion

III.1 wTF-μ\muSR

The magnetic transition temperatures TNT_{\mathrm{N}} and the evolution with temperature of the magnetic volume fraction in EuAl4 and EuGa4 single crystals were established by means of wTF-μ\muSR measurements. A weak transverse field of 5 mT was applied perpendicular to the initial muon-spin direction in the PM state, where it leads to oscillations, as shown in Fig. I. In the long-range ordered AFM state (i.e., 1.5 K), the applied 5-mT field is much smaller than the internal fields. As a consequence, upon entering the AFM state, muon spins precess with frequencies that reflect the internal fields at the muon-stopping sites rather than the weak applied field. Normally, the magnetic order leads to a very fast muon-spin depolarization in the first tenths of μ\mus (see also the ZF-μ\muSR spectra in the insets of Fig. III.2). Therefore, the wTF-μ\muSR spectra can be described by the function:

AwTF​(t)=ANM​cos⁡(γμ​Bint​t+ϕ)⋅e−λ​t,A_{\mathrm{wTF}}(t)=A_{\mathrm{NM}}\cos(\gamma_{\mu}B_{\mathrm{int}}t+\phi)\cdot e^{-\lambda t}, (1)

where ANMA_{\mathrm{NM}} is the initial muon-spin asymmetry (i.e., the amplitude of the oscillation) for muons implanted in the nonmagnetic (NM) or PM fraction of EuAl4 and EuGa4 single crystals; γμ\gamma_{\mu}BintB_{\mathrm{int}} is the muon-spin precession frequency, with γμ=2​π×135.5\gamma_{\mu}=2\pi\times 135.5 MHz/T the muon gyromagnetic ratio and BintB_{\mathrm{int}} the local field sensed by muons (here almost identical to the applied magnetic field, i.e., Bint∼5B_{\mathrm{int}}\sim 5 mT); ϕ\phi is the initial phase, and λ\lambda is the muon-spin relaxation rate. Note that, in the AFM state, the very fast μ\muSR relaxation was excluded and only the residual slow-relaxing asymmetry was analyzed (see the 1.5-K dataset in Fig. I).

Figure II summarizes the resulting wTF-μ\muSR asymmetry values ANMA_{\mathrm{NM}} as a function of temperature. In the PM state, all the implanted muons precess at the same frequency γμ\gamma_{\mu}BintB_{\mathrm{int}}. As the temperature approaches TNT_{\mathrm{N}}, only the muons implanted in the remaining PM/NM phase precess at the frequency γμ\gamma_{\mu}BintB_{\mathrm{int}}, here reflected in a reduced oscillation amplitude. The PM (or NM) sample fraction is determined from the oscillation amplitude. In both EuAl4 and EuGa4, ANMA_{\mathrm{NM}} starts to decrease near the onset of AFM order, where also the magnetic susceptibilities show clear transitions. Although EuAl4 undergoes four successive AFM transitions [indicated by vertical arrows in Fig. II(a)], ANM​(T)A_{\mathrm{NM}}(T) does not capture them individually, as it is sensitive only to the global PM (or NM) volume fraction. The temperature evolution of the magnetic volume fraction can be derived from Vmag​(T)=1−ANM​(T)/ANM​(T>TN)V_{\mathrm{mag}}(T)=1-A_{\mathrm{NM}}(T)/A_{\mathrm{NM}}(T>T_{\mathrm{N}}). The Vmag​(T)V_{\mathrm{mag}}(T) values are summarized in the insets of Fig. II(a) and II(b) for EuAl4 and EuGa4, respectively. To determine the magnetic volume fraction VmagV_{\mathrm{mag}}, the average magnetic transition temperature TNT_{\mathrm{N}}, and the transition width Δ​T\Delta T, Vmag​(T)V_{\mathrm{mag}}(T) data were fitted using the phenomenological function:

Vmag​(T)=Vmag​(0)​12​[1−erf​(T−TN2​Δ​T)],V_{\mathrm{mag}}(T)=V_{\mathrm{mag}}(0)\;\frac{1}{2}\left[1-\mathrm{erf}\left(\frac{T-T_{\mathrm{N}}}{\sqrt{2}\Delta T}\right)\right], (2)

where erf(TT) is the error function. As shown by solid lines in the insets of Fig. II, for EuAl4, we obtain TN=13.9​(2)T_{\mathrm{N}}=13.9(2) K, Δ​T=1.4​(2)\Delta T=1.4(2) K, and Vmag​(0)=91​(2)%V_{\mathrm{mag}}(0)=91(2)\%; while for EuGa4, TN=15.2​(3)T_{\mathrm{N}}=15.2(3) K, Δ​T=1.8​(2)\Delta T=1.8(2) K, and Vmag​(0)=95​(2)%V_{\mathrm{mag}}(0)=95(2)\%. Both samples show sharp transitions and can be considered as fully magnetically ordered at low temperatures, indicative a high sample quality. Note also that, the transition temperatures, as determined from Vmag​(T)V_{\mathrm{mag}}(T), have their onset at ∼16.5\sim 16.5 K and ∼16.7\sim 16.7 K for EuAl4 and EuGa4, both in very good agreement with the magnetometry data.

III.2 ZF- and LF-μ\muSR

To investigate the local magnetic order of EuAl4 and EuGa4 single crystals, ZF-μ\muSR spectra were collected at different temperatures, covering both the PM and AFM states. The time evolution of ZF-μ\muSR asymmetry, AZF​(t)A_{\mathrm{ZF}}(t), encodes the local magnetic fields and their distribution at the muon-stopping site. If the electronic magnetic moments fluctuate very fast (typically above 101210^{12} Hz in the PM state), they do not influence the muon-spin polarization. Randomly oriented slow fluctuating or static moments (below 10410^{4} Hz, such as nuclear spins, or electronic moments in spin glasses), give rise to incoherent precessions and a slow depolarization. Conversely, in case of ordered static moments, a fast depolarization and superimposed oscillations, reflecting the coherent precession of the muon spins, are observed [34]. This is clearly demonstrated in Fig. III.2, where the time evolution of selected ZF-μ\muSR spectra for EuAl4 and EuGa4 are presented.

Refer to caption
Figure 3. : Representative ZF-μ\muSR spectra collected in a transverse muon-spin configuration (𝒑μ⟂𝑺μ\bm{p}_{\mu}\perp\bm{S}_{\mu}) at temperatures covering both the PM and AFM states for (a) EuAl4 and (b) EuGa4, respectively. Insets highlight the short-time spectra, illustrating the coherent oscillations caused by the long-range AFM order. Solid lines through the data are fits to Eq. (3) (see text for details).

