Transitions from a Kondo-like diamagnetic insulator into a modulated ferromagnetic metal in FeGa_3−yGe_y

Significance

A unifying mechanism for the origin of ferromagnetism which takes into account both well-known and polar extreme theories, namely, the localized- and itinerant-electron regimes, remains a longstanding mystery. Here, we study a particularly interesting system of FeGa_3−yGe_y, within which the magnetic orderings can be adjusted by the Ge-electron filling control, turning from itinerancy to adequate localized through a ferromagnetic quantum critical transition with increasing $y$. By involving the effects of the spin fluctuations in general, the theoretical estimations of the magnetic properties at the intermediate state of magnetic orderings formulated in terms of the correlation between itinerancy electrons and the spin fluctuations are in quantitative agreements with experimental observations. Our analysis shows a potential universality to all itinerant ferromagnets.

Abstract

One initial and essential question of magnetism is whether the magnetic properties of a material are governed by localized moments or itinerant electrons. Here, we expose the case for the weakly ferromagnetic system FeGa_3−yGe_y, wherein these two opposite models are reconciled, such that the magnetic susceptibility is quantitatively explained by taking into account the effects of spin–spin correlation. With the electron doping introduced by Ge substitution, the diamagnetic insulating parent compound FeGa₃ becomes a paramagnetic metal as early as at y=0.01, and turns into a weakly ferromagnetic metal around the quantum critical point y=0.15. Within the ferromagnetic regime of FeGa_3−yGe_y, the magnetic properties are of a weakly itinerant ferromagnetic nature, located in the intermediate regime between the localized and the itinerant dominance. Our analysis implies a potential universality for all itinerant-electron ferromagnets.

Magnetic materials are of particular importance in fundamental theoretical studies as well as advanced technical applications due to the subtle correlations within the systems (1–5). Understanding the entanglement of microscopic alignment of spin moments, i.e., the ingredients of magnetic mechanisms, is crucially important. Spin fluctuations in many-body systems are of such importance, as they have proven to play key roles leading to the formation of “strange metal,” non-Fermi liquids, and quantum effects extending to high temperatures (6, 7). In the cuprate and iron-based superconductors, the essential pairing interaction is proved to be mediated by the spin fluctuations as a common thread in the unconventional superconductors (8, 9).

Although ferromagnetism is one of the oldest observed and studied quantum phenomena, the exact mechanism through which it emerges is not fully understood. Well-established theories are restricted to two narrow extremes, i.e., the localized and itinerant-electron regimes. Although great effort including Hartree–Fock approximation (HFA) and random phase approximation (RPA) in magnetic theories has been made to elucidate the magnetic properties in the intermediate range of the two opposite extremes (10–14), a successful theory remains elusive. The HFA underestimates the amplitude of fermion thermal excitations and the spin density fluctuations due to its oversimplified assumption that the thermal spin-flip excited electrons and holes move independently under a static mean field (15, 16). The RPA theory in terms of an oscillating molecular field, however, only takes into proper account the effects of spin waves as elementary excitations around the equilibrium state, missing the correlations between the excited modes of fluctuations (1, 17, 18). A recent picture of the hybrid nature of localized moments and itinerant electrons was explored in several systems, on which the hybrid model in a two-band approximation was proposed to illustrate the magnetism and, in some cases, the origin of superconductivity (19–24). The self-consistent renormalization (SCR) theory of spin fluctuations and related theories successfully approaches the localized regime based on the itinerant picture as an intermediate mechanism in the one-band model by mediating the magnetic momentum of itinerant electrons in terms of wave-number-dependent spin fluctuation and generalized dynamical fluctuations (18, 25–27). Despite this, however, a unified dynamical theory is still being debated, particularly due to the limited diversity of materials for further study, as well as the difficulty in reconciling these two polar extremes (28).

