Magnetic-field–induced Zeeman excitation mode in paramagnetic NdGaO₃single-crystal probed by magneto-terahertz spectroscopy

Transition-metal–based complex oxides have long been the cynosure of technological developments owing to their applied potential in areas of contemporary interest such as spintronics, photonics, optoelectronics, data storage, etc. Oxide materials thin-film heterostructures and interfaces comprise one such area that allows dimensionality as well as structural control of those exotic phases which are not inherent to individual constituents of heterostructures. Utilizing these phenomena to design magnetic- and electronic-properties–based devices with improved functionalities is highly desirable in emerging oxide electronics. These properties/ functionalities are controlled by external parameters such as pressure, temperature, epitaxial strain, magnetic and electric fields, etc. [1–4]. One of the most important ingredients of a thin film heterostructure is the single crystal substrate on which it is formed and structurally integrated. While the lattice symmetry and cell constants of the substrate are important in realizing a structural control, the knowledge of its magnetic properties and electronic structure is required for designing the architecture and desired functionalities. There is a wide range of oxide crystal substrates with different crystal structures such as cubic, tetragonal, pseudocubic, orthorhombic, etc., which are being used in the development of emergent heterostructures [5]. Among all, the rare-earth gallate NdGaO³ (NGO) imparts unique controls of epitaxial strain and orthorhombic distortion. It has been successfully used as a substrate for growth of epitaxial thin films of high-T_c superconductors such as YBa₂Cu₃O₇ (YBCO), high conducting oxides, ferroelectrics, magnetic, magneto-electric, and magneto-resistive materials [6–12].

The NGO crystallizes in GdFeO₃-type orthorhombic symmetry with Pbnm space group at room temperature [13–15]. The orthorhombic lattice constants are $a = 5.43\,\unicode{8491}$, $b = 5.50\,\unicode{8491}$, and $c = 7.71\,\unicode{8491}$, whereas pseudo-cubic (pc) lattice parameters obtained using the following equations: $a_{\text{pc}}= b_{\text{pc}} = \frac{\sqrt{a^{2}+b^{2}}}{2}$ and $c_{\text{pc}} = \frac{c}{2}$ are $a_{\text{pc}} = b_{\text{pc}} = 3.864\,\unicode{8491}$, and $c_{\text{pc}} = 3.855\,\unicode{8491}$. NGO crystals lack any ordered/long-range magnetic ordering down to a temperature range of 2–300K along all major crystal orientations ((100), (110), and (111)) [6,16]. Optical studies revealed several absorption peaks corresponding to the transitions of Nd atoms in the far- and mid-far infrared regime of the electromagnetic spectrum [17,18]. The temperature-dependent THz studies by Ludwig et al. showed that NGO has low loss and its transmission is independent of THz frequencies, which suggests it could be an ideal substrate for thin films with optical properties targeted in the spectral range 0.2–2THz [19]. The crystal field splitting of ground state J-multiplet of Nd³⁺ (⁴I_9/2) into five Kramer's doublets was observed by inelastic neutron scattering measurements [16]. Such crystal field splitting of the paramagnetic rare-earth ions has been observed in various orthoferrites systems in which the magnetic excitations are modified by low-energy crystal field transitions [20–23]. In these studies, the spin wave excitations are modified by the resonant pumping of rare-earth ions by THz pulses. The NGO, however, possesses a large intrinsic paramagnetic magnetic moment owing to rare-earth Nd³⁺ ions. This can render two important implications in the case of heterostructures: i) large Nd³⁺ ions moments of the substrate can significantly distort the magnetic moment contribution of thin films, and ii) the response of Nd³⁺ paramagnetic phase to ultrafast optical pulses probing the magnetic order such as THz spin resonances with controls of temperature and magnetic field relevant to the contemporary context of oxide thin films is not known. This poses a need to evaluate the magnetic field dependence of NdGaO₃ crystal in the THz frequency regime.

In this letter, we show low-energy dynamic response of NGO crystal using magneto-THz spectroscopy in transmission mode. We observed a field-induced resonance mode at 0.2THz in the paramagnetic phase of NGO which has potential implications in THz magnetism. This magnetic-field–induced mode appears with the application of a magnetic field in two NGO crystals along different crystallographic orientations. This mode has been assigned to the splitting of ground state Kramer's doublet of Nd³⁺, and the strength of this magnetic mode decreases with increasing the temperature and almost disappears at around 50 K.

