1. Introduction

Progress in pulsed magnetic field techniques has enabled the study of a rich diversity of phenomena in rare-earth-doped magnetic ceramic materials that exhibit photoluminescence (PL) [1–5]. Generally, 4f electron wave functions play an important role in such systems in the presence of a magnetic field, and Zeeman splitting [6,7] dominates the physics of these materials. It has been shown, for example, that there is Zeeman splitting of the higher ⁵D₀ energy level in Eu³⁺-doped NaGdF₄ nanocrystals in a magnetic field, which results in a blueshift of intra-4f-shell PL (the ⁵D₀→⁷F₀ transition) in Eu³⁺ [8]. This blueshift can be used to modulate color by controlling the magnetic field. In addition, the intra-4f-shell PL intensities (both the ⁵D₀→⁷F₀ and ⁴S_3/2→⁴I_15/2 transitions) in Er³⁺-doped PL crystals depend on the magnetic field, and these crystals can be used to detect magnetic fields [8–11]. Most of these results are in conflict with Judd-Ofelt theory (J-O theory) [12,13], according to which the magnetic field should have no effect on the intensity of the intra-4f-shell luminescence bands of rare-earth ions. Previous studies [14,9,11] have investigated the magnetic-field-dependent shifts of PL peaks, but have not provided specific explanations for the mechanism of Zeeman splitting or for the magnetic-field dependence of the intra-4f-shell PL intensities. Therefore, further study of 4f-electron wave functions is of great importance. The perturbation Hamiltonian for the 4fⁿ configuration of a rare-earth ion in an external magnetic field can be written in the form [15].

Gd₃Ga₄FeO₁₂ is a well-known wide-gap magnetic ceramic [19]. Er³⁺ and Yb³⁺ are widely used dopants that are known to activate PL and to act as efficient luminescence centers (for the intra-4f-shell transition) [20,21]. Consequently, replacement of the Gd³⁺ ion by Er³⁺ and Yb³⁺ ions give rise to important changes in the Zeeman splitting and superexchange interaction of Er³⁺ in Gd₃Ga₄FeO₁₂: Yb³⁺, Er³⁺ ceramic oxides.

In this work, the Zeeman splitting of the intra-4f-shell luminescence band (⁴I_13/2→⁴I_15/2) of Er³⁺ is studied experimentally in Gd₃Ga₄FeO₁₂: Yb³⁺, Er³⁺ in external magnetic fields up to 38 T, and the f-f superexchange interaction of Er³⁺ is calculated by using the DFT. It is found that Zeeman splitting increases with increasing f-f superexchange interaction parameters obtained using DFT and that the intensity of the ⁴I_13/2→⁴I_15/2 emission band depends strongly on the magnetic field, which disagrees with the predictions of J-O theory. The spd-f hybridization between the occupied Er 4f orbitals and the valence states can induce changes in the ⁴I_13/2→⁴I_15/2 luminescence intensities in a magnetic field, which is a likely explanation for the conflict with J-O theory. The work described here provides further understanding of the f-f superexchange interaction and Zeeman splitting of the 4fⁿ configuration. In addition, the materials under study exhibit high intra-4f-shell PL sensitivity to magnetic fields, and therefore have potential applications in magnetic field sensors and magnetic guidance systems in aircraft, for example.

2. Experimental and computational details

A series of Gd_3-xGa₄FeO₁₂: 0.042Yb³⁺, xEr³⁺ (x = 0.021, 0.042, 0.084, and 0.126) samples were synthesized using the sol-gel method [20]. X-ray diffraction (XRD) data confirmed that the preparations were successful. Temperature-dependent magnetization (M-T) studies were carried out in a magnetic measuring system (Quantum Design) with a magnetic field intensity of 0.5 T. Magneto-photoluminescence (MPL) measurements were carried out in a pulsed magnet with a peak magnetic field intensity of 38 T and a pulse duration of 100 ms. The samples were excited by a solid state laser with a wavelength of 532 nm, and the MPL spectra were analyzed by a spectrometer (S500i, Andor) equipped with a charge-coupled device (Du970P, Newton).