The heavy-fermion Kondo insulators provide a good platform to explore the physical properties, including magnetic ordering due to the coupling of the charge dynamics to the component ordering associated with its related fluctuations during a metal–insulator transition. The Kondo-insulator-like semiconductor, FeGa₃, which has a larger pseudogap compared with the typical Kondo insulators, is such an ideal system owing to its expected valence admixture (29, 30). The energy gap of FeGa₃ is ∼0.4 eV, and its pseudogap is assumed to be formed by the strong hybridization between the $3d$ band of Fe and $4p$ band of Ga (31, 32). No magnetic ordering is detected in FeGa₃ by ${}^{57}\text{Fe}$ Mössbauer experiments (32). The nonmagnetic FeGa₃ is reminiscent of another Fe-based Kondo insulator, FeSi, which has drawn attention for decades. Interesting physical properties, including anomalous Hall effect, strong magnetic resistivity, chiral magnetic nature, and reentrant spin glass behaviors, are induced by electron doping in FeSi, whose novel phenomena are represented by the Dzyaloshinski–Moriya interaction, conventional isotropic exchange, anisotropic exchanges, the Zeeman interaction under applied field, and cubic anisotropy effects (33, 34). In this work, we follow the phase transitions by increasing the Ge doping of FeGa_3−yGe_y, through all of which the magnetic properties are significantly affected. The marginal Fermi-liquid behavior is observed on the verge of the transition. The magnetic properties in the ferromagnetic region crossing over between the localized model and itinerant regimes of FeGa_3−yGe_y are reconcilably interpreted by mainly taking into account the effects of temperature-dependent spin fluctuations in general. Spin fluctuations may play a key role in approaching a unified theory for localized and itinerant ferromagnets, and FeGa_3−yGe_y seems to be one of the best candidates for probing the unified theory for itinerant magnetism.

Results

Phase Transitions.

Fig. 1 displays the phase diagram of FeGa_3−yGe_y obtained on the basis of the magnetic and transport measurements (Figs. S1–S3, and S7). Increasing Ge substitution for Ga resulted in FeGa_3−yGe_y turning from a diamagnetic insulator into a paramagnetic metal as early as at $y=0.01$, and eventually into a ferromagnetic metal from $y>0.15$. The relatively small spontaneous magnetic moment $P_{\mathrm{S}}$ indicated a weakly ferromagnetic nature in FeGa_3−yGe_y (Fig. S1); the effective magnetic moment $P_{\mathrm{eff}}$ displayed a weak $y$ dependence as an itinerant ferromagnet (Fig. S2). The marginal Fermi-liquid state for composition on the border of d-metallic ferromagnetism is observed in Figs. S3 and S4, as the electrical resistivity and the specific heat divided by temperature for $y=0.15$ followed the $T^{5/3}$ and $-\ln T$ dependences, when the temperature was sufficiently lower than the characteristic spin fluctuation energy (7, 27, 35). The marginal Fermi-liquid behavior was led by the scattering of fermions from the spin fluctuation-induced exchange magnetic field on the verge of metallic ferromagnetism. As the metallization of FeSi_1−xGe_x occurring at $x=0.25$ was orders of magnitude larger than that of $y=0.01$ in FeGaFeGa_3−yGe_y, and the energy gap of FeSi was ∼1/10th of the gap of FeGa₃, the Ge substitution considerably affected the electronic state in FeGa_3−yGe_y (36).

Fig. 1.

Phase diagram of FeGaFeGa_3−yGe_y. Open squares represent the ferromagnetic transition temperature $T_{\mathrm{C}}$, and the bold arrow shows the quantum critical point $(\text{at}y=0.15)$. Solid line at $y=0.01$ corresponds to the critical transition edge between the Kondo-like insulator and the paramagnetic metal. Solid curve starting from $y=0.15$ is the fitting line of the critical temperature $T_{\mathrm{C}}$. Color scale represents magnetization of FeGa_3−yGe_y measured at $H=1\mathrm{T}$ at various temperatures.

Magnetic Orderings and Universality of Spin Fluctuations.

Conventionally, the critical temperature $T_{\mathrm{C}}$ of itinerant ferromagnets is determined from the $M^2$ vs. $H/M$ relation. According to the mean-effective-field solution of an arbitrary spin Ising model, if the Gaussian distribution of exchange coupling intensity is considerably greater than the mean value of exchange bonds, Arrott plots ($M^2$ vs. $H/M$ plots) should show straight lines, and one plot must pass the origin at $T_{\mathrm{C}}$. In the case of FeGa_3−yGe_y however, only the Arrott plots for $y=0.32$ showed good linear behavior (Fig. 2A and Fig. S6). The other samples showed convex curvatures, even at $T_{\mathrm{C}}$, and the curvature decreased with the increasing electron doping. In contrast, all of the $M^4$ plots (Fig. 2B and Fig. S7) for ferromagnetic FeGa_3−yGe_y showed good linear behavior, especially at $T_{\mathrm{C}}$, where the $M^4$ plot passed the origin (0,0), nonsignificant deviations from linear behavior around QCP were observed in $M^4$ plots (Discussion). In these cases, the critical temperature $T_{\mathrm{C}}$ could still be estimated by the low-field data of the isothermal Arrott plots, approximating the arbitrary spin Ising model. The QCP at $y=0.15$ inferred from the magnetic measurement as $T_{\mathrm{C}}$ reaching 0 was consistent with that derived from the resistivity and specific heat measurements.

