Doping-controlled spin reorientation transition and spin switching of Tm_1−xPr_xFeO₃ single crystals

Abstract

Rare-earth orthoferrites (RFeO₃), which are canted antiferromagnets exhibiting weak ferromagnetism, are potential materials for spintronic devices. The temperature of spin reorientation transition (SRT) in {\mathrm{T}\mathrm{m}}_{1-x}{\mathrm{P}\mathrm{r}}_x{\mathrm{F}\mathrm{e}\mathrm{O}}_3 (x = 0, 0.15, 0.25, 0.5) single crystals shifts to a lower region with the increase of x, attributing to the exchange energy of {\mathrm{P}\mathrm{r}}^{3+}-{\mathrm{F}\mathrm{e}}^{3+} being less than that of {\mathrm{T}\mathrm{m}}^{3+}-{\mathrm{F}\mathrm{e}}^{3+}. Both type-I of spin switching (SSW-I) and the type-II of spin switching (SSW-II) are observed along the a-axis when x = 0.15 and 0.2. The changes of magnetic interaction are analyzed through structural distortion and transforming Curi−Weiss fitting. A rare SSW-II phenomenon is observed along the c-axis during both the cooling and warming processes when x = 0.15, 0.25, 0.5. In addition, a weak ferromagnetic moment (WFM) component is observed despite the antiferromagnetic behavior at high temperature in {\mathrm{T}\mathrm{m}}_{0.5}{\mathrm{P}\mathrm{r}}_{0.5}{\mathrm{F}\mathrm{e}\mathrm{O}}_3 single crystal. The WFM decreases gradually with the increase in temperature, indicating the occurrence of SRT. And the field-induced spin switching (SSW) is observed along the a-axis at 70 K when x = 0.5. The magnetic properties of {\mathrm{P}\mathrm{r}}^{3+} doped {\mathrm{T}\mathrm{m}\mathrm{F}\mathrm{e}\mathrm{O}}_3 single crystals are obviously different from TmFeO₃ and PrFeO₃ single crystals, indicating the complex magnetic interaction among {\mathrm{P}\mathrm{r}}^{3+}, {\mathrm{T}\mathrm{m}}^{3+} and {\mathrm{F}\mathrm{e}}^{3+}. The complex and abundant magnetic phenomena in {\mathrm{T}\mathrm{m}}_{1-x}{\mathrm{P}\mathrm{r}}_x{\mathrm{F}\mathrm{e}\mathrm{O}}_3 single crystals offer significant potential for studying altermagnet and magneto-optical coupling, and are expected to become a new generation of spintronic devices.

Graphical abstract

Keywords

spin reorientation transition / spin switching / Curie−Weiss law / rare-earth orthoferrites / spintronic

1 Introduction

Due to the cancellation of adjacent antiparallel magnetic moments, the antiferromagnetic material has the advantages of no stray field and the intrinsic frequency up to terahertz order etc., making them ideal candidates for high-density and high-speed storage. As a special antiferromagnet, rare-earth orthoferrites (RFeO₃, R = Y, La−Lu) have been extensively studied in quantum optics [1, 2], spin transport [3] and spin mechanics [4, 5], etc., and still has the potential to explore future applications, which is expected to become the next generation of spintronic devices. Recently, the discovery of collinear magnetic structure known as altermagnets [6−9] has opened the way to explore the third kind of magnetic order, which combines the advantages of ferromagnetic and antiferromagnetic, providing a promising platform for high-speed information processing and terahertz communication, showing a unique transport phenomenon. RFeO₃, as a canted antiferromagnet, also possesses both antiferromagnetic properties and weak ferromagnetic characteristics, which still holds great research potential.

