Dzyaloshinsky–Moriya Interaction Induced Anomalous g Behavior of Sr₂IrO₄ Probed by Electron Spin Resonance

Abstract

Among the 5d transition metal iridates, Sr₂IrO₄, which has a layered chalcogenide structure, has received much attention due to its strong spin–orbit coupling (SOC), which produces Mott insulating states and anomalous physical behaviors. In this paper, the microscopic magnetism of Sr₂IrO₄ is studied with electron spin resonance (ESR) measurements. The Lande factor g of the ferromagnetic resonance signal of Sr₂IrO₄ shows anomalous behavior compared to typical ferromagnets. It gradually decreases, and the corresponding resonance field H_r increases, with decreasing temperature. The various physical parameters. including the saturated magnetic field H_s derived from M-H, H_r, ΔH_pp, the g factor and the intensity I extracted from ESR spectra, are analyzed in detail. Eventually, it is revealed that the anomalous behavior of the g-factor is induced by in-plane Dzyaloshinsky–Moriya interaction (DMI) rather than the SOC effect.

1. Introduction

Spin–orbit coupling (SOC) and the Dzyaloshinsky–Moriya interaction (DMI) have gained significant attention in the field of condensed matter physics due to their roles in creating chiral spin textures and exotic magnetic phases [1,2]. It is well known that SOC plays a critical role in determining a material’s band structure and electronic properties, such as topological insulators, Dirac semimetals, Weyl semimetals with band inversion and novel Mott insulators for 5d transition-metal oxides (TMOs) [3,4]. Orbital characters with different strengths and Bloch states generate different Mott insulating states in 3d, 4d and 5d TMOs. The Dzyaloshinsky–Moriya interaction (DMI) and its accompanied weak ferromagnetism (WFM) are generally believed to be very sensitive to crystal symmetry and antisymmetric spin coupling [5,6,7,8,9,10,11]. The manipulations needed to enhance DMI strength or understand the underlying mechanism has been a hot topic during recent years in the field of spintronics [6]. The study of DMI and external stimuli, such as temperature (T) or magnetic field (H), can produce valuable insights into the fundamental physics underlying peculiar phenomenon in a single-phase compound, such as α-Fe₂O₃, CuCl₂•2H₂O, Sr₂IrO_4, etc. [7,8,9,10,11].

The studies on anomalous transport and its physics in 5d transition-metal oxide (TMO) iridates Sr_{n + 1}Ir_nO_{3n + 1} (n = 1, 2) have attracted significant attention due to their intriguing insulating nature, possessing a novel Mott ground state with J_eff = 1/2 quantum spin rather than the expected metallic state with S = 1/2 [12,13]. So far, it is known that a strong SOC energy ζ_SO~0.4 eV is responsible for the peculiar electronic and magnetic properties in Sr₂IrO₄ [14]. It generates an effective total angular momentum J_eff = 1/2 state with an isotropic mixture of xy, yz and zx orbitals and hybrid up-down spin characters [15,16]. As a result, J_eff = ½ forms a narrow Hubbard band due to reduced hopping factors, such that a small U opens a Mott gap and an unusual Mott insulator is realized [16]. Much work has been performed on understanding the novel electronic structure and unusual transport in Sr₂IrO₄, such as ARPES, optical conductivity, resonant X-ray scattering (RXS) techniques and first-principles band calculations [12,13,14,15,16,17,18].

However, less work has been focused on studying its large antisymmetric DMI effect on magnetism properties, such as the in-plane WFM arising from the antiphase rotations of the IrO₆ octahedra. A zigzag internal magnetic field is generated by the superexchange interaction of the Ir-O-Ir-O-Ir chain [18]. As shown in Figure 1a,b, DMI always causes neighboring magnetic moments to tend to be perpendicular to each other, rather than parallel or antiparallel [19]. Instead of just being parallel or antiparallel to each other, under the competing effects of AFM and FM, the magnetic moments will tend to form an angle that minimizes the energy of the system [20]. By introducing the DMI action constant D (the direction of D in Sr₂IrO₄ is parallel to the c-axis), the representation of the Hamiltonian quantity is of the form