Fig. 2.

Arrott plots and $M^4$ plots. $M^2$ vs. $H/M$ (Arrott plots) and $M^4$ vs. $H/M$ for FeGa_3−yGe_y with $y=0.14$, $0.15$, $0.16$, $0.20$, $0.24$, and $0.32$, respectively, as A and B. Dashed lines in B are the description of Eq. 4 and should be where $M^4$ plots shown up at the critical temperature $T_{\mathrm{C}}$ (text).

The Rhodes–Wohlfarth and Deguchi–Takahashi plots are drawn in Fig. 3 A and B, respectively (37, 38). $(1/2)P_{\mathrm{C}}$ in Fig. 3A represents the effective spin per atom, whose value can be derived from $P\mathrm{eff}2=P_{\mathrm{C}}(P_{\mathrm{C}}+2)$. The largest magnitude of the magnetic ordering parameter $P_{\mathrm{C}}/P_{\mathrm{S}}$ obtained in this work was 2.6 at $y=0.16$, corresponding to a weakly itinerant nature. The smallest $P_{\mathrm{C}}/P_{\mathrm{S}}$ of 1.8 at $y=0.32$ indicated an adequate localized nature within the system, which is comparable with nickel’s value of 1.5. In the Deguchi–Takahashi plot, where $T_0$ represents the energy width of the dynamical spin fluctuation spectrum in frequency space corresponding to the stiffness of spin density in amplitude, the localized nature of electrons became dominant in the systems if $T_0$ was comparable with $T_{\mathrm{C}}$ in magnitude (18, 38, 39). The right side of the abscissa in Fig. 3B, where $T_{\mathrm{C}}/T_0\sim 1$, represents the localized regime. The left side, where $T_{\mathrm{C}}/T_0\ll 1$, represents the extreme of itinerancy. Fig. 3B consistently shows FeGa_3−yGe_y spread from the localized regime toward the itinerant one with increasing $y$, suggesting a modulated state of magnetic moments in the cross-over region of the two pole extremes. For magnetic orderings in terms of the model of closed Kondo–Heisenberg approximation, the increasing electron doping caused the effects of the Kondo interaction $⟨J_K{\sum }_i{S}_i\cdot s_{ci}⟩$ to become relatively weaker than the Heisenberg interaction $⟨(J_H/z){\sum }_{(ij)}{S}_i{S}_j⟩$ did in the system, i.e., the itinerancy acquired from the Kondo effect in $d$ electrons through intersite exchange became less significant by the continuous electron doping in the FeGa_3−yGe_y.

Fig. 3.

Rhodes–Wohlfarth and Deguchi–Takahashi plots. $P_{\mathrm{C}}/P_{\mathrm{S}}$ vs. $T_{\mathrm{C}}$ plot and $P_{\mathrm{eff}}/P_{\mathrm{S}}$ vs. $T_{\mathrm{C}}/T_0$ plot for FeGa_3−yGe_y and various ferromagnets, as A and B, respectively. Data are reproduced from refs. 42–53. (A) Parameters of ${\text{FeGa}}_{3-y}$Ge_y do not follow the universal line, and $P_{\mathrm{C}}/P_{\mathrm{S}}$ of FeGa_3−yGe_y is relatively small compared with other ferromagnets with same magnitude of $T_{\mathrm{C}}$. (B) Red straight line represents Takahashi’s theoretical line, $P_{\mathrm{eff}}/P_{\mathrm{S}}=1.4\times {(T_{\mathrm{C}}/T_0)}^{-2/3}$, which roughly describes FeGa_3−yGe_y.