In the PM state (T>TNT>T_{\mathrm{N}}), the μ\muSR spectra still exhibit a relatively fast muon-spin depolarization (∼2\sim 2 μ\mus-1), implying the existence of strong spin fluctuations, here further confirmed by LF-μ\muSR measurements (see below). In absence of spin fluctuations, the muon-spin depolarization is usually due to the nuclear dipole fields [34], with a typical value of less than 0.1 μ\mus-1 in EuAl4 and EuGa4 [35]. The μ\muSR spectra in the AFM state (T≤TNT\leq T_{\mathrm{N}}) are characterized by highly damped oscillations, typical of long-range magnetic order (see insets in Fig. III.2), superimposed on a slowly decaying relaxation, observable only at long times. To track these changes across the whole temperature range, the ZF-μ\muSR spectra of EuAl4 and EuGa4 were analyzed using the following model:

AZF​(t)=A1⋅[α​cos⁡(γμ​Bint​t+ϕ)⋅e−λT​t+(1−α)⋅e−λL​t]+A2⋅e−λtail​t.\begin{split}A_{\mathrm{ZF}}(t)=&\,A_{1}\cdot\left[\alpha\cos(\gamma_{\mu}B_{\mathrm{int}}t+\phi)\cdot e^{-\lambda_{\mathrm{T}}t}+(1-\alpha)\cdot e^{-\lambda_{\mathrm{L}}t}\right]+\\ &A_{2}\cdot e^{-\lambda_{\mathrm{tail}}t}.\end{split} (3)

Here, α\alpha and 1−α1-\alpha are the oscillating (i.e., transverse) and nonoscillating (i.e., longitudinal) fractions of the μ\muSR signal, respectively, whose initial total asymmetry is equal to A1A_{1}. λT\lambda_{\mathrm{T}} and λL\lambda_{\mathrm{L}} represent the transverse and longitudinal relaxation rates, while A1A_{1} and A2A_{2} represent the asymmetries of the two nonequivalent muon-stopping sites. In EuAl4, muons stopping at the second site do not undergo any precession, but show only a slow relaxation, here described by λtail\lambda_{\mathrm{tail}}. In EuGa4, a single muon-stopping site is sufficient to describe the ZF-μ\muSR spectra. Finally, BintB_{\mathrm{int}}, ϕ\phi, and γμ\gamma_{\mu} are the same as in Eq. (1). Similar expressions have been used to analyze the μ\muSR data in other Eu-based magnetic materials, most notably, in the Eu122 iron pnictides [36, 37].

In polycrystalline materials with a long-range magnetic order, one expects α=2/3\alpha=2/3, since statistically one third of the muon spins are aligned parallel to the local field direction (i.e., 𝑺μ∥Bint\bm{S}_{\mu}\parallel B_{\mathrm{int}}) and, hence, do not precess. In EuAl4 and EuGa4 single crystals, we find α\alpha to be 0.87 and 0.46, respectively. Since the ZF-μ\muSR spectra were collected in a rotated muon-spin configuration (i.e., 𝑺μ\bm{S}_{\mu} ⟂\perp 𝒑μ\bm{p}_{\mu}), and the cc-axis is parallel to the muon momentum (i.e., c∥𝒑μc\parallel\bm{p}_{\mu}), the internal

Refer to caption
Figure 4. : Temperature dependence of (a) internal field Bint​(T)B_{\mathrm{int}}(T), (b) transverse muon-spin relaxation rate (known also as damping rate) λT​(T)\lambda_{\mathrm{T}}(T), and (c) longitudinal muon-spin relaxation rate λL​(T)\lambda_{\mathrm{L}}(T) for EuAl4, as derived from ZF-μ\muSR analysis. The muon-spin relaxation rate of the tail of the ZF-μ\muSR spectra, λtail\lambda_{\mathrm{tail}}, is shown in panel (d). Solid lines are fits to the equations described in the text; dash-dotted lines are guides to the eyes.

magnetic fields at the muon-stopping sites should be mostly aligned along the [001]-direction in EuAl4, but along the [111]-direction in EuGa4.

The derived fit parameters for both cases are summarized in Fig. III.2 and Fig. III.2. As can be clearly seen in the top panels, EuAl4 and EuGa4 show rather different Bint​(T)B_{\mathrm{int}}(T) behaviors. In EuAl4, the Bint​(T)B_{\mathrm{int}}(T) undergoes a sudden drop at ∼\sim13 K, which corresponds to the second AFM transition in the magnetic susceptibility [see Fig. II(a)]. Conversely, in EuGa4, Bint​(T)B_{\mathrm{int}}(T) resembles the typical mean-field type curve below TNT_{\mathrm{N}}. Since BintB_{\mathrm{int}} is directly proportional to the magnetic moment, the evolution of BintB_{\mathrm{int}} reflects that of the magnetic structure. According to neutron scattering studies, in EuAl4, the magnetic qq-vector changes from 𝒒1\bm{q}_{1} = (0.085, 0.085, 0) at TNT_{\mathrm{N}} = 13.5 K to 𝒒2\bm{q}_{2} = (0.170, 0, 0) at 11.5 K and slightly to 𝒒3\bm{q}_{3} = (0.194, 0, 0) at 4.3 K [31]. Therefore, we identify the drop of BintB_{\mathrm{int}} at 13 K with the critical temperature where the magnetic structure changes from q1q_{1} to q2q_{2}. At the same time, the modification of magnetic structure from q2q_{2} to q3q_{3} is too tiny to have a measurable effect on BintB_{\mathrm{int}}. By contrast, the AFM structure of EuGa4 is rather simple [its magnetic vector being 𝒒\bm{q} = (0, 0, 0)] and it persists down to 2 K [32]. As a consequence, in EuGa4, BintB_{\mathrm{int}} decreases monotonically as the temperature increases. In both compounds, Bint​(T)B_{\mathrm{int}}(T) can be modeled by the phenomenological equation:

Bint​(T)=Bint​(0)⋅[1−(TTN)γ]δ.B_{\mathrm{int}}(T)=B_{\mathrm{int}}(0)\cdot\left[1-\left(\frac{T}{T_{\mathrm{N}}}\right)^{\gamma}\right]^{\delta}. (4)

Here, Bint​(0)B_{\mathrm{int}}(0) is the internal magnetic field at zero temperature, while γ\gamma and δ\delta are two empirical parameters.