The structure of RFeO₃ is derived from the distortion of ideal perovskite structure. Fe³⁺ and six O²⁻ form FeO₆ octahedra, and R³⁺ is located in the voids between FeO₆ octahedra. Due to the large radius of R³⁺, in order to maintain the structural stability, the FeO₆ octahedron will undergo a certain degree of twisting and tilting, and R³⁺ may leave their ideal positions, thereby causing structural distortion and changing the cubic crystal system space group from Pm3m to Pbnm [10]. The R³⁺ and the Fe³⁺−magnetic sublattice [11] of RFeO₃ gives rise to three magnetic interactions: R³⁺−R³⁺ interaction, R³⁺−Fe³⁺ interaction, and Fe³⁺−Fe³⁺ interaction. Since the far distance between two R³⁺ or Fe³⁺, they need to achieve indirect super-exchange interaction through O²⁻ [12]. According to the Goodenough−Kanamori−Anderson rule [13−15], the strength of the super-exchange interaction is maximized when Fe³⁺ and O²⁻ bond angle is 180° [16], and minimized when the angle approaches 90°. Variations in Fe−O_I bond length and Fe−O_I−Fe bond angle will influence the overlap of Fe³⁺ and O²⁻ electron clouds, thereby affecting the strength of the super-exchange interaction. Meanwhile, the Dzyaloshinskii−Moriya interaction [17, 18] results in an inclination angle of the magnetic moments of Fe³⁺, causing the magnetic moments become canted antiferromagnetic with weak ferromagnetic moment (WFM) [19]. The distortions of the lattice and the complex magnetic interactions in RFeO₃ have triggered a variety of complex magnetic phenomena, such as ultrafast photomagnetic effects [5], spin reorientation transitions (SRT) [20−24], spin switching (SSW) [25−28], and ferroelectricity [29]. In our study, we mainly focus on the SRT and the SSW.

The first Neel temperature (T_N1) of the RFeO₃ is high, typically ranging from 600 K to 700 K. In the high-temperature region below T_N1, R³⁺ is paramagnetic state (above second Neel temperature (T_N2 < 10 K)), and the Fe³⁺−O²⁻−Fe³⁺ interaction is the strongest. The WFM arising from the spin-canted state of Fe³⁺ aligns along the c-axis of the crystal, while the antiferromagnetic moment (AFM) aligns along the a-axis, corresponding to Γ₄ (G_x, A_y, F_z) spin configuration. With the decrease of temperature, the R³⁺−Fe³⁺ interaction enhanced, and the AFM undergoes a flip, causing the spin configuration to change and the occurrence of SRT. Since RFeO₃ only has three spin configurations [10], the temperature-induced SRT can be mainly classified into two types: (i) Γ₄ (G_x, A_y, F_z) → Γ₂ (F_x, C_y, G_z), as observed in PrFeO₃ [21], SmFeO₃ [22], TmFeO₃ [27], etc.; (ii) Γ₄ (G_x, A_y, F_z) → Γ₁ (A_x, G_y, C_z), such as CeFeO₃ [20] and DyFeO₃ [24]. When the temperature is above the R³⁺ magnetic ordering temperature (T_N2), the magnetic moments of R³⁺ and Fe³⁺ are arranged in a parallel or antiparallel coupling state. In some RFeO₃, at specific temperature or external magnetic field (H), due to the flipping of the magnetic moments of R³⁺ or the simultaneous flipping of the magnetic moments of R³⁺ and Fe³⁺, the magnetization undergoes a sudden change, and this change is called SSW. If the magnetic moments of R³⁺ and Fe³⁺ flip simultaneously, resulting in a reversal of the magnetization sign, this is type-I of SSW (SSW-I), such as NdFeO₃ [25], ErFeO₃ [26] and Nd_0.8Pr_0.2FeO₃ [28]. If only R³⁺ magnetic moment reverses, it is type-II of SSW (SSW-II), as seen in PrFeO₃ [21], TmFeO₃ [27] and Tm_0.8Sm_0.2FeO₃ [23].

PrFeO₃ single crystal exhibits unique SRT properties. Due to the complex coupling effect between the 4f electrons of Pr³⁺ and the 3d electrons of Fe³⁺, SRT is highly sensitive to the H and gradually gets suppressed as the H increases. SRT can be observed at 6.5 K only at a small external field (H = 5 Oe) [21]. Furthermore, in the undoped RFeO₃ system, only TmFeO₃ [27] and PrFeO₃ [21] single crystals have observed SSW-II in both field-cooled cooling (FCC) and field-cooled warming modes along the c-axis. Besides the SSW-II observed along the c-axis, SSW-II is also observed along the a-axis in TmFeO₃ [27] single crystal in the FCC mode, and the temperature range of SRT (T_SRT) is 82 K−95 K. The peculiar interaction among Tm³⁺, Pr³⁺ and Fe³⁺ leads to more interesting phenomena when Tm³⁺ and Pr³⁺ are doped with other R³⁺.