NGO single crystal substrates of thickness 0.5mm with (001) and (110) crystallographic orientations named as NGO (001) and (110) substrates were used for THz spectroscopic investigations in the temperature range 1.5–50K and applied magnetic field up to 5T. The measurements were performed in transmission geometry on a THz time-domain spectrometer based on 1560 nm femtosecond fiber coupled TERA K-15 spectrometer (Menlo systems GmBH) integrated with a 7T spectromagPT Oxford cryostat. In this spectrometer, a femtosecond laser source T-light FC* generates a femtosecond laser pulse (of wavelength ${\lambda} = 1560\ \text{nm}$, pulse width ${<}90\ \text{fs}$, and pulse repetition rate of 100MHz) which is used to generate and detect THz radiation. The terahertz emitter (Fe: InGaAs) and detector (LT-InGaAs) are based on photoconductive antenna switches. Both the fibre-coupled emitter and detector antennas are triggered simultaneously by the laser beam. The THz beam which is generated by the terahertz emitter is focused on the sample by combinations of THz polymer (TPX) lenses. The detector antenna consists of a dipole shape with a small gap in the micrometer range. When excited with a femtosecond laser, the antenna works as a dipole and incoming THz radiation coming from the emitter induces a small current. This photocurrent is measured by a lock-in amplifier. The arrival of the laser pulse with respect to the THz pulse was changed by the optomechanical delay line and the time profile of the THz electric field was recorded. To improve the signal-to-noise ratio (SNR), 10000 single-shot pulses were averaged out. To eliminate the unwanted absorption of THz pulses by water and to enhance the SNR, the THz optical transmission studies were performed in a dry nitrogen atmosphere. THz-TDS measurements were performed in the following way: first the THz pulse was passed through the vacuum at a given magnetic field and temperature and then the THz pulse was passed through a NGO single crystalline substrate under the same experimental conditions and the THz temporal waveform was collected. The THz pulse passing through the vacuum acts as a reference. Fast Fourier transforms (FFT) of THz temporal waveforms were obtained to get the amplitude and phase information in the frequency domain. From FFT data, various optical constants such as dielectric constant, transmission coefficient, refractive index, etc. were obtained. Magnetic measurements were performed on both NGO (001) and (110) substrates using Quantum Design superconducting quantum interference device (SQUID-VSM) in the temperature range 2–300 K.

In order to understand the magnetic properties of both NGO (001) and NGO (110) single crystalline substrates, we performed temperature-dependent DC magnetization measurements in the presence of an external magnetic field of strength 500 Oe in both zero-field cooled (ZFC) and field cooled (FC) protocols. Here, during the measurements, an external magnetic field was applied perpendicular to the surface of the sample. Figure 1(a) shows the DC magnetization as a function of temperature for both NGO (001) and NGO (110) substrates. The magnetic moment for both substrates increases as we decrease the temperature which is a typical paramagnetic behavior. In our case, both ZFC and FC curve revealed similar features down to low temperature for both the substrates, therefore, we have presented ZFC data only for both the substrates. However, a discernible difference between the magnetic moments of both the substrates was observed at low temperature (see fig. 1(a)) which suggests the anisotropic nature of the magnetic moment. This anisotropic nature of magnetic moment has been previously observed in NGO [6,13,16]. To gain further insight, the inverse magnetic susceptibility of both the substrates was plotted as a function of temperature and is shown in fig. 1(b). A slight deviation from the Curie law was observed at low temperatures for both substrates and is presumably due to crystal electric field effects [16].

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Figure 2(a) shows the schematic representation of the THz time-domain spectrometer, as explained in the experimental details section. All measurements were performed in the Faraday geometry, in which the directions of an applied magnetic field and the incident THz pulse beam are orthogonal to the surface of the sample. Figure 2(b) and (c) shows the THz waveform in the time-domain and its corresponding FFT spectrum in the frequency domain of NGO (001) substrate at 1.8K in the presence of the magnetic field, respectively. A slight absorption arises towards the higher time scale in THz time-domain waveform with the application of the magnetic field, which is also clearly visible in the FFT spectrum towards the lower frequency side. This absorption becomes more prominent with an increase in the magnetic field. The evolution of this absorption feature as a function of magnetic field and the temperature is evaluated systematically in the subsequent sections. Further, to eliminate the possibility of the effect of the external magnetic field on THz emitters, we have plotted the temporal THz waveform in free space in the presence of the magnetic field (see fig. 2(d)) which excludes any possibility of an effect of the magnetic field on THz emitters.