The many-body problem for the electrons around the nuclei was solved based on DFT. The electronic states of Gd_3-xGa₄FeO₁₂: 0.042Yb³⁺, xEr³⁺ (x = 0.021, 0.042, 0.084, and 0.126) were obtained from the projector augmented wave method as implemented in the VASP code [22]. The calculations were conducted using the supercell method. A 160-atom supercell for Gd₃Ga₄FeO₁₂ was used for the study of the Yb³⁺ and Er³⁺ co-doped material. Geometry optimization of Gd₃Ga₄FeO₁₂ was performed with eight Fe ions replacing eight Ga ions at different positions (octahedral and tetrahedral sites). One Er atom replaced one Gd atom in the 160-atom supercell of Gd₃Ga₄FeO₁₂ to give Gd₃Ga₄FeO₁₂: 0.042Er³⁺. Two Yb atoms and one Er atom replaced three Gd atoms in the 320-atom supercell of Gd₃Ga₄FeO₁₂ to give Gd₃Ga₄FeO₁₂: 0.042Yb³⁺, 0.021Er³⁺. One Yb atom and one Er atom replaced two Gd atoms in the 160-atom supercell of Gd₃Ga₄FeO₁₂ to give Gd₃Ga₄FeO₁₂: 0.042Yb³⁺, 0.042Er³⁺. One Yb atom and two Er atoms replaced three Gd atoms in the 160-atom supercell of Gd₃Ga₄FeO₁₂ to give Gd₃Ga₄FeO₁₂: 0.042Yb³⁺, 0.084Er³⁺. One Yb atom and three Er atoms replaced four Gd atoms in the 160-atom supercell of Gd₃Ga₄FeO₁₂ to give Gd₃Ga₄FeO₁₂: 0.042Yb³⁺, 0.084Er³⁺. To account for the f-f hybridization, the 4f spin orbitals of the Er impurities were included in the calculations. In our application of DFT, we dealt with exchange correlation on the basis of the generalized-gradient approximation (GGA) [23,24]. However, the interactions among the 3d electrons of Fe and the 4f electrons of Er, Yb, and Gd in the GGA approximation are described only partially, owing to their strong localization. Thus, we used the so-called GGA + U scheme [25] and introduced into the calculations two common correction parameters U and J to correct the Coulomb and exchange interactions, respectively, among the 3d electrons at Fe sites and the 4f electrons at Er, Yb, and Gd sites. We chose the U and J values by comparison with experimentally determined band structures. For the Fe and Gd atoms in Gd₃Fe₅O₁₂, we took U_Fe = 2, 4, 6, 8 eV and U_Gd = 2, 4, 6, 8 eV, respectively, together with J = 0.5 eV, in order to obtain a more accurate electronic structure. The cutoff energy in the plane-wave basis set was taken as 450 eV.

3. Results and discussion

3.1 Magnetism

Figure 1 shows the temperature dependence of magnetization (M-T) for Gd₃Ga₅O₁₂ (GGG) and Gd_3-xGa₄FeO₁₂: 0.042Yb³⁺, xEr³⁺ (x = 0.021, 0.042, 0.084, and 0.126) in a field of 0.5 T. It can clearly be seen that there is a sudden increase in magnetization below 30 K for all the compositions compared with that of GGG. This increase is related to magnetic ordering of rare-earth ions. The M-T curves show increased magnetization for all the Gd₃Ga₄FeO₁₂: Yb³⁺, Er³⁺ samples, resulting from magnetized Fe³⁺ and Er³⁺, which is in agreement with the results of a previous study [19]. The magnetization increases with increasing Er³⁺ concentration between 0 and 0.042, and then decreases above 0.084, because Yb³⁺ is a nonmagnetic ion and Er³⁺ a magnetic ion.

 

Fig. 1 Temperature dependence of magnetization in a field of 0.5 T for GGG and Gd_3-xGa₄FeO₁₂: 0.042Yb³⁺, xEr³⁺ (x = 0.021, 0.042, 0.084, and 0.126). Inset (a) shows the Curie-Weiss law fit of the dc susceptibility data.

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The magnetic moment increases with increasing Er³⁺ concentration between 0 and 0.042, and then decreases above 0.084 because the magnetic moment of Fe³⁺ (∼4.33 µ_B) is higher than that of Ga³⁺ (0 µ_B) and the magnetic moment of Gd³⁺ (∼7.9 µ_B) is higher than those of Yb³⁺ (0 µ_B) and Er³⁺ (6.6 µ_B) [26]. The maximum magnetic moment is obtained when the Er concentration reaches 0.042. The inverse susceptibilities of GGG and Gd_3-xGa₄FeO₁₂: 0.042Yb³⁺, xEr³⁺ were calculated from the M-T data, and their temperature dependence is plotted in inset (b) of Fig. 1. The Curie temperature θ and the effective magnetic moment µeff were extracted from the Curie-Weiss fit in the temperature region 0-300 K and are given in Table 1. The negative value of the paramagnetic Curie temperature (−3.8 K) for GGG suggests the presence of low-temperature antiferromagnetic interactions in GGG, while the negative value of the Weiss constant can be assigned to the crystal field effect of the rare earth without the need to invoke the onset of any antiferromagnetic interactions [27]. In addition, the Gd₃Ga₄FeO₁₂: Yb³⁺, Er³⁺ system shows ferromagnetic behavior below 7.76 K for all the other samples (Table 1).