In our study, high-quality Tm_1−xPr_xFeO₃ (x = 0, 0.15, 0.25, 0.5) single crystals are grown by the optical floating zone method, and the zero-field-cooled warming (ZFC), FCC curves and the isothermal magnetization (M−H) curves are measured. Unlike PrFeO₃ single crystal which are only found at low temperature and low field, with the increase of x, T_SRT shifts towards lower temperature, moving from 82 K−95 K to 72 K−92 K. When x = 0.15 and 0.25, SSW-I is observed along the a-axis, which is absent in TmFeO₃ and PrFeO₃ single crystals. However, when the doping ratio reached 50%, SSW-I disappears. We discuss the changes in magnetic interactions based on the structural variations, and then analyze the reasons for the appearance of SSW-I. Furthermore, the transforming Curie−Weiss law is employed to fit the data before and after the SSW-I in the Tm_0.85Pr_0.15FeO₃ single crystal at H = 120 Oe in the FCC mode, and the data after SRT in the Tm_0.5Pr_0.5FeO₃ single crystal at H = 100 Oe is also fitted. The magnetic moment of Fe³⁺ and the internal field arising from the magnetic moment tilt of Fe³⁺ at the rare earth sites are calculated. The changes in magnetic interactions during the SSW-I transition are further analyzed. Apart from a-axis, SSW-II is also observed along the c-axis, and triggering temperature of both SSW-I and SSW-II can be regulated by H. Finally, we calculated the weak ferromagnetic saturation magnetization of M−H curve in the Tm_0.5Pr_0.5FeO₃ single crystal along the a-axis. The WFM part gradually decreases while the antiferromagnetic contribution gradually increases, also indicating the occurrence of SRT.

2 Experimental method

The medicinal powder is weighed according to the ratio specified in Eq. (1) and thoroughly mixed,

(1)\begin{array}{rl}& \frac{1-x}{2}{\mathrm{T}\mathrm{m}}_2{\mathrm{O}}_3+\frac{x}{6}{\mathrm{P}\mathrm{r}}_6{\mathrm{O}}_{11}+\frac{1}{2}{\mathrm{F}\mathrm{e}}_2{\mathrm{O}}_3\\ & \triangleq {\mathrm{T}\mathrm{m}}_{1-x}{\mathrm{P}\mathrm{r}}_x{\mathrm{F}\mathrm{e}\mathrm{O}}_3+\frac{x}{6}{\mathrm{O}}_2.\end{array}

x represents the content of Pr³⁺. RFeO₃ polycrystalline powders can be synthesized though the solid-state reaction method followed by multiple sintering steps. Large-sized single crystals can be grown with the diameter of 5 mm and the length exceed 50 mm using the optical float zone method, as shown in Fig.1(a). The single crystals are characterized by X-ray diffraction and structural refinement. The weighted profile R-factors (R_wp = 10.9%, 3.88%, 4.21%, and 6.38%) obtained from the refinement [Fig.1(a)] confirm the high quality of the samples. In addition, the Fe−O_I bond length and Fe−O_I−Fe bond angle [Fig.1(b)] are obtained through refinement to analyze the changes in the super-exchange interaction. The single crystal is cut into test cube using a Laue camera with a cutting machine. The Laue back-reflection patterns confirm the accurate orientation and high quality of the cut crystals [Fig.1(c)]. Subsequently, the ZFC, FCC and M−H curves are tested using the magnetic property measurement system (MPMS 3) from Quantum Design. Selected magnetic measurement results are analyzed using transforming Curie–Weiss fitting to investigate the complex magnetic interactions in RFeO₃.