Refer to caption
Figure 5. : Temperature dependence of (a) internal field Bint​(T)B_{\mathrm{int}}(T), (b) transverse muon-spin relaxation rate (i.e., damping rate) λT​(T)\lambda_{\mathrm{T}}(T), and (c) longitudinal muon-spin relaxation rate λL​(T)\lambda_{\mathrm{L}}(T) for EuGa4, as derived from ZF-μ\muSR analysis. Solid lines are fits to the equations described in the text; dash-dotted lines are guides to the eyes.

As indicated by the solid lines in Fig. III.2(a) and Fig. III.2(a), the above model describes the data reasonably well, yielding the parameters listed in Table III.2. In EuAl4, the first AFM phase (AFM1) is characterized by Bint​(0)=0.57​(5)B_{\mathrm{int}}(0)=0.57(5) T. The change in magnetic structure lowers Bint​(0)B_{\mathrm{int}}(0) down to 0.33(2) T in the second AFM phase (AFM2). In EuGa4, Bint​(T)B_{\mathrm{int}}(T) follows the typical mean-field type curve, yielding Bint​(0)=0.89​(2)B_{\mathrm{int}}(0)=0.89(2) T. Considering the presence of the same magnetic Eu2+ ions in both cases and the similar lattice parameters, the significantly different Bint​(0)B_{\mathrm{int}}(0) values are most likely attributed to the different muon-stopping sites or to different magnetic structures in EuAl4 and EuGa4, the latter having been proved by neutron scattering studies. Indeed, at base temperature, EuAl4 exhibits a complex incommensurate magnetic structure, while this is commensurate in EuGa4 [31, 32].

The temperature dependence of the transverse and longitudinal μ\muSR relaxation rates λT​(T)\lambda_{\mathrm{T}}(T) and λL​(T)\lambda_{\mathrm{L}}(T) are summarized in Figs. III.2(b) and (c) for EuAl4 and in Figs. III.2(b) and (c) for EuGa4, respectively. The transverse relaxation rate λT\lambda_{\mathrm{T}} is a measure of the width of static magnetic field distribution at the muon-stopping site and is also affected by dynamical effects, as e.g., spin fluctuations. The longitudinal relaxation rate λL\lambda_{\mathrm{L}} is determined solely by spin fluctuations. In EuAl4 and EuGa4, λT​(T)\lambda_{\mathrm{T}}(T) exhibits completely opposite behaviors. In EuAl4 [see Fig. III.2(b)], λT\lambda_{\mathrm{T}} is zero in the PM state, and becomes increasingly prominent as the temperature decreases below TNT_{\mathrm{N}}, reflecting a more disordered field distribution well inside the AFM state. Such a large λT\lambda_{\mathrm{T}} at temperatures far below TNT_{\mathrm{N}} is unusual for an antiferromagnet, and implies an increasingly inhomogeneous distribution of local fields in the AFM state of EuAl4. Thus, at 1.5 K, λT∼58​(10)\lambda_{\mathrm{T}}\sim 58(10) μ\mus-1, which implies a half-width at half-maximum (HWHM) of field distribution Δ=68​(12)\Delta=68(12) mT (here, Δ=λT/γμ\Delta=\lambda_{\mathrm{T}}/\gamma_{\mu}). Such enhanced local-field distribution might be related to the complex spatial arrangement of the Eu magnetic moments in EuAl4, where the magnetic propagation vector is incommensurate with the crystal lattice [31]. By contrast, in EuGa4, λT​(T)\lambda_{\mathrm{T}}(T) follows the typical behavior of materials with a long-range (anti)ferromagnetic order [36], i.e., diverging at TNT_{\mathrm{N}} and continuously decreasing at T<TNT<T_{\mathrm{N}}. Such λT​(T)\lambda_{\mathrm{T}}(T) suggests a very homogeneous distribution of local fields, consistent with the commensurate magnetic propagation vector in EuGa4 [32]. At TT = 1.5 K, in EuGa4, λT\lambda_{\mathrm{T}} is found to be ∼\sim0.72 μ\mus-1, a value which is almost three orders of magnitude smaller than that of EuAl4. This is also reflected in the ZF-μ\muSR spectra shown in the insets of Fig. III.2, where the damping of the muon-spin precession is much weaker in EuGa4 than in EuAl4.

The longitudinal μ\muSR relaxation rates λL\lambda_{\mathrm{L}} shown in Fig. III.2(c) and Fig. III.2(c) are much smaller than the transverse relaxation rates λT\lambda_{\mathrm{T}}. At 1.5 K, λL\lambda_{\mathrm{L}}/λT∼0.15\lambda_{\mathrm{T}}\sim 0.15 and 0.02 for EuAl4 and EuGa4, respectively. In contrast to λT​(T)\lambda_{\mathrm{T}}(T) [see Fig. III.2(b) and Fig. III.2(b)], EuAl4 and EuGa4 exhibit a similar temperature-dependent λL​(T)\lambda_{\mathrm{L}}(T), typical of materials with long-range magnetic order. In both cases, λL​(T)\lambda_{\mathrm{L}}(T) diverges near TNT_{\mathrm{N}}, followed by a significant drop at T<TNT<T_{\mathrm{N}}, indicating that spin fluctuations are the strongest close to the onset of the AFM order. At 1.5 K, λL\lambda_{\mathrm{L}} is 8.2 and 0.01 μ\mus-1 for EuAl4 and EuGa4, respectively. In EuAl4, at temperatures well inside the AFM state, λL\lambda_{\mathrm{L}} is hundreds of times larger than in EuGa4, thus suggesting much stronger spin fluctuations in the AFM state of EuAl4 than in EuGa4. Conversely, in the PM state, both EuAl4 and EuGa4 exhibit similar λL\lambda_{\mathrm{L}} values. Note that, in EuAl4, as shown in Fig. III.2(d), muons implanted in the second site experience only the spin fluctuations. Consequently, λtail​(T)\lambda_{\mathrm{tail}}(T) in EuAl4 shows similar features to λL​(T)\lambda_{\mathrm{L}}(T), i.e., it exhibits a maximum near the onset of the AFM order and it, too, decreases as the temperature is lowered. Future calculations of the muon-stopping sites, might be helpful to better appreciate the differences between EuAl4 and EuGa4. In the fast-fluctuation limit (typical of magnetically ordered materials), the zero-field longitudinal muon-spin relaxation rate is described by:

λL=2​γμ2​Δ2ν,\lambda_{\mathrm{L}}=\frac{2\gamma_{\mu}^{2}\Delta^{2}}{\nu}, (5)

where Δ\Delta is the amplitude of field fluctuations, while ν\nu is their correlation frequency (i.e., 1/ν=τ1/\nu=\tau, is the spin-correlation time) [34]. The estimated spin-correlation times are τ\tau = 1.3 and 8.2 ns for EuAl4 and EuGa4, respectively.

