1. Introduction

Terahertz (THz) technology has become central in work involving far-infrared light because of its many uses in fields such as biology [1], medicine [2] and THz imaging [3]. Since its first reported application in measuring water vapor in 1989 [4] and semiconductors and dielectrics in 1990 [5], THz spectroscopy has attracted significant interest for use in studying molecular resonances [6]; as well as in probing into rotational reorientations in the rare-earth orthoferrites [7, 8], which are generally continuous phase transitions Γ₄→Γ₂₄→Γ₂ [9, 10]. Using THz pulses, A. V. Kimel et al [8] achieved ultrafast control of spin reorientation, then later reported the optical excitation of ferromagnetic resonance and antiferromagnetic resonance in TmFeO₃ [11].

Recently, THz time domain spectroscopy has been used to investigate the spin orientation during the spin reorientation phase transition under zero or weak external magnetic field [12]. However, for THz spectroscopy to find broader applications in work on magnetic resonance, a few issues must be addressed. First, THz spectroscopic observation of the continuous and discontinuous spin reorientation transitions through shifts in resonant frequencies must be verified; Secondly, we need to do THz spectroscopic research on the thermodynamics of the continuous and discontinuous spin reorientations in the rare-earth orthoferrites. Several recent attempts have been done in the THz spectroscopic applications. R. V. Mikhaylovskiy et al introduced THz emission spectroscopy to obtain ultrafast laser-induced spin dynamics in TmFeO₃ and ErFeO₃ [13]. J. Jiang et al used THz time domain spectroscopy to study the temperature-induced spin reorientation transition in NdFeO₃ [14]. Beyond these attempts, THz time domain spectrum has attracted much attention [15–18]. The distinctive feature of the THz time domain spectrum is its obvious time delay which contains novel information about the magnetic materials. Correspondingly, resonant absorption peaks can be found in the frequency domain spectra (fast Fourier transform (FFT) of the THz time domain spectra). K. Yamaguchi et al [7] observed the spin reorientation transition using the THz time domain spectra. Temperature dependence of antiferromagnetic and ferromagnetic resonant frequencies results from the changes in the anisotropy energy. However, his theory would be more direct if the experimental results are more quantitative. TmFeO₃ and NdFeO₃ orthoferrites are typical magnetic materials which show continuous spin reorientation transitions. The temperature in R. V. Mikhaylovskiy’s work seems to be not low enough to detect discontinuous spin reorientation transitions in ErFeO₃ orthoferrite [13]. A. M. Balbashov et al believed that HoFeO₃ orthoferrite is the exception to Γ₄→Γ₂₄→Γ₂ sequence [19]. The first-order phase transitions Γ₂₄→Γ₁₂→Γ₂ have been observed experimentally [20] and analyzed theoretically in HoFeO₃ orthoferrite [19, 21, 22]. T₁, T₂ and T₃ (T₃<T₂<T₁) correspond to the characteristic temperature points of magnetic phase transitions Γ₄ → Γ₂₄, Γ₂₄ → Γ₁₂, and Γ₁₂ → Γ₂. Using HoFeO₃ orthoferrite, we have an intriguing insight on both continuous and discontinuous transitions through the shifts in THz frequencies.

Here, we introduce THz time domain spectroscopy to observe both continuous and discontinuous magnetic phase transitions using THz ferromagnetic and antiferromagnetic resonances. The thermodynamics of the continuous and discontinuous spin reorientation transitions is given in anisotropy energy forms. We fit the antiferromagnetic resonant frequencies to gain quantitative insight on the changes in anisotropy energy. The measured M-H (moment–magnetic field) curves further demonstrate the thermodynamics of the spin reorientation transitions.

2. Experimental Methods

To prepare HoFeO₃ ceramics, we used a conventional solid state reaction of dried Ho₂O₃ (99.9%) and Fe₂O₃ (99.9%) powders. A stoichiometric composition of HoFeO₃ (Ho₂O₃: Fe₂O₃ = 1:1) was ball-milled for 24 h and dried at 70 °C for more than 12 h. After that, the mixed powders were calcined at 1100 °C for 3 h. These mixed powders were reground, pressed into wafers under a pressure of 4 MPa, and sintered into pellets at 1400 °C for 3 h.

The phase structures of the HoFeO₃ ceramics were analyzed using X-ray diffraction (XRD; Rigaku D/max 2500, Japan, Cu K_α radiation) at room temperature. The diffraction angle ranged from 20° to 80° at a scanning speed of 6°/min.

A THz-TDS system with a temperature control accessory was introduced to measure THz time domain spectra with a time resolution of 0.01 ps. The collected temperature ranged from 270 K to 2.8 K with temperature steps of 0.5–10 K. The temperature-dependent spin reorientation transitions were analyzed by performing a fast Fourier transform of the THz time domain spectra.

3. Results and discussions

3.1 Material characterization

Figure 1(a) shows an indexed XRD pattern of the HoFeO₃ ceramics. Figure 1(b) shows the canted perovskite lattice structure of the HoFeO₃ ceramics, which contains four iron ions and four rare-earth ions per crystallographic unit cell. Each iron ion is situated at the center of the octahedron and there are six oxygen ions at the corners of the octahedron.

 

Fig. 1 Crystal structure of HoFeO₃ ceramics. Figure 1. (a). Indexed XRD pattern of HoFeO₃ ceramics. (b) Locations of various ions.