3 Results and discussion

The magnetic characteristics of Tm_1−xPr_xFeO₃ (x = 0, 0.15, 0.25, 0.5) single crystals along the three crystal axes are shown in Fig.2. The different magnetic behaviors along the three axes highlight the presence of magnetic anisotropy in these single crystals. In the high-temperature region (T_N2 < T < T_N1), the antisymmetric exchange interaction between R³⁺ and Fe³⁺ is relatively weak, allowing the Fe³⁺−Fe³⁺ interaction to dominate (in the following text, R³⁺ = Tm³⁺ and Pr³⁺). With the decreasing temperature, the R³⁺−Fe³⁺ interaction gradually strengthens. and becomes sufficiently strong to overcome the magnetic anisotropy energy. The AFM reorients from the G_x to G_z [30], and the WFM shifts from the F_z to F_x, resulting in the SRT of Γ₄→Γ₂. The Pr³⁺ doping does not alter the type of SRT, which is the same as TmFeO₃ single crystal [27]. However, with the increase of x, T_SRT shifts to lower temperature, moving from 82 K−95 K to 72 K−92 K. According to the phase diagram of the RFeO₃ summarized by Li et al. [21], the T_SRT of PrFeO₃ single crystal is much lower than TmFeO₃, suggesting that the Pr³⁺−Fe³⁺ exchange interaction energy is weaker than Tm³⁺−Fe³⁺. As a result, PrFeO₃ can overcome the anisotropic Fe³⁺−Fe³⁺ interaction at a lower temperature. Therefore, the increase of Pr³⁺ content gradually diluted the Tm³⁺−Fe³⁺ exchange interaction energy [31], resulting in the decrease of T_SRT. In the TmFeO₃ single crystal, a magnetic compensation phenomenon is observed at 15 K [as shown in Fig.2(a)]. As the temperature increases to 15 K, the magnetic moment of Tm³⁺ decreases while the magnetic moment of Fe³⁺ gradually increases, resulting in a reduction of the magnetization. When the temperature reaches 15 K, the magnetic moment of partial Tm³⁺, under the induction of the Fe³⁺ sublattice, flips and couples parallel to the magnetic moment of Fe³⁺, causing the increase of magnetization. This negative magnetization trend may lead to the emergence of SSW-I. This phenomenon is also observed in TmFe_0.8Mn_0.2O₃ single crystal [23]. As shown in Fig.2(b) and (c), when x = 0.15 and x = 0.25, the SSW-I and SSW-II are observed simultaneously along the a-axis in the FCC mode, which is not present in the parent phase TmFeO₃ single crystal [27]. In addition, SSW-I in the ZFC and FCC modes exhibit a butterfly-shape [26]. concerning T₀ symmetry.

In the ZFC mode, SSW-I is observed along the a-axis when x = 0.15 and x = 0.25, resulting in a butterfly-shaped SSW-I Compared with TmFeO₃ and PrFeO₃ single crystals (the PrFeO₃ data are from the standard CIF file), Tm_0.85Pr_0.15FeO₃ exhibits a longer Fe−O_I bond length and a greater deviation of the Fe−O_I−Fe bond angle from 180° [Fig.1(b)], leading to the weakening of the Fe³⁺−Fe³⁺ interaction. As shown in Fig.3(a), in the ZFC mode, the isotropic R³⁺−Fe³⁺ interaction results in an intrinsic parallel or antiparallel alignment between the R³⁺ and Fe³⁺ magnetic moments. Meanwhile, before SSW-I, the molecular field of Fe³⁺ induces the polarization of the R³⁺ magnetic moment, resulting in a WFM parallel to H [16]. Therefore, the coupling mode is ↑↓↓ (the arrows from left to right represent the magnetic moment directions of Tm³⁺, Pr³⁺, and Fe³⁺ respectively). Due to weakening of the Fe³⁺−Fe³⁺ interaction, the R³⁺−Fe³⁺ interaction can be further enhanced with the increase of temperature. The R³⁺ magnetic moment further decreases, resulting in zero magnetization (T₀) and negative magnetization. However, with the temperature further rises, the Fe³⁺−Fe³⁺ interaction will be strengthened, and R³⁺−Fe³⁺ interaction will be unable to resist the influence of H, resulting in the R³⁺ and Fe³⁺ magnetic moments to flip simultaneously. The coupling mode change from ↑↓↓ to ↓↑↑. For example [Fig.3(a)], in the ZFC mode, when H = 180 Oe, the magnetization along the a-axis in the Tm_0.85Pr_0.15FeO₃ single crystal changes from −0.36 emu/g to 0.44 emu/g at T_SI-2 = 78 K. When x = 0.25, butterfly-shaped SSW-I is also observed. Although the distortion degree is smaller compared to TmFeO₃ single crystal, it is larger than PrFeO₃ single crystal [Fig.1(b)], thus explains the occurrence of butterfly-shaped SSW-I. The variations in x = 0.15 and x = 0.25 single crystals reflect the complex magnetic interactions between R³⁺ and Fe³⁺. It is worthy to further exploration through various methods. When x = 0.5, SSW-I disappears. Compared with other single crystals, the Fe−O_I−Fe bond angle becomes 163.93(5)° [Fig.1(b)], the distortion degree of the octahedron is smaller, and the Fe³⁺−O²⁻−Fe³⁺ super-exchange interaction is enhanced, inhibiting the generation of SSW-I. In the FCC mode, it also shows SSW-I. When SSW-I occurs, the magnetic moment reversal of R³⁺ and Fe³⁺ is opposite to the ZFC mode. Moreover, it can be clearly seen that T_SI-1/ T_SI-2 has a positive/negative correlation with H. In the FCC mode, T_SI-1 becomes larger with the increase of H. Tm³⁺ is in a paramagnetic state above T_N2, and the magnetic moment direction tends to be parallel to H. Therefore, with the increase of H, it will flip at higher the temperature. In the ZFC mode, a larger H also leads to the earlier flipping of the Tm³⁺ magnetic moment.