The vigorous spin fluctuations in these compounds are further supported by LF-μ\muSR measurements. As shown in Fig. III.2, the μ\muSR spectrum in a 0.7-T longitudinal field is almost identical to that collected in a zero-field condition, suggesting that muon spins cannot be decoupled, hence, that spin fluctuations survive even in a field of 0.7 T in both EuAl4 and EuGa4. Note that, such spin fluctuations are robust against external magnetic fields, both in the AFM- (e.g., 1.5 K) and in the PM state (i.e., 50 K) far above TNT_{\mathrm{N}} (see details in Fig. A in the Appendix). Similar μ\muSR results have been reported in other Eu-based materials, e.g., EuCd2As2, where the strong spin fluctuations cause the breaking of time-reversal symmetry and lead to the formation of magnetic Weyl fermions [38].

Table 1.: Summary of the EuAl4 and EuGa4 single-crystal parameters obtained by means of magnetization- and μ\muSR measurements.
TNχT_{\mathrm{N}}^{\chi}(K) TNμ​SRT_{\mathrm{N}}^{\mathrm{{\mu}SR}}(K)111Determined from the asymmetry of wTF-μ\muSR spectra. TNμ​SRT_{\mathrm{N}}^{\mathrm{{\mu}SR}}(K)222Determined from fits of ZF-μ\muSR spectra. BintB_{\mathrm{int}}(T) γ\gamma δ\delta
EuAlAFM14{}_{4}^{\mathrm{AFM1}} 15.6(2) 13.9(1) 16.0(2) 0.57(5) 2.50(5) 0.50(5)
EuAlAFM24{}_{4}^{\mathrm{AFM2}} 12.3(3) — 13.7(4) 0.33(2) 0.92(5) 0.17(3)
EuGa4 16.5(2) 15.2(3) 16.5(4) 0.89(2) 1.50(5) 0.50(5)
Refer to caption
Figure 6. : LF-μ\muSR time-domain spectra collected at 18 K (i.e., slightly above TNT_{\mathrm{N}}) in an applied magnetic field of 0 and 0.7 T in (a) EuAl4 and (b) EuGa4. Both spectra were collected in a longitudinal muon-spin configuration, i.e., 𝒑μ\bm{p}_{\mu} ∥\parallel 𝑺μ\bm{S}_{\mu}. The applied magnetic field is parallel to the muon-spin direction. In either case, no appreciable decoupling of muon spins with field can be identified.

III.3 Discussion

First we discuss why the successive magnetic transitions of EuAl4 remain undetected by μ\muSR, an absence which might be due to different reasons. Firstly, the asymmetries obtained from wTF-μ\muSR (see Figs. I and II) reflect the internal fields sensed by the implanted muons. However, when the applied transverse field is much smaller than the internal fields, the wTF signal is mostly determined by the muons implanted in the residual NM (or PM) fraction of a magnetically ordered sample. This is reflected in a significant drop in the temperature-dependent asymmetry A​(T)A(T). In EuAl4, below the onset of AFM order, the internal fields are hundreds of times larger than the applied wTF. Although changes in the magnetic structure, detected as successive transitions in the EuAl4 magnetometry data, decrease the internal field from ∼0.4\sim 0.4 T to 0.33 T, this still remains much larger than wTF. Therefore, the successive magnetic transitions of EuAl4 are not easily detectable via wTF-μ\muSR measurements. Secondly, a slight change/rearrangement of the magnetic structure does not have a large impact on the internal field. According to neutron scattering studies, in EuAl4, the magnetic qq-vector changes from 𝒒1\bm{q}_{1} = (0.085, 0.085, 0) at TN=13.5T_{\mathrm{N}}=13.5 K to 𝒒2\bm{q}_{2} = (0.170, 0, 0) at 11.5 K and slightly to 𝒒3\bm{q}_{3} = (0.194, 0, 0) at 4.3 K [31]. Therefore, we identify the drop of BintB_{\mathrm{int}} at 13 K with the critical temperature where the magnetic structure changes from q1q_{1} to q2q_{2}. At the same time, the modification of magnetic structure from q2q_{2} to q3q_{3} with the magnetic moments pointing at the same direction is too tiny to have a measurable effect on BintB_{\mathrm{int}}. Thirdly, changes in magnetic structure have little effect on the longitudinal relaxation rates λL\lambda_{\mathrm{L}}, which reflect solely the spin fluctuations in EuAl4. In general, spin fluctuations decrease significantly as the temperature moves away from TNT_{\mathrm{N}}, but they diverge near the onset of the magnetic transition. Hence, in the magnetically ordered state, changes in λL\lambda_{\mathrm{L}} caused by slight modifications of the magnetic structure are negligible compared to the temperature driven effects.

Since most of the skyrmion phases appear in a field range not easily accessible by standard μ\muSR instruments, up to now, only a handful of results have been reported where LF-μ\muSR is used to study the skyrmion compounds. These include GaV4(S,Se)8 [39], Cu2OSeO3 [40], and the Co-Zn-Mn alloy [40, 41], whose skyrmion phases are stabilized by a relatively small field (¡ 0.1 T). While for many newly discovered skyrmion systems, i.e., GdRu2Si2 and Gd3Ru4Al12 (as well as for EuAl4 and EuGa4 studied here) [6, 7, 30, 29], the critical field required for stabilizing the skyrmion phase is above 1 T. In their AFM state, EuAl4 and EuGa4 exhibit comparable spin fluctuations to other well-studied skyrmion compounds. For instance, the muon-spin relaxation rates extracted from LF-μ\muSR measurements in the skyrmion phases of Cu2OSeO3 and GaV4(S,Se)8 are ∼0.2\sim 0.2–0.8 μ\mus-1, similar to those of Eu(Al,Ga)4 (see Figs. III.2 and III.2). All these skyrmion compounds exhibit similar temperature-dependent muon-spin relaxation rates λL​(T)\lambda_{\mathrm{L}}(T), with an enhanced and broadened peak in λL​(T)\lambda_{\mathrm{L}}(T) at temperatures just below the critical temperature. Muon-spin relaxation rates also increase when entering the skyrmion phase by applying longitudinal magnetic fields, thus providing another method for identifying the presence of magnetic skyrmions. In the EuAl4 and EuGa4 case, where there is no skyrmion phase in zero field, the relaxation rates diverge at TNT_{\mathrm{N}}, followed by a significant drop at T<TNT<T_{\mathrm{N}} due to the slowing down of spin fluctuations, a typical feature of magnetically ordered materials. A similar behavior is observed in Co10Zn10 [40], a parent compound of the Co-Mn-Zn alloys, which lacks any skyrmion phases. According to Hall-resistivity measurements, the skyrmion phase may exist in a field range ∼1\sim 1–2.5 T in EuAl4 and ∼4\sim 4–7 T in EuGa4 [6, 7]. Aimed at investigating the intrinsic magnetic properties of both compounds, most of the current μ\muSR studies are performed in zero-field conditions. To compare the muon-spin relaxation rates of EuAl4 and EuGa4 with those of other skyrmion compounds, and check if there are any skyrmion phases, further temperature-dependent μ\muSR measurements under high magnetic fields are required.