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3.2 Terahertz measurements

We present the temporal THz waveforms and their FFTs at several temperature points in Fig. 2. Figure 2(a) shows parts of the collected temporal THz waveforms. The maximum peak reflects THz pulses transmitting through the HoFeO₃ ceramics, followed by some periodic oscillations caused by the induced magnetic dipole transitions. The reference signal is collected without the sample. Zeeman splitting can be induced by the internal magnetic field in a fine latticed structure which contains the excited level m_J’ and the ground level m_J. A magnetic dipole transition is excited when |m_J –m_J’| = ± 1 [23]. Most of the spins are in the ground states and spins can be pumped into excited levels. The spins then return to their original states, producing single-frequency radiation. The transitions between the levels can be detected by the THz pulses as the temperature is lowered. Thus, the THz waveforms contain new magnetic information about the HoFeO₃ ceramics, showing an obvious time delay in the time domain spectra compared with the reference spectrum. Our previous work has detailed the process [23].

 

Fig. 2 THz spectra at several temperature points. Figure 2. (a). The temporal waveforms from 270 K to 2.8 K. Reference (Ref.) spectrum is collected without the sample. There is noticeable time delay and signal attenuation in the temporal spectra. (b). The corresponding THz frequency domain spectra (FFT of the THz time domain spectra). The shifts in the resonant frequencies originate from the new magnetic information.

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Figure 2(b) shows the corresponding frequency domain spectra. The noticeable dips in the frequency domain spectra originate from the new magnetic information in the HoFeO₃ ceramics. The absorption peak at 0.57 THz corresponds to antiferromagnetic resonance at 270 K. Figure 3 summarizes the locations of the ferromagnetic and antiferromagnetic resonances at all temperatures in the frequency domain spectra, with T₁≈70 ( ± 5) K, T₂≈50 ( ± 2.5) K, and T₃≈37.5 ( ± 2.5) K.

 

Fig. 3 The locations of ferromagnetic resonances (FMr) and antiferromagnetic resonances (AFM) in the frequency domain spectra at all temperatures, with characterized temperature points T₁, T₂, and T₃.

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3.3 Anisotropy energy mechanism

To explain the variations of the ferromagnetic and antiferromagnetic resonant frequencies in Fig. 3, we introduce an anisotropy energy mechanism. The ferromagnetic resonant frequency and antiferromagnetic resonant frequency forms of the Γ₄ and Γ₂ magnetic phases are given as ${\upsilon }_{\text{FM}}$ and ${\upsilon }_{\text{AFM}}$ in Eq. (1) and Eq. (2), where gyromagnetic ratio $\gamma$, exchange field $H_{\text{E}}$, saturated magnetic moment $M_0$, and anisotropic energy constants $K_{\text{ac}}$ and $K_{\text{ab}}$are introduced [24].

Figure 4 shows the entire process description of the spin reorientations and magnetic phase transitions. Γ₄ → Γ₂₄, Γ₂₄ → Γ₁₂, and Γ₁₂ → Γ₂ are shown in different curves. The antiferromagnetic moment G deviated from a at T<T₁ and reoriented in the (a, c) plane because of the softened K_ab. However, Γ₁₂ was more favored than Γ₂₄ at T = T₂. Then, G jumped into the (b, c) plane, though there was still 10–20°between the jumping point and the c axis [21]. Finally, G began to approach c at T = T₃, which was driven by the hardening K_cb(T).

 

Fig. 4 Temperature-driven spin reorientations and magnetic phase transitions in HoFeO₃. The blue solid curve, blue dotted curve, and red dotted curve are Γ₄ → Γ₂₄, Γ₂₄ → Γ₁₂, and Γ₁₂ → Γ₂, respectively.

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3.4 Discussion on the thermodynamics

To gain more quantitative insight, we fit the $\upsilon -\frac{1}{T}$ and $\upsilon -\ln (\frac{1}{T})$ curves with a piecewise linear fit for the Γ₄ and Γ₂, respectively. Figure 5 gives these fitting results and experimental data for comparison. The fitting equations for the antiferromagnetic resonant frequencies of Γ₄ and Γ₂ are Eq. (3) and Eq. (4), respectively:

 

Fig. 5 The fitted antiferromagnetic resonant frequencies and experimental results at (a) 211–70 K and (b) 37.5–2.8 K. The triangles and solid lines are the experimental data and the fitting results, respectively.

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Fig. 6 The macroscopic magnetization of a HoFeO₃ ceramic at (a) T = 300 K and (b) T = 5 K in a varying external magnetic field.

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4. Conclusion

We introduced a THz time domain spectroscopy to study continuous and discontinuous spin reorientation transitions by using ferromagnetic and antiferromagnetic resonances. New magnetic information of HoFeO₃ ceramics is reflected by the obvious time delay in the time domain spectra and the absorption peak in the frequency domain spectra. The changes in anisotropy energy result in the continuous and discontinuous spin reorientation transitions in the HoFeO₃ ceramics. The fitting antiferromagnetic resonant frequencies help us to achieve quantitative insight on the changes in the anisotropy energy. M-H curves further verify the thermodynamics of the spin reorientation transitions. We believe that THz time domain spectroscopy holds a promise for broader applications in studying molecular resonances, such as spin reorientation transitions.