In addition to SSW-I, SSW-II is also observed in the FCC mode along the a-axis when x = 0.15, 0.25, 0.5. The Pr³⁺ magnetic moment is antiparallel to Fe³⁺ below 13 K. In contrast, the Tm³⁺ magnetic moment is antiparallel to Fe³⁺ below 80 K. Therefore, the occurrence of SSW-II is attributed to the flipping of the Pr³⁺ magnetic moment under the influence of Fe³⁺ and Tm³⁺ magnetic moments. When x = 0.25, SSW-II exhibits multi-step characteristics. In Fig.3(d), at 40 Oe, the first magnetization jump occurred at 30 K. At this point, Pr³⁺ magnetic moment partially flipped and the system was not in a steady state. As the temperature was further decreased, Pr³⁺ magnetic moment continued to flip, causing fluctuations in the magnetization, until the system reached a steady state. In addition, with the increase of H, T_SII-1 gradually moves towards low temperature and is suppressed at H = 120 Oe (x = 0.15) and 100 Oe (x = 0.25). When x = 0.5, the situation of SSW-II is similar (as shown in Fig. S1). The triggering temperatures of both SSW-I and SSW-II can be significantly regulated by H and are expected to be applied in spin switching devices.

Structural changes is one of the many reasons contributing to the complex magnetic changes. The transforming Curi−Weiss law is applied to verify the magnetic interaction further. Eq. (2) obtains the experimental Curie constant [32-34],

(2)\chi ={\chi }_0+{\displaystyle \frac{C_{\mathrm{e}\mathrm{x}\mathrm{p}}}{T-\theta }}.

χ₀ is a parameter independent of temperature, C_exp is the experimental Curie constant and θ is the Weiss temperature. The theoretical Curie constant is obtained through Eq. (3) [32],

(3)C={\displaystyle \frac{N{\mu }_{\mathrm{e}\mathrm{f}\mathrm{f}}^2}{{3k}_B}}.

C, N, k_B are the theoretical Curie constant, Avogadro constant and Boltzmann constant respectively. The theoretical effective magnetic moment (μ_eff) is given by Eq. (4) [32],

(4){\mu }_{\mathrm{e}\mathrm{f}\mathrm{f}}=\sqrt{\left(1-x\right){\mu }_{\mathrm{T}\mathrm{m}}^2+x{\mu }_{\mathrm{P}\mathrm{r}}^2+{\mu }_{\mathrm{F}\mathrm{e}}^2}.