The observation of a topological Hall effect in the magnetic state is usually attributed to noncoplanar spin textures, such as magnetic skyrmions, characterized by a finite scalar spin chirality in real space. These magnetic skyrmions are stabilized by the Dzyaloshinskii-Moriya interaction, often observed in noncentrosymmetric materials [42, 43, 44, 45, 46, 47, 48]. Conversely, magnetic materials with a centrosymmetric crystal structure that still host magnetic skyrmions are rare. To date, only a few systems have been reported, including some gadolinium intermetallic compound [19, 29, 30], Fe3Sn2 [49], and possibly, also EuCd2As2 [50]. In centrosymmetric systems, skyrmions can be stabilized, for instance, by magnetic frustration (e.g., in Gd3Ru4Al12, Gd2PdSi3, and Fe3Sn2), or by the competition between the magnetic interactions and magnetic anisotropies (e.g., in GdRu2Si2) [51, 29, 19, 49, 30]. According to magnetization and nuclear magnetic resonance studies, the magnetic anisotropy is moderate in EuAl4 and EuGa4 [6, 7, 52]. Since both EuAl4 and EuGa4 adopt the same crystal structure of GdRu2Si2, skyrmions might be stabilized by the same mechanism. In addition, a four-spin interaction, mediated by itinerant electrons, has also been proposed as an important ingredient for the formation of skyrmions in centrosymmetric materials [53, 51, 54]. Very recently, the chiral magnet Co7Zn7Mn6 was found to host a skyrmion phase far below the magnetic ordering temperature, where spin fluctuations are believed to be the key for stabilizing the magnetic skyrmions [41]. Our μ\muSR results reveal that both EuAl4 and EuGa4 exhibit robust spin fluctuations against external magnetic fields, which analogously might be crucial for understanding the origin of topological Hall effect and of possible skyrmions in both materials.

IV Conclusion

In summary, we investigated the temperature evolution of the local magnetic properties of EuAl4 and EuGa4 by means of μ\muSR spectroscopy. wTF-μ\muSR measurements confirm that EuAl4 and EuGa4 undergo an AFM transition at TNT_{\mathrm{N}} ∼\sim 16 and 16.5 K, which are consistent with the magnetization data. The magnetic volume fractions, as determined from wTF-μ\muSR asymmetry, are 91% and 95% for EuAl4 and EuGa4, respectively, implying a good sample quality in both cases. By using ZF-μ\muSR measurements, we could follow the temperature evolution of the local magnetic fields and of spin fluctuations. The estimated internal fields at zero temperature are 0.33 and 0.89 T for EuAl4 and EuGa4, respectively. EuAl4 exhibits a more disordered internal field distribution than EuGa4, reflected in a large transverse muon-spin relaxation rate λT\lambda_{\mathrm{T}} far below TNT_{\mathrm{N}}, most likely related to its complex magnetic structure. The vigorous spin fluctuations revealed by both ZF-μ\muSR and LF-μ\muSR might be crucial for understanding the origin of topological Hall effect and of possible skyrmions in EuAl4 and EuGa4. In future, it might be interesting to investigate the magnetic properties of EuAl4 and EuGa4 using the μ\muSR technique under high magnetic fields, where the topological Hall effect appears.

Acknowledgements.
T.S. acknowledges support from the Natural Science Foundation of Shanghai (Grant Nos. 21ZR1420500 and 21JC1402300) and the Schweizerische Nationalfonds zur Förderung der Wissenschaftlichen Forschung (SNF) (Grant Nos. 200021_188706 and 206021_139082). Y.X. acknowledges support from the Shanghai Pujiang Program (Grant No. 21PJ1403100). This work was also financially supported by the National Natural Science Foundation of China (NSFC) (Grant Nos. 12174103 and 11874150) and the Sino-Swiss Science and Technology Cooperation (Grant No. IZLCZ2-170075). We thank G. Lamura for the assistance during some phases of the LF-μ\muSR experiments.

Appendix A Longitudinal-field μ\muSR in EuAl4

Refer to caption
Figure 7. : LF-μ\muSR time-domain spectra collected at 1.5 K (a) (far below TNT_{\mathrm{N}}) and 50 K (b) (far above TNT_{\mathrm{N}}) in an applied magnetic field of 0 and 0.78 T in EuAl4. Both spectra were collected in a longitudinal muon-spin configuration, i.e., 𝒑μ\bm{p}_{\mu} ∥\parallel 𝑺μ\bm{S}_{\mu}. The applied magnetic field is parallel to the muon-spin direction.

In Fig. A we present the ZF- and LF-μ\muSR spectra of EuAl4, collected at temperatures well inside the AFM state (1.5 K) and far above TNT_{\mathrm{N}}, in the PM state (i.e., 50 K). In the AFM state [see Fig. A(a)], the fast drop of the μ\muSR asymmetry reflects a very fast muon-spin depolarization in the first tenths of μ\mus [see also ZF-μ\muSR data in Fig. III.2(a)]. A 0.78-T longitudinal magnetic field has negligible effects on the long-time μ\muSR spectra. Indeed, both the ZF- and LF-μ\muSR spectra are almost identical, implying that the spin fluctuations persist deep inside the AFM state of EuAl4. Surprisingly, similar features are observed also in the PM state, as clearly demonstrated in Fig. A(b) [see also Fig. III.2]. Since the data suggest that muon spins cannot be decoupled neither in the AFM nor in the PM state, this implies that, in this type of materials, spin fluctuations exist over a wide temperature range, well above the AFM transition. We recall that, according to previous μ\muSR studies on EuCd2As2, spin fluctuations are strongly enhanced below 100 K, thus causing the breaking of time-reversal symmetry and leading to the formation of magnetic Weyl fermions [38]. Further measurements at higher temperatures, including both ZF- and LF-μ\muSR, are highly desirable to check if a similar phenomenology occurs also in the BaAl4-type family of materials.

