Ultralow temperature terahertz magnetic thermodynamics of perovskite-like SmFeO3 ceramic

The terahertz magnetic properties of perovskite-like SmFeO3 ceramic are investigated over a broad temperature range, especially at ultralow temperatures, using terahertz time-domain spectroscopy. It is shown that both resonant frequencies of quasi-ferromagnetic and quasi-antiferromagnetic modes have blue shifts with the decreasing temperature due to the enhancement of effective magnetic field. The temperature-dependent magnetic anisotropy constants are further estimated using the resonant frequencies, under the approximation of omitting the contribution of Sm3+ magnetic moments to the effective field. Specially, the effective anisotropy constants in the ca and cb planes at 3 K are 6.63 × 105 erg/g and 8.48 × 105 erg/g, respectively. This thoroughly reveals the terahertz magnetic thermodynamics of orthoferrites and will be beneficial to the application in terahertz magnetism.

Rare earth orthoferrites with distorted perovskite structure have received lots of attention in recent years [1][2][3] . This series of compounds have been found to possess G-type antiferromagnetic ordering formed by Fe 3+ ions spins and the precession frequency of magnetic moments can extend to terahertz regime due to the strong internal magnetic field 4,5 . Besides, the canted spins also induce weak macroscopic magnetization and ferroelectricity in some members 6 . Therefore, ReFeO 3 -type oxides exhibit abundant physical properties such as the terahertz magnetic response, the muitiferroic and magneto-optical effect 7-12 . There are usually three competitive exchange interactions in orthoferrites induced by Fe-Fe, Re-Fe, and Re-Re, respectively. The Fe-Fe interaction determines the formation of antiferromagnetic ordering in the high temperature region, while the Re-Fe exchange effect will lead to magnetic anisotropy and further induce the spin reorientation (SR) 13 . However, the Re-Re interaction will be activated at very low temperature, which contributes to the long range magnetic ordering of rare earth ions. For example, SmFeO 3 exhibits antiferromagnetic ordering along a axis with a net spontaneous magnetization along c axis below 670 K (Neel temperature). Then, the magnetic moments continuously rotate from a axis to c axis during 450 ~ 480 K due to the Sm-Fe interaction. The formation of long range magnetic ordering in Sm sublattice plays an important role in determining the macroscopic magnetic properties at very low temperature. Magnetization reversal was observed in SmFeO 3 below 5 K under a magnetic field about 300 ~ 500 Oe, which can be ascribed to the antiparallel ferromagnetic moments of Fe sublattice and Sm sublattice 6,14 . This interesting phenomenon may have potential applications in the magnetic switch under a weak applied field.
Despite the terahertz antiferromagnetic resonances and potential physical phenomena, SmFeO 3 has not been investigated in the terahertz regime. In this work, we fabricate the SmFeO 3 ceramic samples and characterize their terahertz magnetic properties in a wide temperature range. We will discuss the magnetic thermodynamics of the SmFeO 3 ceramic in details, including the temperature-dependent ferromagnetic and antiferromagnetic resonant frequencies of Fe sublattice, as well as the contribution of Sm spins to macroscopic magnetization and magnetic resonance at the ultralow temperatures. Figure 1 shows the terahertz transmission frequency-domain spectra (normalized to the reference spectrum) of the SmFeO 3 ceramic between 3 K and 292 K. Only partial curves are presented to keep the tendency observable. Below 200 K, two dips are observed on the transmission curves, which can be ascribed to the so-called quasi-ferromagnetic mode (F mode) and quasi-antiferromagnetic mode (AF mode) of SmFeO 3 , respectively 15 . The resonant frequencies of F mode and AF mode are 0.34 THz and 0.62 THz at 200 K, respectively. As the temperature decreases, both the resonant frequencies of two modes exhibit blue shift. At 40 K, the respective frequencies are 0.55 THz and 0.70 THz. Below 40 K, the effect of temperature on the resonant frequencies becomes much more significant. When temperature lowers to 10 K, the frequencies of two modes increase to 0.67 THz and 0.80 THz, respectively. At 3 K, F mode and AF mode further harden, whose frequencies are 0.84 THz and 0.95 THz, respectively. It is worth noting that the resonant strength weakens at high temperatures. Specially, the dip attributed to F mode cannot be resolved from the background above 200 K, while AF mode also gets very weak at room temperature (RT), with a frequency of 0.57 THz.