μ_Tm, μ_Pr, μ_Fe is Tm³⁺, Pr³⁺ and Fe³⁺ theoretical effective magnetic moment, respectively. The difference between the C_exp and C is negligible (Fig.4), further confirming the high quality of the single crystals and the reliability of the calculated results. Subsequently, the magnetic interactions in the FCC mode below T_SRT are analyzed according to Eq. (5) [35, 36],

(5)M=M_{\mathrm{F}\mathrm{e}}+{\displaystyle \frac{C_{\mathrm{e}\mathrm{x}\mathrm{p}}(H_{\mathrm{i}}+H_{\mathrm{a}})}{T-\theta }}.

M_Fe is Fe³⁺ canted magnetic moment, H_i is internal field generated by Fe³⁺ canted magnetic moment at R site and H_a is applied magnetic field. The results of x = 0.15 and x = 0.5 are shown in Fig.4. The FCC curves of Tm_0.85Pr_0.15FeO₃ single crystal along the a-axis at 120 Oe are fitted before and after SSW-I. Before SSW-I [the pink curve of Fitting1 in Fig.4(a)], M_Fe1 = 498.90 emu/mol, H_i1 = −3597.63 Oe, θ₁ = −15.44 K. M_Fe1 > 0 indicates that the magnetic moment of Fe³⁺ is aligned with H [37]. H_i1 < 0 suggests that the R³⁺ total magnetic moment is opposite to H. Meanwhile, |H_i1| > H_a indicates that the R³⁺ total magnetic moment is antiparallel to Fe³⁺, and θ₁ < 0 reveals the antiferromagnetic interaction between R³⁺ and Fe³⁺. Based on the phase diagram [21], the magnetic moments of Tm³⁺ and Fe³⁺ are antiparallelly coupled, while those of Pr³⁺ and Fe³⁺ are parallelly coupled. These correspond to the coupling mode of ↓↑↑. With the decrease of temperature, the magnetization decreases and the negative magnetization strengthens, corresponding to |H_i1+H_a| > H _i2+H_a. |H_i1+H_a| > |H _i2+H_a| induces to the enhancement of the R³⁺−Fe³⁺ interaction [37], leading to the negative magnetization. After completing SSW-I, M_Fe2 = −458.15 emu/mol, H_i2 = 3417.17 Oe. The Fe³⁺ magnetic moment is opposite to H, while the R³⁺ total magnetic moment aligns with H. leading to the coupling mode of ↑↓↓. In the Tm_0.5Pr_0.5FeO₃ single crystal [Fig.4(b)], after SRT, fitting results are M_Fe = −461.58 emu/mol, H_i = 5626.34 Oe, θ = −12.41 K. The Fe³⁺ magnetic moment remains antiparallel to H, while the R³⁺ total magnetic moment keeps aligning with H. Therefore, SSW-I is not observed when x = 0.5.

In addition to the SSW observed along the a-axis, SSW-II is also observed along the c-axis. As shown in Fig.5(a)−(c), SSW-II are observed in the FCC mode along the c-axis for x = 0.15, 0.25, and 0.5. In the high-temperature region, the R³⁺ total magnetic moment is intrinsically parallel to the Fe³⁺ magnetic moment corresponding to the coupling mode of ↑↑↑. With the temperature decrease, the R³⁺ effective magnetic moment increases, leading to an increase in the total magnetization. As the temperature further decreases, the Tm³⁺ magnetic moment tends to be antiparallel to the Fe³⁺ magnetic moment. Therefore, the Tm³⁺ magnetic moment flips, resulting in SSW-II, and the coupling mode shifts to ↓↑↑. With the increase of H, T_SII-2 gradually moves towards the lower temperature, and SSW-II is suppressed at 80 Oe, 70 Oe, and 50 Oe respectively. In the high-temperature region, the Tm³⁺ magnetic moment is aligned with H. Therefore, with the increase of H, Tm³⁺ magnetic moment is more difficult to flip. When H becomes sufficiently large to maintain the Tm³⁺ magnetic moment aligned with H, SSW-II disappears. Therefore, in the FCC mode, with the increase of H, Tm³⁺ will flip at a lower temperature, leading to the decrease of T_SII-2, which is regulated by the magnetic field. Due to the reduction of Tm³⁺ content, the suppressed H of SSW-II decreases with the increase of x. In the ZFC mode [Fig.5(d)], with the increase of temperature, magnetization along the c-axis of Tm_0.5Pr_0.5FeO₃ single crystal fluctuates upward, resulting in a weak SSW-II. The continuous flipping of Tm³⁺ causes the coupling mode changes from ↓↑↑ to ↑↑↑. With the increase of H, the trigger temperature of SSW-II gradually decreases until 70 Oe. SSW-II along the c-axis at x = 0.15 and 0.25 is similar to x = 0.5 in ZFC mode (as shown in Fig. S2).