H_DMI ~ −D₁₂·(S₁ × S₂),

(1)

which allows the two spins S₁ and S₂ of Ir⁴⁺ to be aligned orthogonally, resulting in a lower energy state [21]. Previous studies have shown that the magnitude of the DMI obtained by DFT calculations in Sr₂IrO₄ is 0.25 meV along the c-axis [21]. The DMI increases with the rotation angle (θ_Oct) of the IrO₆ octahedron around the c-axis and reaches a maximum at θ_Oct ≈ 11° (Figure 1c) [18]. Furthermore, it is reported that the appearance of DMI stems from crystal structural distortion and leads to WFM at T_c in Sr₂IrO₄ [12]. The above results suggest that the large DMI should produce exotic magnetism, rather than SOC, in iridate Sr₂IrO₄.

To clearly address the DMI’s effect on the magnetic properties of Sr₂IrO₄, in this article, an ESR technique is used as a measurement probe. ESR is a powerful tool for studying dynamical microscopic magnetism with unpaired electrons, especially in paramagnetic states [22]. Our results show that the g value extracted from ESR spectra is proportional to the DMI. The other derivative magnetic parameters from ESR are also dependent on DMI someway. As a consequence, the DMI of Sr₂IrO₄ is demonstrated as having a crucial role in the variation of magnetism parameters with T.

2. Experimental

Single crystals of Sr₂IrO₄ were grown with a flux method by using SrCl₂ flux [23]. A mixture of SrCO₃ (Innochem (Beijing, China), 99.9%), IrO₂ (Western Element (San Jose, CA, USA), 99.99%) and SrCl₂ (Aladdin (Hangzhou, China), 99.99%) with a molar ratio of 1.8:1.0:15 was melted at 1300 °C. The melt was slowly cooled down to 900 °C at a rate of 8 °C per hour and then furnace-cooled to room temperature. Plate-like single crystals with typical dimensions of 1 mm × 1 mm × 0.1 mm were found in the solidified melt. The results of X-ray measurement by using a Rigaku-RINT 2400 diffractometer at room temperature were fully consistent with the space group I4₁/acd previously reported for Sr₂IrO₄ [12]. The ESR experiments were carried out with a JEOL JES-FA200 ESR spectrometer at X-band frequencies γ ≈ 9.1 GHz in temperatures ranging from 5 to 290 K. Microwave power was set to 1 mW and the magnetic field was parallel to the natural cleavage crystal plane (001). Since the in-plane anisotropy of the J_eff = 1/2 moments is extremely small [12], there is no significant difference in the in-plane magnetism along the a- or b-axis. The in-plane and out-of-plane magnetism were measured by using a commercial superconducting quantum interference device magnetometer (MPMS-XL, Quantum Design, San Diego, CA, USA).

3. Results and Discussions

The compound Sr₂IrO₄ crystallizes in the layered perovskite structure (space group: I4₁/acd), consisting of four IrO₂ staggered layers (in a unit cell) of corner-sharing IrO₆ octahedra with rotational distortion interleaved by Sr-O interlayers [24], as illustrated in Figure 2a. A nonlinear antiferromagnetic spin structure with WFM from a spin canting in-plane ground state has been proposed (Figure 2b). A metamagnetic transition from in-plane interlayer AFM to a WFM is revealed at low temperatures and critical magnetic fields above 2 kOe, as shown in Figure 2c.