Results and Discussion
The resonant frequencies of F mode and AF mode at various temperatures are extracted from the frequency-domain spectra and presented in Fig. 2. As mentioned above, the resonant frequencies for both modes undergo a sharp decrease over the range of 3 ~ 40 K, while above 40 K, the frequency-temperature curves slope gently downward, especially for the AF mode. Besides, the F mode data between 200 K and RT are not shown since it almost disappears in this temperature interval.  Next, let us consider the physical origin of the resonant modes and the corresponding magnetic thermodynamics. The crystal structure of SmFeO 3 is shown in Fig. 3. As can be seen, Fe 3+ ions occupy the (0 0 0.5) sites, of which, there are eight edge sites and four face center sites, according to the symmetry of Pbnm space group. Besides, the eight nearest Fe 3+ ions constitute a cubic and the spin orientations for adjacent ions are opposite, that is, G-type antiferromagnetic ordering is formed 13 . In fact, the spins of adjacent Fe 3+ ions are not strictly antiparallel. Specifically, just below the Neel temperature (T N = 670 K), the canted spin mainly orient along a axis and also have a small component along c axis. Therefore, the magnetic structure can be denoted as , where G is the antiferromagnetic vector and F is the ferromagnetic vector. Like most other rare earth orthoferrites, SmFeO 3 undergoes a spin reorientation transition due to the interaction between rare earth ions and Fe 3+ ions. However, difference is that the transition temperature of SmFeO 3 is the highest in the family of rare earth orthoferrites and much higher than RT. At about 480 K, the Γ 4 phase changes to Γ ( , ) G F z x 2 through a mesophase Γ 42 14 . Thus, as seen in Fig. 3, the Fe 3+ spins orient along c axis with a weak macroscopic magnetization along a axis below the transition temperature. The canted spins induce the weak macroscopic magnetism in SmFeO 3 and the magnitude of magnetization depends on the ferromagnetic component of magnetic moments, while the terahertz magnetic resonances caused by the spin precession under an internal magnetic field relate to the F and G vectors.
According to some previous studies, the magnetic moments of Sm 3+ ions play an important role on the magnetic properties of SmFeO 3 at low temperature. Owing to the relative strong Re-Fe exchange interaction, SmFeO 3 possess a high SR transition temperature, while the Re-Re interaction leads to a high magnetic ordering temperature for Sm 3+ ions. At about 140 K, the Sm 3+ ions spins are activated in the ab plane. As seen in Figs 3 and 4(a), Sm 3+ ions exhibit the ( ) , F C x y symmetry, that is, the spins satisfy the following equations: S 1x = S 2x = S 3x = S 4x and S 1y = S 2y = −S 3y = −S 4y . Thus, Sm 3+ ions possess C-type antiferromagnetic ordering along b axis and also a ferromagnetic moment along a axis. Moreover, macroscopic magnetization orients along the −a direction, antiparallel with the one of Fe 3+ ions 16 . During the cooling process, the remarkably increased net magnetic moment of Sm 3+ ions will cancel with the opposite contribution from Fe 3+ ions, which leads to a zero macroscopic magnetization at the temperature called compensation point (about 5 K). Below this temperature, magnetic reversal is observed when magnetic field is applied parallel to a axis in the SmFeO 3 crystal 6 . To verify this phenomenon in the ceramic sample, we test the − M T curve under an applied magnetic field of 1000 Oe. As shown in Fig. 4(b), the magnetization increases first during the cooling process due to the increased ferromagnetic component of Fe 3+ spins, then begins to decrease gradually because of the activation of Sm 3+ spins at about 170 K (different from the crystal sample), companied by a sharp decline below 40 K. Nevertheless, magnetic reversal does not appear in the SmFeO 3 ceramic even when the temperature is lowered to 2 K. The possible reason is as follows. Crystal sample has a long range ordering and the macroscopic magnetization is measured when applied field is parallel to a axis. However, for the SmFeO 3 ceramic, magnetic ordering is formed in a single crystal grain and the orientation of crystal grains is random, and therefore, the measured magnetization is an average value of various orientations between the magnetic field and crystal axis. Now, let us further consider the temperature dependent magnetic resonant frequencies based on the foregoing discussions about the magnetic structure in SmFeO 3 . In antiferromagnetic materials, the resonant frequency can be described by 4,17 Figure 3. The crystal structure, atom arrangements, and spin orientations of SmFeO 3 crystal below 140 K. The eight nearest Fe 3+ ions constitute a cubic, whose spins orient along c axis with a weak macroscopic magnetization along a axis. By contrast, the spins of Sm 3+ activated below 140 K locate in the ab plane and increase during cooling process.
where H ca eff and H cb eff , K ca eff and K cb eff are the effective second-order anisotropy fields and anisotropy constants in the ca and cb planes respectively, and M 0 is the saturation magnetic moment. Moreover, the exchange field H E is proportional to the magnetic moment ( ) where λ is the molecular field coefficient 4 . Since the temperature region considered in this work is much lower than the Neel temperature and no SR transition occurs, the exchange field can be regarded as nearly temperature independent 20 . However, the anisotropy field changes influenced by the Fe-Fe exchange, magnetic dipole interaction, and crystal field will change with temperature 21 . According to Eqs (2) and (3), it can be found that the square of resonant frequency is proportional to the anisotropy constant. As a consequence, we can obtain the temperature dependent anisotropy constants using the frequency data.