In order to further understand the anisotropic magnetic behavior, the M−H curves are measured at different temperatures (as shown in Fig. S3). The transition between the weak ferromagnetic state and the antiferromagnetic state indicates that Tm_0.5Pr_0.5FeO₃ single crystal is undergoing SRT when T_SRT = 72 K−92 K, which is correspond to Fig.2(d). As previously discussed, RFeO₃ is not a strictly antiferromagnetic material. Due to the tilt induced by the distortion of the FeO₆ octahedral structure, the incline of Fe³⁺ magnetic moment causes the WFM in RFeO₃. Weak ferromagnetism along the a-axis of Tm_0.5Pr_0.5FeO₃ single crystal is observed in the antiferromagnetic state, according to Eq. (6) [35],

(6){\sigma }_s=M-{\chi }_{\mathrm{A}\mathrm{F}\mathrm{M}}H.

{\sigma }_s is the weak ferromagnetic saturation magnetization, and {\chi }_{\mathrm{A}\mathrm{F}\mathrm{M}} is the antiferromagnetic contribution. At a higher temperature (95 K), the curve shows a tilted straight line (Fig. S3), presenting obvious antiferromagnetic behavior. As shown in Fig.6(e), after deducting {\chi }_{\mathrm{A}\mathrm{F}\mathrm{M}}, the curve exhibits weak ferromagnetic behavior. At low temperature, after deducting {\chi }_{\mathrm{A}\mathrm{F}\mathrm{M}}, the magnetization remains essentially unchanged, indicating that the sample has no antiferromagnetic component at 70 K. With the increase of temperature, {\sigma }_s gradually decreases [as shown in Fig.6(f)], indicating the decrease of WFM, and the AFM along the a-axis gradually domains, proving the occurrence of SRT. In addition, as the temperature increases, the coercive field (H_C) gradually increases from 20 Oe to 100 Oe [as shown in Fig.6(f)], which is attributed to the hindered rotation of Fe³⁺ spin with the increase of temperature. The field-induced SSW is also observed along the a-axis in Tm_0.5Pr_0.5FeO₃ single crystal. For example, at 70 K and 110 Oe [as shown in Fig.6(a)], the magnetization changed from −0.94 emu/g to 0.97 emu/g, showing a distinct jump.

4 Conclusion

Tm_1−xPr_xFeO₃ (x = 0, 0.15, 0.25, 0.5) single crystals are successfully grown by optical float zone method and their magnetic properties are measured. With the increase of x, T_SRT shifts to lower temperature. SSW-I was absent in TmFeO₃ and PrFeO₃ single crystals. SSW-I and SSW-II is observed along the a-axis when x = 0.15 and 0.25, but SSW-I disappears at x = 0.5. The variations of magnetic interactions in x = 0.15 and 0.5 single crystals are analyzed through structural distortion and transforming Curi−Weiss fitting, further discussing the reasons for the generation of SSW-I. A rare phenomenon of SSW-II existing simultaneously during cooling and warming process along the c-axis of Tm_1−xPr_xFeO₃ (x = 0.15, 0.25, 0.5) single crystals. Both SSW-I and SSW-II can be regulated by H. Although Tm_0.5Pr_0.5FeO₃ single crystal exhibits antiferromagnetic behavior at higher temperatures along the a-axis, there is still a WFM. With the increase of temperature, the weak ferromagnetic part gradually decreases, proving the occurrence of SRT. Meanwhile, field-induced SSW is observed along the a-axis in Tm_0.5Pr_0.5FeO₃ single crystal. The complex and rich magnetic phenomena observed in Tm_1−xPr_xFeO₃ (x = 0.15, 0.25, 0.5) single crystals hold significant potential for advancing research in altermagnetism and magneto-optical coupling, making them promising candidates for the development of next-generation spintronic devices.

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