Figure 3a presents the temperature (T) dependence of in-plane magnetization (M) at 2–360 K under a magnetic field of 0.1 kOe. The magnetization discrepancy between ZFC and FC curves below T_c suggests the existence of magnetic glassy-like behavior [25]. According to the mean-field model of magnetic glasses, this implies that there exists a competition of magnetic interactions between AFM and FM [26]. The Curie temperature is 226 K, defined by the derivative maximum of |dM/dT|, as shown in Figure 3b. The effective magnetic moment P_Meff is around 0.64 μ_B/f.u., determined by using the Curie–Weiss law with the expression ${\chi }^{-1}=(T-\theta )/C$ in the temperature regime from 240 to 255 K (Figure 3b), where θ denotes the Weiss constant and C is the Curie constant, which is consistent with the reported effective moment [27,28]. The Curie–Weiss temperature is around 240 K. Instead of a collinear type AFM, this DM type antisymmetric exchange interaction is originated from the rotation of octahedra IrO₆ about the c-axis by ~11°, which leads to a transition to a WFM phase at T_c, as previously reported [27]. With decreasing temperature, the value of magnetization is gradually reduced. This implies the DMI (WFM strength) is simultaneously diminished, reflecting the change of structure with decreasing temperature [27]. As T is lowered, the basal bond angle <Ir–O–Ir> increases, the rotation angle (θ_Oct) of IrO₆ octahedra around the c axis decreases to ~10.3° at 20 K and the basal bond length d_Ir−O2 decreases, while the apical bond length d_Ir−O1 increases [27]. Upon decreasing T to around 100 K, the magnetic-structural coupling occurs as reported by Imtiaz Noor Bhatti [27]. However, there is no trace on this phenomenon in any macromagnetization measurements, such as M-T or M-H, due to the limited detection sensitivity of the magnetic flux method. With the temperature below 11 K, the magnetization rises with lowering T, which might be related to magnetic fluctuation at low temperatures [29].

Figure 3c illustrates the temperature dependence of magnetization, measured in magnetic fields of 1 kOe, 2 kOe and 10 kOe parallel to [010], respectively. It can be seen that the higher H of 2 kOe, corresponding to the critical field-driven in-plane metamagnetic transition, induces a WFM transition and shifts the subsequent AFM transition to a lower temperature of around 150 K. Furthermore, the magnetic field of 10 kOe shifts the Curie temperature to 236 K from 226 K (at H = 0.1 kOe). It realizes the WFM transition and remains in this magnetic state until 2 K. These results indicate that the maximum strength of DMI or WFM is located at T_c.

Figure 4a presents the magnetization curves with in-plane H at temperatures of 2, 10, 50, 100, 150 and 200 K, and out-of-plane H at 2, 10, 50 and 100 K, respectively. An in-plane saturated moment of 0.09 µB/Ir at 2 K is obtained by unit conversion in the WFM state, which is close to the previous report [12]. It is understandable that the in-plane saturated magnetization decreases with temperature due to phonon thermal disturbance [30]. The out-of-plane magnetization is quite negligible and irregular since the c-axis is the hard magnetization direction. However, it is peculiar that the in-plane saturated magnetic field H_s decreases with the increasing temperature, as shown in Figure 4b. This indicates that the DMI strength is enhanced by the temperature increasing, which is consistent with the M-T measurement results (Figure 3a). Furthermore, from Figure 4b, we can see that the change trend of H_s with temperature is similar to the behavior of resonance magnetic field H_r extracted from the ESR measurements, implying there is a close correlation between H_s and H_r. Here, H_r is defined by the magnetic field, corresponding to the center between the lowest and highest values of dP/dH.

Figure 5 shows the ESR spectra for Sr₂IrO₄ in the temperature range from 11 to 230 K with the measurement geometry of the H//ab plane. There is no ESR signal detected for H//c-axis since there is a large magnetic anisotropy mediated by DMI in Sr₂IrO₄ [13]. Above 230 K (T_c = 226 K), there is no ESR signal observed for the H//ab plane, which should be ascribed to the strong spin–lattice relaxation in the paramagnetic region [12].

For all the ESR spectra, the calculated asymmetry ratio A/B of the absorption derivative is less than 2.6. This is indicative of the insulating character of Sr₂IrO₄ [31,32]. At high temperatures, each spectrum consists of a single line with a Dysonian line shape, indicating the microwave skin depth is smaller than the single crystalline size [31,32], which holds up to the temperature of about T* ≈ 105 K. Below T*, more distortions occur and the resonance line largely broadens, which indicates there emerges more complex magnetic interactions and magnetic inhomogeneity. T* corresponds to the reported temperature where the magneto-structural coupling appears [27].