References

  • Armitage et al. [2018] N. P. Armitage, E. J. Mele, and A. Vishwanath, Weyl and Dirac semimetals in three-dimensional solids, Rev. Mod. Phys. 90, 015001 (2018).
  • Lv et al. [2021] B. Q. Lv, T. Qian, and H. Ding, Experimental perspective on three-dimensional topological semimetals, Rev. Mod. Phys. 93, 025002 (2021).
  • Wang et al. [2021] K. Wang, R. Mori, Z. Wang, L. Wang, J. H. S. Ma, D. W. Latzke, D. E. Graf, J. D. Denlinger, D. Campbell, B. A. Bernevig, A. Lanzara, and J. Paglione, Crystalline symmetry-protected non-trivial topology in prototype compound BaAl4, npj Quantum Mater. 6, 28 (2021).
  • Kneidinger et al. [2014] F. Kneidinger, L. Salamakha, E. Bauer, I. Zeiringer, P. Rogl, C. Blaas-Schenner, D. Reith, and R. Podloucky, Superconductivity in noncentrosymmetric BaAl4 derived structures, Phys. Rev. B 90, 024504 (2014).
  • Nakamura et al. [2016] A. Nakamura, T. Uejo, H. Harima, S. Araki, T. C. Kobayashi, M. Nakashima, Y. Amako, M. Hedo, T. Nakama, and Y. Ōnuki, Characteristic Fermi surfaces and charge density wave in SrAl4 and related compounds with the BaAl4-type tetragonal structure, J. Alloys Compd. 654, 290 (2016).
  • Shang et al. [2021] T. Shang, Y. Xu, D. J. Gawryluk, J. Z. Ma, T. Shiroka, M. Shi, and E. Pomjakushina, Anomalous Hall resistivity and possible topological Hall effect in the EuAl4{\mathrm{EuAl}}_{4} antiferromagnet, Phys. Rev. B 103, L020405 (2021).
  • Zhang et al. [2021] H. Zhang, X. Y. Zhu, Y. Xu, D. J. Gawryluk, W. Xie, S. L. Ju, M. Shi, T. Shiroka, Q. F. Zhan, E. Pomjakushina, and T. Shang, Giant magnetoresistance and topological Hall effect in the EuGa4 antiferromagnet, J. Phys.: Condens. Matter 34, 034005 (2021).
  • Araki et al. [2014] S. Araki, Y. Ikeda, T. C. Kobayashi, A. Nakamura, Y. Hiranaka, M. Hedo, T. Nakama, and Y. Ōnuki, Charge density wave transition in EuAl4, J. Phys. Soc. Jpn. 83, 015001 (2014).
  • Nakamura et al. [2014] A. Nakamura, Y. Hiranaka, M. Hedo, T. Nakama, Y. Miura, H. Tsutsumi, A. Mori, K. Ishida, K. Mitamura, Y. Hirose, K. Sugiyama, F. Honda, T. Takeuchi, T. D. Matsuda, E. Yamamoto, Y. Haga, and Y. Ōnuki, Unique Fermi surface and emergence of charge density wave in EuGa4 and EuAl4, Jpn. Phys. Soc. Conf. Proc. 3, 011012 (2014).
  • Nakamura et al. [2015] A. Nakamura, T. Uejo, F. Honda, T. Takeuchi, H. Harima, E. Yamamoto, Y. Haga, K. Matsubayashi, Y. Uwatoko, M. Hedo, T. Nakama, and Y. Ōnuki, Transport and magnetic properties of EuAl4 and EuGa4, J. Phys. Soc. Jpn. 84, 124711 (2015).
  • Shimomura et al. [2019] S. Shimomura, H. Murao, S. Tsutsui, H. Nakao, A. Nakamura, M. Hedo, T. Nakama, and Y. Ōnuki, Lattice modulation and structural phase transition in the antiferromagnet EuAl4, J. Phys. Soc. Jpn. 88, 014602 (2019).
  • Kobata et al. [2016] M. Kobata, S. Fujimori, Y. Takeda, T. Okane, Y. Saitoh, K. Kobayashi, H. Yamagami, A. Nakamura, M. Hedo, T. Nakama, and Y. Ōnuki, Electronic structure of EuAl4 studied by photoelectron spectroscopy, J. Phys. Soc. Jpn. 85, 094703 (2016).
  • Moya et al. [2021] J. M. Moya, S. Lei, E. M. Clements, K. Allen, S. Chi, S. Sun, Q. Li, Y. Y. Peng, A. Husain, M. Mitrano, M. J. Krogstad, R. Osborn, P. Abbamonte, A. B. Puthirath, J. W. Lynn, and E. Morosan, Incommensurate magnetic orders and possible field-induced skyrmions in the square-net centrosymmetric EuGa2Al2 system, arXiv: 2110.11935  (2021).
  • Tokura and Kanazawa [2021] Y. Tokura and N. Kanazawa, Magnetic skyrmion materials, Chem. Rev. 121, 2857 (2021), and references therein.
  • Neubauer et al. [2009] A. Neubauer, C. Pfleiderer, B. Binz, A. Rosch, R. Ritz, P. G. Niklowitz, and P. Böni, Topological Hall effect in the AA phase of MnSi, Phys. Rev. Lett. 102, 186602 (2009).
  • Gayles et al. [2015] J. Gayles, F. Freimuth, T. Schena, G. Lani, P. Mavropoulos, R. A. Duine, S. Blügel, J. Sinova, and Y. Mokrousov, Dzyaloshinskii-Moriya interaction and Hall effects in the skyrmion phase of Mn1-xFexGe, Phys. Rev. Lett. 115, 036602 (2015).
  • Kanazawa et al. [2011] N. Kanazawa, Y. Onose, T. Arima, D. Okuyama, K. Ohoyama, S. Wakimoto, K. Kakurai, S. Ishiwata, and Y. Tokura, Large topological Hall effect in a short-period helimagnet MnGe, Phys. Rev. Lett. 106, 156603 (2011).
  • Franz et al. [2014] C. Franz, F. Freimuth, A. Bauer, R. Ritz, C. Schnarr, C. Duvinage, T. Adams, S. Blügel, A. Rosch, Y. Mokrousov, and C. Pfleiderer, Real-space and reciprocal-space Berry phases in the Hall effect of Mn1-xFexSi, Phys. Rev. Lett. 112, 186601 (2014).