It is worth noting that the above discussion have not taken account of the contribution of Sm 3+ magnetic moments to the effective field. This approximation is valid, especially for the temperature region above 40 K, as the magnetic moment of Sm 3+ ion is much weaker than that of Fe 3+ ion. The ground state levels for Sm 3+ ([Xe]4f 5 ) and Fe 3+ ([Ar]3d 5 ) in an octahedral crystal field are 6 H 5/2 and 6 S 5/2 , respectively. According to the Hund's rules 22 , we may conclude that the saturation magnetic moment of Fe 3+ (5.92 µ B , close to 5 µ B in the orthoferrites system 18 , where µ B is Bohr magneton.) is large enough compared to the one of Sm 3+ (0.85 µ B , actually less than this value even at 5 K 6 ), and that the effective field is mainly contributed by the magnetic moments of Fe 3+ ions.
We fit the ν − T curve using the nonlinear curve fitting method. The fitting results are also presented in Fig. 2 together with the experimental data for comparison. The equations used for fitting the frequencies of F mode and AF mode can be expressed by Eq. (4) and Eq. (5), respectively. As shown in Fig. 2, the fitting curves agree well with the experimental points for both F mode and AF mode, thus, the proposed equation is applicable in the temperature range from 3 K to RT. However, the fitting curve can be divided into three intervals due to the different tendencies. Between 40 K and RT, the resonant frequencies and temperature satisfy the linear relationship; the items T 1 and T 1 2 can be omitted since they are small enough compared to the linear item. The blue shift of resonant frequencies can be attributed to the increase of anisotropy constants, and hence the enhancement of effective magnetic field. It is noted that the F mode hardens fast than AF mode, which implies that growth rate of K ca eff is larger than that of K cb eff . The second region is during 5 ~ 40 K. Since both the T 1 item and the linear item work in this interval, the resonant frequencies remarkably increase with decreasing temperature. The addition of T 1 item implies that the effective anisotropy constants increase more quickly during cooling, compared to the first process. Then, below 5 K, the linear item can be deleted. However, the T 1 item is not enough to depict the rapidly increased frequencies, so we introduce the T 1 2 item, with which we get a good fitting (see Fig. 2). Furthermore, the effective anisotropy constants K ca eff and K cb eff are calculated according to Eqs (2) and (3), using the resonant frequency data. The amplitudes of the anisotropy constants are normalized to the one at 3 K, and both the experimental and fitting values have been obtained and presented in Fig. 5. According to some previous studies, the exchange field in rare earth orthoferrites is about 6.4 × 10 6 Oe 19,20 , and M 0 is calculated as 109.85 emu/g, and hence, the effective anisotropy constants K ca eff and K cb eff at 3 K can be estimated as 6.63 × 10 5 erg/g and 8.48 × 10 5 erg/g, respectively. Thus, we have obtained the temperature dependent anisotropy constants which essentially determine the magnetic resonant frequency of orthoferrites.

Conclusions
In summary, the terahertz magnetic thermodynamics of the SmFeO 3 ceramic have been investigated over a wide temperature region from 3 K to 292 K. The macroscopic magnetization is measured and the magnetic reversal does not occur even at 2 K for the ceramic sample. Additionally, both the F mode and AF mode of the SmFeO 3 orthoferrite harden during the cooling process, which can be attributed to the increase of anisotropy constants, and hence the enhancement of the effective magnetic field. The resonant frequencies of both two modes can be well fitted with a nonlinear equation of temperature, which clearly describes the temperature dependence of the resonant frequencies in different temperature region. With the frequency values, we also estimate the anisotropy constants at various temperatures.  . The schematic diagram of the terahertz time-domain measurement system. Terahertz pulse is excited by a 780 nm near-infrared femtosecond laser in the emitter component, and first passes through the quartz window of the liquid helium cryostat, then interacts with the sample, followed by the other quartz window, and lastly arrives at the detector component. The cooling system with liquid Helium circulation can realize the precise temperature control between 3 K and RT.