Importantly, $P_{\mathrm{C}}/P_{\mathrm{S}}$ of FeGa_3−yGe_y were not described by the fitting curve and had much smaller values than other ferromagnetic metals and alloys with the same $T_{\mathrm{C}}$ in Fig. 3A. Unlike the majority of ferromagnetic metals or alloys, FeGa_3−yGe_y contains a considerably low effective Fermi energy caused by its sharp density of states at $E_{\mathrm{F}}$ (40), resulting in its $T_{\mathrm{C}}$ to vary considerably less rapidly than the $E_{\mathrm{F}}$ and the failure to follow the Rhodes–Wohlfarth curve, which well describes the behavior of most other metals and alloys (41). However, FeGa_3−yGe_y with various amplitudes of dynamical spin fluctuations, corresponding to different $T_0$ values as shown in Table 1, roughly satisfied the generalized Rhodes–Wohlfarth theoretical equation, $P_{\text{eff}}/P_{\text{S}}=1.4\times {(T_{\mathrm{C}}/T_0)}^{-2/3}$ in Fig. 3B, and relatively widely spread along the line with its $T_{\text{C}}$ increasing from 0 at QCP to a considerably high value of 53.1 K. The good fitting of the equation for the entire range of weak ferromagnets implies a great reliance on spin fluctuations in reconciling the ferromagnets with different electron itinerancy from a localized regime to an itinerant regime.

Table 1.

Spin-fluctuation parameters

$y$	$P_{\mathrm{eff}}$	$P_{\mathrm{C}}$	$P_{\mathrm{S}}$	$T_{\mathrm{C}}$	$T_{\mathrm{A}}({10}^4)$	$T_0({10}^2)$	$\overline{F_1}({10}^5)$
0.16	0.71	0.226	0.087	7.2	7.56	1.10	1.39
0.18	0.74	0.244	0.112	14.3	9.67	1.87	1.33
0.20	0.79	0.274	0.133	24.8	1.23	2.99	1.35
0.21	0.80	0.281	0.136	32.6	1.42	3.48	1.59
0.24	0.90	0.345	0.156	36.4	1.33	4.23	1.11
0.27	0.91	0.352	0.187	46.9	1.29	4.30	1.03
0.32	0.96	0.386	0.216	53.1	1.18	4.56	0.73

Spin-fluctuation parameters estimated from magnetic measurements for $y=0.16,0.18,0.20,0.21,0.24,0.27,$ and $0.32$. $P_{\mathrm{eff}},P_{\mathrm{S}},T_{\mathrm{C}},T_{\mathrm{A}},T_0$, and $\overline{F_1}$, represent effective magnetic moment (${\mu }_{\mathrm{B}}$/Fe), spontaneous magnetic moment at ground state (${\mu }_{\mathrm{B}}$/Fe), Curie temperature (K), the width of the distribution of the dynamical susceptibility in the $q$ space (K), the energy width of the dynamical spin fluctuation spectrum (K), and fourth-order expansion coefficients of magnetic free energy (K), respectively. $\frac{1}{2}{P}_{\mathrm{C}}$ represents effective spin per atom (${\mu }_{\mathrm{B}}$).

Experiment vs. Theory.

Experimental results of inverse susceptibilities ${\chi }^{-1}$ vs. temperature $T$ and those of the theoretical reconstruction are shown in Fig. 4 (see also Fig. S8). The reasonable consistency between experimental observations and theoretical calculations evidenced the precision of the spin-fluctuation parameters we estimated in this work and also the success of our analysis for the modulated ferromagnetic FeGa_3−yGe_y.

Fig. 4.

Temperature dependence of inverse susceptibility. $T$ dependences of ${\chi }^{-1}$ for FeGa_3−yGe_y with $y=0.16$, $0.20$, $0.24$ and $0.32$. Black lines and squares represent experimental results. Red lines represent reconstructed results based on the theories of spin fluctuations (text).

Discussion

Magnetic behaviors that are intermediate between localized and itinerant nature in FeGa_3−yGe_y imply great difficulty in explaining the magnetic properties within a unified theory. Additionally, celebrated models properly describing the ground state should be taken beyond to involve the temperature-dependent effects of spin fluctuations. Starting by dealing with the intrinsic free energy $F$ in magnetization, which can be expanded in powers of magnetization $M$ by tracking the splitting in band calculation:

$$F(M,T)=F(0,T)+\frac{1}{2}{a}_1(T)M^2+\frac{1}{4}{a}_2(T)M^4+\mathrm{\cdots },$$ [1]

Converted as the magnetic field $H$-dependent $M$ equation:

$$H=\frac{\partial F}{\partial M}=a_1(T)M+a_2(T)M^3+a_3(T)M^5+\mathrm{\cdots },$$ [2]

where $F(0,T)$ is the free energy at $M=0$, and $a_i(T)$ are expansion coefficients related with the electron density of states and its derivatives near $E_{\mathrm{F}}$.