  • Kurumaji et al. [2019] T. Kurumaji, T. Nakajima, M. Hirschberger, A. Kikkawa, Y. Yamasaki, H. Sagayama, H. Nakao, Y. Taguchi, T.-h. Arima, and Y. Tokura, Skyrmion lattice with a giant topological Hall effect in a frustrated triangular-lattice magnet, Science 365, 914 (2019).
  • Lee et al. [2009] M. Lee, W. Kang, Y. Onose, Y. Tokura, and N. P. Ong, Unusual Hall effect anomaly in MnSi under pressure, Phys. Rev. Lett. 102, 186601 (2009).
  • Li et al. [2013] Y. Li, N. Kanazawa, X. Z. Yu, A. Tsukazaki, M. Kawasaki, M. Ichikawa, X. F. Jin, F. Kagawa, and Y. Tokura, Robust formation of skyrmions and topological Hall effect anomaly in epitaxial thin films of MnSi, Phys. Rev. Lett. 110, 117202 (2013).
  • Huang and Chien [2012] S. X. Huang and C. L. Chien, Extended skyrmion phase in epitaxial FeGe(111) thin films, Phys. Rev. Lett. 108, 267201 (2012).
  • Schulz et al. [2012] T. Schulz, R. Ritz, A. Bauer, M. Halder, M. Wagner, C. Franz, C. Pfleiderer, K. Everschor, M. Garst, and A. Rosch, Emergent electrodynamics of skyrmions in a chiral magnet, Nat. Phys. 8, 301 (2012).
  • Qin et al. [2019] Q. Qin, L. Liu, W. Lin, X. Shu, Q. Xie, Z. Lim, C. Li, S. He, G. M. Chow, and J. Chen, Emergence of topological Hall effect in a SrRuO3 single layer, Adv. Mater. 31, 1807008 (2019).
  • Matsuno et al. [2016] J. Matsuno, N. Ogawa, K. Yasuda, F. Kagawa, W. Koshibae, N. Nagaosa, Y. Tokura, and M. Kawasaki, Interface-driven topological Hall effect in SrRuO3-SrIrO3 bilayer, Sci. Adv. 2, e1600304 (2016).
  • Jonietz et al. [2010] F. Jonietz, S. Mühlbauer, C. Pfleiderer, A. Neubauer, W. Münzer, A. Bauer, T. Adams, R. Georgii, P. Böni, R. A. Duine, K. Everschor, M. Garst, and A. Rosch, Spin transfer torques in MnSi at ultralow current densities, Science 330, 1648 (2010).
  • Nagaosa and Tokura [2013] N. Nagaosa and Y. Tokura, Topological properties and dynamics of magnetic skyrmions, Nat. Nanotechnol. 8, 899 (2013), and references therein.
  • Fert et al. [2017] A. Fert, N. Reyren, and V. Cros, Magnetic skyrmions: Advances in physics and potential applications, Nat. Rev. Mater. 2, 17031 (2017).
  • Hirschberger et al. [2019] M. Hirschberger, T. Nakajima, S. Gao, L. Peng, A. Kikkawa, T. Kurumaji, M. Kriener, Y. Yamasaki, H. Sagayama, H. Nakao, K. Ohishi, K. Kakurai, Y. Taguchi, X. Yu, T. Arima, and Y. Tokura, Skyrmion phase and competing magnetic orders on a breathing kagomé lattice, Nat. Commun. 10, 5831 (2019).
  • Khanh et al. [2020] N. D. Khanh, T. Nakajima, X. Yu, S. Gao, K. Shibata, M. Hirschberger, Y. Yamasaki, H. Sagayama, H. Nakao, L. Peng, K. Nakajima, R. Takagi, T. Arima, Y. Tokura, and S. Seki, Nanometric square skyrmion lattice in a centrosymmetric tetragonal magnet, Nat. Nanotechnol. 15, 444 (2020).
  • Kaneko et al. [2021] K. Kaneko, T. Kawasaki, A. Nakamura, K. Munakata, A. Nakao, T. Hanashima, R. Kiyanagi, T. Ohhara, M. Hedo, T. Nakama, and Y. Ōnuki, Charge-density-wave order and multiple magnetic transitions in divalent europium compound EuAl4, J. Phys. Soc. Jpn. 90, 064704 (2021).
  • Kawasaki et al. [2016] T. Kawasaki, K. Kaneko, A. Nakamura, N. Aso, M. Hedo, T. Nakama, T. Ohhara, R. Kiyanagi, K. Oikawa, I. Tamura, A. Nakao, K. Munakata, T. Hanashima, and Y. Ōnuki, Magnetic structure of divalent europium compound EuGa4 studied by single-crystal time-of-flight neutron diffraction, J. Phys. Soc. Jpn. 85, 114711 (2016).
  • A. Suter and Wojek [2012] A. A. Suter and B. M. Wojek, Musrfit: A free platform-independent framework for μ\muSR data analysis, Phys. Procedia 30, 69 (2012).
  • Yaouanc and de Réotier [2011] A. Yaouanc and P. D. de Réotier, Muon Spin Rotation, Relaxation, and Resonance: Applications to Condensed Matter (Oxford University Press, Oxford, 2011).
  • Fujita et al. [2020] M. Fujita, K. M. Suzuki, S. Asano, H. Okabe, A. Koda, R. Kadono, and I. Watanabe, Magnetic behavior of T’-type Eu2CuO4 revealed by muon spin rotation and relaxation measurements, Phys. Rev. B 102, 045116 (2020).
  • Tran et al. [2018] L. M. Tran, M. Babij, L. Korosec, T. Shang, Z. Bukowski, and T. Shiroka, Magnetic phase diagram of Ca-substituted EuFe2As2, Phys. Rev. B 98, 104412 (2018).
  • Guguchia et al. [2013] Z. Guguchia, A. Shengelaya, A. Maisuradze, L. Howald, Z. Bukowski, M. Chikovani, H. Luetkens, S. Katrych, J. Karpinski, and H. Keller, Muon-spin rotation and magnetization studies of chemical and hydrostatic pressure effects in EuFe2(As1-xPx)2, J. Supercond. Nov. Magn. 26, 285 (2013).