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Figure 3(a) and (b) shows THz transmission of both NGO (001) and (110) substrates as a function of frequency at 1.8K in different applied magnetic fields. With the application of magnetic field, a dip corresponding to resonance absorption towards the low-frequency side manifests in the THz transmission spectrum. This mode strengthens with increasing the magnetic field. This mode clearly has a magnetic origin for two reasons; in addition to the fact that it manifests as well as strengthens with the magnetic field, there is also a blue shift as it shifts to higher frequencies on increasing the magnetic field. To quantify this, the peak frequency of resonant modes was plotted as a function of magnetic field for both substrates (see fig. 3(c)). As can be seen from fig. 3(c), for a given magnetic field, the resonant frequency of (110) oriented substrate is higher as compared with (001) oriented substrate which suggests that the magnetic anisotropy is inherent to the magnetic structure, therefore confirming a different g-factor associated with different crystallographic orientations in NGO [13]. From fig. 3(c), it can be observed that the peak resonant frequency shows a non-linear dependency on the applied magnetic field. In addition to this, we have plotted the peak THz transmission as a function of magnetic field along both the crystallographic orientations (see fig. 3(d)). Here, the peak THz transmission corresponds to a dip in the transmission spectrum. More absorption was observed for NGO (110) as compared to that of NGO (001) substrate which could be because of the magnetic anisotropy in NGO [6,13,16].

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Inelastic neutron diffraction measurements performed by Podlesnyak et al. on NGO single crystal showed that the crystal field which arises from the distorted octahedral formed by oxygen ions around Nd³⁺ ions split the 10-fold degeneracy of ground state multiplet of Nd³⁺ (4I_9/2) into five Kramer's doublets. The energy of all the four excited doublets is above 10 meV [16] (see fig. 4(a)). Since the energy of the ground state doublet lies with the THz regime, therefore, THz spectroscopy can be used as an ideal probe to explore the ground state doublet because of its high accuracy as well as non-invasive attributes. Recently, THz technology has been used to probe various low-energy excitations such as magnons in antiferromagnets, superconducting gaps, charge-density waves, intra-level transitions, etc. [24–28]. The external magnetic field interacts with the spin of the electronic system via Zeeman interaction. In a non-zero magnetic field, the Zeeman interaction can lift the degeneracy of the ground state Kramer's doublet and can induce magnetic allowed transitions between magnetic sublevels and shift the energies of the transitions.

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The blue shift of magnetic-field–induced mode and the splitting of the ground state doublet with the application of magnetic field can be described by the following equation [27]: $\Delta E=(\Delta E_{0}^{2}+\mu^{2} B_{\text{ext}}^{2}{g^{2}})^{1/2}$; where the symbols $\Delta E_{0}$, g, $B_{\text{text}}$, and ${\mu}$ represent the initial splitting in the absence of magnetic field, the Lande g-factor, the external applied magnetic field, and the total magnetic moment, respectively. This relation suggests that the splitting of the level increases with increasing the magnetic field. This increases $\Delta E$, i.e., the spacing between the intra-level doublets, due to which the magnetic-field–induced modes appear at higher frequency and result in the blue shift (see fig. 3(a) and (b)). In the present case, the splitting between the intra-level doublet starts at ${\sim}1\ \text{T}$ for both the substrates. However, this type of Zeeman splitting should have started at lower non-zero fields. The absence of this feature at lower magnetic fields can be attributed to two facts as either the response at low fields falls at the lower frequency side which is beyond the spectral range of our spectrometer or it is quite weak to show any discernible feature. Zeeman splitting is more prominent for NGO (110) as compared to that of NGO (001) substrate. This could be because of the larger Coulombic repulsion among 4f orbitals along (110) crystallographic orientations as compared to (001) crystallographic orientations due to magnetic anisotropy which could lift the degeneracy at a much faster rate compared to (001) crystallographic orientations.

The splitting of Kramer's doublet levels has previously been reported for antiferromagnetic crystal NdFe₃(BO₃)₄ explored by optical absorption techniques which were attributed to the exchange interactions provided by Fe and the external magnetic field [29,30]. The magnetic-field–induced Zeeman's splitting in complex systems such as hemin, CoX₂ (PPh₃)₂ where (X = Cl or Br), $\text{Fe}(\text{H}_{2}\text{O})_{6}^{2+}$ and NiCl₂(PPh₃)₂ were also explored by using THz time-domain electron paramagnetic resonance (EPR) spectroscopic measurements [31]. However, in the present case of NGO, the electronic configuration of Ga³⁺ is 3d₁₀ which rules out the possibility of any exchange interactions due to Ga³⁺ ions, therefore we considered this splitting to be the fine structure splitting of the ground state Kramer's doublet which was degenerate in the absence of the magnetic field (see fig. 4(a)).