The thermodynamic state of the free energy is determined by the association of the hopping conduction electrons with the repulsion by electrons with opposite spin directions on site. For an itinerant ferromagnetic system, where its thermodynamic state becomes stable at finite magnetization, its magnetic properties can be described by the linear Arrott plot within coefficients $a_1$ and $a_2$ neglecting higher power terms, since the conduction electron density is fairly restricted around the Fermi energy $E_{\mathrm{F}}$ in the ferromagnets, which leads to the famous equation:

$$M^2(M,T)=-\frac{a_1(T)}{a_2(T)}+\frac{1}{a_2(T)}\frac{H}{M(H,T)}.$$ [3]

Numerous systems are governed by Eq. 3. As examples, some weakly ferromagnetic compounds similar to FeGa_3−yGe_y are ZrZn₂ (42), Sc₃In (54), ZrTiZn₂ (55), ZrZn_1.9 (55), and Ni-Pt alloys (43). However, Arrott plots of ferromagnetic FeGa_3−yGe_y are not linear around the Curie points, especially when $y$ is positioned close to the critical point of 0.15. This suggests the need for a higher power term of free energy $a_3(T)M^5$, which is not contemplated by the ground-state-based magnetic theories such as HFA or RPA (11, 13, 56). Even in the present form of the SCR theory, the fourth expansion coefficient, $a_2(T)$, is assumed to be temperature-independent, resulting in an inaccurate prediction that the spontaneous magnetic moment in ferromagnets vanishes at the Curie temperature. This also implies the need for a higher power of term $a_3(T)M^5$ in the free-energy function for the approximation. Inputting all of the dynamical parameters of $a_i$ for the $M^4$ plots at the critical point $T_{\mathrm{C}}$, we have ref. 38:

$$H/M=\frac{T_{\mathrm{A}}^3}{2{\mu }_{\mathrm{B}}{[3\pi T_{\mathrm{C}}(2+\sqrt{5})]}^2}{\left(\frac{P_{\mathrm{S}}}{M_{\mathrm{S}}}\right)}^5{M}^4.$$ [4]

where the spin-fluctuation parameter $T_{\mathrm{A}}$ represents the width of the distribution of the dynamical susceptibility in wave vector space, and $M_{\mathrm{S}}=N_0{\mu }_{\mathrm{B}}{P}_{\mathrm{S}}$ is the spontaneous magnetization in the ground state, with $N_0$ representing the number of atoms. For $y$ = 0.14, 0.15, and 0.16 in FeGa_3−yGe_y, the small deviation from linear of the $M^4$ plots may be caused by the comparable $a_2(T)M^3$ terms and $a_3(T)M^5$ terms in the vicinity of the QCP, indicating the comparable effects in nonlinear couplings of spin fluctuations to the effects of nonnegligible temperature dependence in general. For $y\ge 0.16$, the term $a_3(T)M^4$ gradually becomes overwhelming compared with $a_2(T)M^2$, ($\frac{T_{\mathrm{A}}^3{\rho }^3}{2N_0^5{\mu }_{\mathrm{B}}^6\left({\rho \prime }^2/{\rho }^2-\rho ″/3\rho \right){[3\pi T_{\mathrm{C}}(2+\sqrt{5})]}^2}{M}^2\gg 1$, where $\rho$ represents the density of states); hence, the $M^4$ plots show much better linear behaviors than the Arrott plots do, and, synchronously, the curvature begins to decrease in the plots with the electron doping. For ferromagnetic ${\text{FeGa}}_{3-y}$Ge_y, we observe that the Arrott plots at $T_{\mathrm{C}}$ nearly pass the origin, indicating the nonnegligible temperature dependence of spin fluctuations is still considerable, even in the case where their $T_{\mathrm{C}}$ vanishes at the critical point.