In Figure 6, the temperature dependences of the intensity I (extracted from double integration of the first derivative ESR spectra dP/dH) and the line width ΔH_pp are quantitatively presented. ΔH_pp is extracted by the H distance recorded between the highest and lowest peaks of dP/dH. Upon lowering T, the I keeps rising from T_c and then approaches maximum at 95 K just below T*. The peak of I below T* with T should arise from the dramatic change of magneto-structural coupling, evidenced by the crystal lattice parameters around T* [27]. This has never been observed by any macromagnetic measurements such as M-T or M-H since it is a second-order-like structural transition at T*. The c axis of Sr₂IrO₄ is compressed and the a-axis becomes longer at T*, leading to a reasonable further additional amount of distortion release of octahedra IrO₆ with T decreasing, accompanied by the progressively broadened ΔH_pp. It is generally believed that the ESR intensity should be proportional to the square of the effective moments of the spins contributing to a given resonance line I_ESR ~ μ²_eff = g²S(S + 1)μ_B² [22,33]. Below T*, the monotonous reduction of I is consistent with the M-T behavior since the DMI effect is reduced abruptly, whereas the AFM becomes dominated [27]. Above T*, the change of I is not obvious in the M-T measurements (Figure 3a) due to the spin–lattice relaxation effect. Similar to I behavior, upon decreasing T, the ΔH_pp rises, which is characteristic of an insulating nature of Sr₂IrO₄ except for a symbol contributed from the complicated magnetic interactions. At 95 K just below T*, the appearance of maximum ΔH_pp is more solid evidence for the strong coupling between structure and magnetism.

The most striking anomalous feature of Sr₂IrO₄ is explored via the H_r and g factor. The temperature dependence of the g factor is exhibited in Figure 3a, which is obtained from the main peak of ESR signals calculated by the resonance condition $\hslash \varpi =g{\mu }_B{H}_{res}$ of the first derivative spectra dP/dH in the temperature range of 11–230 K. The g factor shows identical behavior to the temperature dependence of in-plane M, which is a strong hint of the correlation between the g factor and DMI, since M reflects the DMI change with T. The g factor is also a fingerprint of magnetic resonance field H_r. The temperature dependence of H_r is demonstrated in Figure 4b. It can be seen that the trend of H_r vs. T is similar to the H_s behavior. As we know, in a conventional ferromagnet, the magnetic moment should strengthen and the H_s tends to be smaller at lower temperatures. However, in single crystal Sr₂IrO₄, upon lowering the T, the in-plane magnetic moment decreases when H < 0.2 kOe (Figure 3a) while the H_s becomes larger (Figure 4b). This should be ascribed to the reduction of in-plane DMI with temperature decreasing [6]. Thus, we can conclude that the change of parameters H_r, H_s, M and g factor vs. T are all derived from the DMI ebbing away with T decreasing, rather than SOC interaction. It is well known that the strong SOC results in J_eff = 1/2 as a character of an anomalous Mott insulator in Sr₂IrO₄. Here, we show another abnormal magnetic property of the g factor and its associated physical parameters driven by DMI, probed by ESR measurements. The results have proved that the ESR technique is very sensitive to structural-magnetic coupling and reliable for detecting the slight modification of WFM by external stimulus such as T, H and even crystalline octahedral distortion.

4. Conclusions

In summary, we have prepared layered single crystals of 5d Sr₂IrO₄ and studied its anomalous magnetic behavior by M-T, M-H and X-band ESR measurements. It was demonstrated that the crystal structure induces the presence of the DMI action and changes with T, which in turn affects the magnetic behavior of Sr₂IrO₄. As the temperature decreases, the rotation of IrO₆ octahedra distortion decreases, leading to an attenuated DMI among the IrO₆ octahedral chains, which eventually results in anomalous magnetic parameters, as probed by ESR measurements. The analysis of the typical magnetic parameters of H_r, H_s, I, g factor and ΔH_pp shows that ESR is a powerful tool for characterizing DMI action since it is more sensitive to WFM and even non-perceptual moment change induced by the coupling between structure distortion and magnetism.