To investigate the temperature-dependent dynamics of this magnetic mode, we performed THz time-domain measurements in the presence of a magnetic field of 4.5T for NGO (001) substrate at the temperature of 1.8, 6, 10, 20, and 50K (see fig. 4(b)). The amplitude of this magnetic-field–induced mode decreases as we increase the temperature and almost vanishes at 50K. This behavior could be attributed to the decrease in the magnetic susceptibility with increase in temperature [32]. Also, a red shift of this field-induced mode is observed at 1.8K. This red shift in the resonance peak position at 1.8K might be because of the onset of the antiferromagnetic ordering of Nd³⁺ ions which occurs at around 1K [13,33].

We further analyzed the THz transmission through the NGO (001) substrate in terms of the normalized loss function $\sigma(\omega)$ which accounts for both the absorptive and reflective effects and is defined as

$$\begin{equation*} \sigma (\omega)=-\ln \left(\left| \frac{S(\omega )}{S_{\text{ref}} (\omega )}\right|\right), \end{equation*}$$

where $S(\omega )$ and $S_{\text{ref}}(\omega )$ correspond to the spectrum of the THz signal transmitted through the sample and the reference signal, respectively [32]. The loss function of NGO (001) substrate as a function of frequency at a temperature of 1.8K in various magnetic field strengths is shown in fig. 4(c). It is clearly seen from the fig. 4(c) that the NGO (001) single crystalline substrate exhibits a high loss function in the lower frequency regime, i.e., below 0.2THz in the presence of magnetic fields. On the other hand, towards the higher frequency regime, the loss function shares similar features in the presence of magnetic fields.

The field-induced mode of NGO can have numerous implications in contemporary research in quantum materials including rare-earth orthoferrites as well as antiferromagnetic spintronics. Several systems exhibit THz resonance absorption modes in a frequency range similar to that of NGO. For example, a collective spin precession frequency known as antiferromagnetic resonance in NiO [34], kink bound states in 1D Ising spin chain systems [35], quasi-ferromagnetic and quasi-antiferromagnetic resonances in orthoferrites [20–23,36], electromagnons in multiferroic manganites [37], and charge density waves in oxides [38] fall in the energy range 0.1–1.5THz. The resonance mode arises due to either structural transition or magnetic ordering structures (such as spin reorientation, incommensurate magnetic ordering, etc.) and can be controlled using an external magnetic field. The NGO crystal is a preferred choice as a substrate for the fabrication of epitaxial thin film heterostructure. Its THz resonance mode due to the splitting of Kramer's doublet has a pronounced magnetic field control; these features can coincide with the THz resonance mode of thin films deposited on it. In such cases, the THz transmission spectra need careful evaluation in terms of the contribution of NGO substrate, and the mechanism to separate such contributions from that of the thin film.

In high-T_c unconventional superconductors such as cuprates and multiferroic manganite, the magnetic field suppresses the superconductivity and electromagnons, respectively [37,39,40]. In the case of the AFM NiO system, magnetic-field–induced mode was observed at very high magnetic fields towards low frequency. The magnetic field can alter the magnetic structure in the case of ordered structures. However, in the present case, NGO lacks any magnetic order down to 2 K. The presence of magnetic-field–induced mode has all the attributes of its origin in Nd³⁺ ions.

Terahertz spectroscopy has proved to be an ideal tool to probe collective low-energy dynamics. In the present case, we implemented THz spectroscopy to explore the optical properties of NGO single crystalline substrate with (001) and (110) crystallographic orientations in the presence of a magnetic field. A magnetic-field–induced mode was observed for both NGO (001) and (110) substrates whose strength varies and gets blue-shifted on increasing the magnetic field. This mode strengthens upon increasing the magnetic field while weakens upon increasing the temperature. The strength of the mode is found to be greater for NGO (110) as compared to (001) substrate. The mode disappears at a temperature beyond 50K. This mode arises due to the intra-level transitions in the ground state Kramer's doublet.

DSR thanks the Science and Engineering Research Board (SERB), Department of Science and Technology, New Delhi, for financial support under research Project No. CRG/2020/002338.

Data availability statement: All data that support the findings of this study are included within the article (and any supplementary files).