Well-established approximations such as HFA and RPA only deal with the paramagnetic contributions of spin fluctuations of elementary excitations; however, for ${\text{FeGa}}_{3-y}$Ge_y, effects of temperature-dependent long-range mode–mode coupling spin fluctuations on the thermal equilibrium state is crucial in accounting for its magnetic properties. We take the quantum statistical mechanical theory of SCR approximation of spin fluctuations into consideration, in which two well-known assumptions are inherited: (i) In the ground state, the magnetic properties can be described by the band calculation; and (ii) the effects of spin–spin couplings can be mainly represented by the second expansion coefficient of the free energy. We should mention that the theories of spin fluctuations are then in contrast with the phenomenological–theoretical-based technique of the modified Arrott plot in which arbitrary critical exponents can be applied (18, 57), since the function of free energy in the theories of spin fluctuation is even.

In the weakly ferromagnetic limit of the SCR approximation, the imaginary part of the dynamical spin susceptibility for ferromagnets is described by the double Lorentzian form in the small $q,\omega$ region (18):

$$\mathrm{Im}\chi (q,\omega )=\frac{\chi (\mathrm{0,0})}{1+q^2/{\kappa }^2}\frac{\omega {\mathrm{\Gamma }}_q}{{\omega }^2+{\mathrm{\Gamma }}_q^2}.$$ [5]

where ${\mathrm{\Gamma }}_q$ is the spectral width of the spin fluctuations given by ${\mathrm{\Gamma }}_q=(A/C)q(q^2+{\kappa }^2)={\mathrm{\Gamma }}_{0}q(q^2+{\kappa }^2)$, and ${\kappa }^2=\varrho /2A\chi =N_0/2\overline{A}\chi$, and it leads to:

$$\frac{P_{\mathrm{S}}^2}{4}=\frac{15T_0}{T_{\mathrm{A}}}c{\left(\frac{T_{\mathrm{C}}}{T_0}\right)}^{4/3},$$ [6]

in weakly ferromagnetic systems.

Derived from Eq. 5, the inverse magnetic susceptibility is given by ref. 38:

$$y=\frac{N_0}{2T_A{\eta }^2}\frac{{\kappa }^2}{{\kappa }^2+q^2}{\chi }^{-1}\cong \frac{\overline{F_1}{P}_{\mathrm{s}}^2}{8T_{\mathrm{A}}{\eta }^2}\{-1+\frac{1+\nu y}{c}{\int }_0^{1/\eta }\mathrm{d}z{z}^3[\ln u-\frac{1}{2u}-\mathrm{\Psi }(u)]\}.$$ [7]

With $u=z(y+z^2)/t$, $t=T/T_{\mathrm{C}}$, $\nu ={\eta }^2{T}_{\mathrm{A}}/U$, $\eta ={(T_{\mathrm{C}}/T_0)}^{1/3}$, $c=0.3353$. $\mathrm{\Psi }(u)$ is the digamma function, and parameter $\overline{F_1}$ is the mode–mode coupling constant, representing the fourth-order expansion coefficients of magnetic free energy. $\overline{F_1}=N_A^3{(2{\mu }_{\mathrm{B}})}^4/\zeta k_{\mathrm{B}}$, $\zeta$ is the slope of the Arrott plots at low temperature. $U$ represents the intraatomic exchange energy, and $N_{\mathrm{A}}$ and $k_{\mathrm{B}}$ are Avogadro’s number and the Boltzmann constant (58).

Due to the compensation of the increasing thermal amplitude of spin fluctuation for the suppression of the zero-point spin fluctuation under applied magnetic field with increasing temperature, the sum of total spin amplitude squared at finite temperature can be treated as nearly conserved, which leads to the following equation (38, 39):

$$\overline{F_1}=\frac{4}{15}\frac{k_{\mathrm{B}}{T}_{\mathrm{A}}^2}{T_0}.$$ [8]

The above assumption of total spin amplitude conservation agrees with the first fully quantitative study based on an analogous rotationally invariant Hartree approximation of the effects of spin fluctuations in terms of high-precision studies based on the de Haas–van Alphen effect in conjunction with semiempirical band models or direct experimental measurements such as neutron scattering, as the essential magnetic equations inferred from the above two theories are quantitatively consistent under certain approximations (18, 26, 59, 60). The validity of Eq. 8 is confirmed by the inelastic neutron scattering or nuclear magnetic resonance measurements on the archetypal weak itinerant ferromagnets that are analogous to ${\text{FeGa}}_{3-y}$Ge_y (61–65), since the fourth expansion coefficient $\overline{F_1}$ derived from magnetic measurements is consistent with the estimated values by using spin-fluctuation parameters $T_{\mathrm{A}}$ and $T_0$ inferred from above dynamical measurements by applying Eq. 8. The spin amplitude conservation is also observed in the case of the one-dimensional and 2D Hubbard’s approximate model (66–68).