  • Ma et al. [2019] J.-Z. Ma, S. M. Nie, C. J. Yi, J. Jandke, T. Shang, M. Y. Yao, M. Naamneh, L. Q. Yan, Y. Sun, A. Chikina, V. N. Strocov, M. Medarde, M. Song, Y.-M. Xiong, G. Xu, W. Wulfhekel, J. Mesot, M. Reticcioli, C. Franchini, C. Mudry, M. Müller, Y. G. Shi, T. Qian, H. Ding, and M. Shi, Spin fluctuation induced Weyl semimetal state in the paramagnetic phase of EuCd2As2, Sci. Adv. 5, eaaw4718 (2019).
  • Franke et al. [2018] K. J. A. Franke, B. M. Huddart, T. J. Hicken, F. Xiao, S. J. Blundell, F. L. Pratt, M. Crisanti, J. A. T. Barker, S. J. Clark, A. Stefancic, M. C. Hatnean, G. Balakrishnan, and T. Lancaster, Magnetic phases of skyrmion-hosting GaV4S8-ySey (yy = 0, 2, 4, 8) probed with muon spectroscopy, Phys. Rev. B 98, 054428 (2018).
  • Hicken et al. [2021] T. J. Hicken, M. N. Wilson, K. J. A. Franke, B. M. Huddart, Z. Hawkhead, M. Gomilsek, S. J. Clark, F. L. Pratt, A. Stefancic, A. E. Hall, M. Ciomaga Hatnean, G. Balakrishnan, and T. Lancaster, Megahertz dynamics in skyrmion systems probed with muon-spin relaxation, Phys. Rev. B 103, 024428 (2021).
  • Ukleev et al. [2021] V. Ukleev, K. Karube, P. M. Derlet, C. N. Wang, H. Luetkens, D. Morikawa, A. Kikkawa, L. Mangin-Thro, A. R. Wildes, Y. Yamasaki, Y. Yokoyama, L. Yu, C. Piamonteze, N. Jaouen, Y. Tokunaga, H. M. Rønnow, T. Arima, Y. Tokura, Y. Taguchi, and J. S. White, Frustration-driven magnetic fluctuations as the origin of the low-temperature skyrmion phase in Co7Zn7Mn6, npj Quantum Mater. 6, 40 (2021).
  • Mühlbauer et al. [2009] S. Mühlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch, A. Neubauer, R. Georgii, and P. Böni, Skyrmion lattice in a chiral magnet, Science 323, 915 (2009).
  • Yu et al. [2011] X. Z. Yu, N. Kanazawa, Y. Onose, K. Kimoto, W. Z. Zhang, S. Ishiwata, Y. Matsui, and Y. Tokura, Near room-temperature formation of a skyrmion crystal in thin-films of the helimagnet FeGe, Nat. Mater. 10, 106 (2011).
  • Yu et al. [2010] X. Z. Yu, Y. Onose, N. Kanazawa, J. H. Park, J. H. Han, Y. Matsui, N. Nagaosa, and Y. Tokura, Real-space observation of a two-dimensional skyrmion crystal, Nature 465, 901 (2010).
  • Seki et al. [2012a] S. Seki, X. Z. Yu, S. Ishiwata, and Y. Tokura, Observation of skyrmions in a multiferroic material, Science 336, 198 (2012a).
  • Kézsmárki et al. [2015] I. Kézsmárki, S. Bordács, P. Milde, E. Neuber, L. M. Eng, J. S. White, H. M. Rønnow, C. D. Dewhurst, M. Mochizuki, K. Yanai, H. Nakamura, D. Ehlers, V. Tsurkan, and A. Loidl, Néel-type skyrmion lattice with confined orientation in the polar magnetic semiconductor GaV4S8, Nat. Mater. 14, 1116 (2015).
  • Tokunaga et al. [2015] Y. Tokunaga, X. Z. Yu, J. S. White, H. M. Rønnow, D. Morikawa, Y. Taguchi, and Y. Tokura, A new class of chiral materials hosting magnetic skyrmions beyond room temperature, Nat. Commun. 6, 7638 (2015).
  • Seki et al. [2012b] S. Seki, J.-H. Kim, D. S. Inosov, R. Georgii, B. Keimer, S. Ishiwata, and Y. Tokura, Formation and rotation of skyrmion crystal in the chiral-lattice insulator Cu2OSeO3, Phys. Rev. B 85, 220406(R) (2012b).
  • Li et al. [2019] H. Li, B. Ding, J. Chen, Z. Li, Z. Hou, E. Liu, H. Zhang, X. Xi, G. Wu, and W. Wang, Large topological Hall effect in a geometrically frustrated kagome magnet Fe3Sn2, Appl. Phys. Lett. 114, 192408 (2019).
  • Xu et al. [2021] Y. Xu, L. Das, J. Z. Ma, C. J. Yi, S. M. Nie, Y. G. Shi, A. Tiwari, S. S. Tsirkin, T. Neupert, M. Medarde, M. Shi, J. Chang, and T. Shang, Unconventional transverse transport above and below the magnetic transition temperature in Weyl semimetal EuCd2​As2{\mathrm{EuCd}}_{2}{\mathrm{As}}_{2}, Phys. Rev. Lett. 126, 076602 (2021).
  • Batista et al. [2016] C. D. Batista, S.-Z. Lin, S. Hayami, and Y. Kamiya, Frustration and chiral orderings in correlated electron systems, Rep. Prog. Phys. 79, 084504 (2016).
  • Niki et al. [2015] H. Niki, S. Nakamura, N. Higa, H. Kuroshima, T. Toji, M. Yogi, A. Nakamura, M. Hedo, T. Nakama, Y. Ōnuki, and H. Harima, Studies of 27Al NMR in EuAl4, J. Phys.: Conf. Ser. 592, 012030 (2015).
  • Heinze et al. [2011] S. Heinze, K. von Bergmann, M. Menzel, J. Brede, A. Kubetzka, R. Wiesendanger, G. Bihlmayer, and S. Blügel, Spontaneous atomic-scale magnetic skyrmion lattice in two dimensions, Nat. Phys. 7, 713 (2011).
  • Ozawa et al. [2017] R. Ozawa, S. Hayami, and Y. Motome, Zero-field skyrmions with a high topological number in itinerant magnets, Phys. Rev. Lett. 118, 147205 (2017).