The component of the effects of zero-point spin fluctuations emphasized in the above assumption for low-temperature ferromagnetic order systems is neglected by the former versions of SCR spin fluctuation theories, in which only the local spin amplitude squared is assumed to be conserved. This difference leads to considerable differences between the current analysis and discussions based on former SCR theories. In this work, the fourth expansion coefficient $\overline{F_1}$ under the influence of the zero-point component of spin fluctuations is assumed to be temperature-dependent and associated with the spectral width amplitudes of spin fluctuations. Hence, all necessary spin-fluctuation parameters can be estimated by applying Eq. 8 merely using macroscopic magnetization measurements, without the need of pursuing any dynamical measurements (44, 69). In addition, a sixth expansion term of free energy left absent from the former SCR theories which is related to the critical temperature $T_{\mathrm{C}}$ should also be taken into consideration at relatively large values of $\eta ={(T_{\mathrm{C}}/T_0)}^{1/3}$. The importance of this is shown in the magnetic equations as Arrott plots and $M^4$ plots of ${\text{FeGa}}_{3-y}$Ge_y. The universally satisfied relation between $P_{\mathrm{eff}}/P_{\mathrm{S}}$ and $T_{\mathrm{C}}/T_0$ in the generalized Rhodes–Wohlfarth plot of Fig. 3B is also derived from the total spin amplitude conservation assumption. The quantitative agreements between the theoretical reconstruction and the experimental results indicate the success in elucidating the magnetization mechanisms for the intermediate range ${\text{FeGa}}_{3-y}$Ge_y system. The successful explanation for the ferromagnetic ${\text{FeGa}}_{3-y}$Ge_y system fitting well into the generalized Rhodes–Wohlfarth relation, $P_{\mathrm{eff}}/P_{\mathrm{S}}=1.4\times {(T_{\mathrm{C}}/T_0)}^{-2/3}$, which describes a large variety of ferromagnets, indicates a potential universality in quantitatively explaining the magnetism of weakly ferromagnetic systems in a broad $T_{\mathrm{C}}$ range by involving the effects of spin fluctuations.

In this work, we have shown that electron doping by Ge substitution substantially affects the magnetic ground state and spin–spin correlation in ${\text{FeGa}}_{3-y}$Ge_y, causing phase transitions and significant changes in magnetic orderings, as well as the spin fluctuation mediated marginal Fermi-liquid state on the border of ferromagnetism. We successfully take the temperature-dependent effects of spin fluctuations in general into account for the modulated ferromagnetic ${\text{FeGa}}_{3-y}$Ge_y ranging from an itinerant regime to the adequate localized region, and the theoretical reconstruction agrees well with the experimental observations. Our analysis shows a potential universality for the entire range of weakly itinerant ferromagnetic systems. ${\text{FeGa}}_{3-y}$Ge_y should be a promising model system to help achieve a unified magnetic theory for localized and itinerant electrons.

Methods

Single crystals of ${\text{FeGa}}_{3-y}$Ge_y were synthesized by the Ga self-flux method. Powders of Fe (99.99%), Ge (99.99%), and Ga (99.9999%) ingot with the ratio of Fe:Ge:Ga = 1: $Y$: 9 $(0.01\le Y\le 3)$ were loaded and sealed in an evacuated silica tube. The mixture was melted and homogenized in a furnace and cooled to room temperature slowly. Excess Ga flux was removed with an aqueous solution of H₂O₂ and HCl. X-ray diffraction pattern confirmed the samples are single crystal in FeGa₃ type structure without second phase. The chemical composition of ${\text{FeGa}}_{3-y}$Ge_y was determined by wavelength-dispersive electron microprobe analysis. The magnetization, electrical resistivity, and specific heat of ${\text{FeGa}}_{3-y}$Ge_y were measured by the superconducting quantum interference device, the standard four-probe method in a ⁴He cryostat and quantum design physical property measurement system, respectively.