Table 1. Paramagnetic Curie temperatures θ and effective magnetic moments µeff calculated from the Curie-Weiss law fit for GGG and Gd_3-xGa₄FeO₁₂: Yb³⁺, xEr³⁺ [x = 0.021 (I), 0.042 (II), 0.084 (III), and 0.126 (IV)] in the temperature region 0-300 K. The theoretical magnetic moments µ_theo were calculated from Eq. (2).

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For comparison, Table 1 also shows the theoretical values of the magnetic moment, calculated from the formula [28]

3.2 Temperature dependence of zero-magnetic-field and high-magnetic-field photoluminescence

To better understand the behavior of the PL intensity of the Gd₃Ga₄FeO₁₂: Yb³⁺, Er³⁺ system in a magnetic field, we first measured its temperature dependence in zero magnetic field. Figures 2(a)-(d) show the zero-magnetic-field spectra of the ⁴I_13/2→⁴I_15/2 luminescence bands of Gd_3-xGa₄FeO₁₂: 0.042Yb³⁺, xEr³⁺ (x = 0.021, 0.042, 0.084, and 0.126) at three different temperatures. It can be noted that at a temperature of 5 K, there are are three, two, three, and four Stark levels of the ⁴I_13/2→⁴I_15/2 luminescence bands for Er³⁺ concentrations x = 0.021, 0.042, 0.084 and 0.126, respectively, while at 80 K, there are three, four, three, and three peaks, respectively, for these concentrations. At room temperature, eight Stark components are observed for all samples. These results indicate that the spin-orbit ground state ⁴I_15/2 splits into two to five main Stark levels at low temperature (below 80 K), whereas it splits into eight main Stark levels at room temperature. Therefore, there is a high-symmetry pseudocubic Er³⁺ site at low temperature and a low-symmetry pseudo-cubic site at room temperature. The irreducible representations for pseudo-cubic symmetries are Γ₆ + Γ₇ + 3Γ₈. The ground state ⁴I_15/2 is (2J + 1)-fold degenerate. In high crystal-field symmetry, the lowest Kramers doublet is Γ₆ or Γ₇ of ⁴I_13/2 and ⁴I_15/2. In low crystal-field symmetry, the quartets Γ₈ additionally split, giving rise to a total of eight Stark levels, which are Kramers doublets. The lowest Stark levels for both the ⁴I_13/2 and ⁴I_15/2 states are simple Kramers doublets, permitting only the simplest doublet splitting of these Stark levels in an external magnetic field.

 

Fig. 2 (a)-(d) Zero-magnetic-field spectra of the ⁴I_13/2→⁴I_15/2 luminescence bands of Gd_3-xGa₄FeO₁₂: 0.042Yb³⁺, xEr³⁺ at different temperatures for (a) x = 0.021, (b) x = 0.042, (c) x = 0.084, and (d) x = 0.126. (e)-(l) Photoluminescence spectra at magnetic fields 0-38 T at 5 K and 80 K, showing different splitting behaviors of Gd_3-xGa₄FeO₁₂: 0.042Yb³⁺, xEr³⁺; (e) x = 0.021 at 5 K; (f) x = 0.042 at 5 K; (g) x = 0.084 at 5 K; (h) x = 0.126 at 5 K; (i) x = 0.021 at 80 K; (j) x = 0.042 at 80 K; (k) x = 0.084 at 80 K; (l) x = 0.126 at 80 K.

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The purple arrows at the ⁴I_13/2→⁴I_15/2 bands (the highest peak at 1533 nm) in Figs. 2(a), 2(b), and 2(d) indicate the 0-0 transition between the lowest Stark levels of the ⁴I_13/2 and ⁴I_15/2 states of Er³⁺. This transition does not involve the absorption or emission of phonons; rather it is a so-called zero-phonon transition or pure electronic transition. The intensities of all other transitions (include small peaks at the left and right and highest middle peak) in Figs. 2(a), 2(b), and 2(d) greatly decrease with decreasing temperature, indicating that these transitions are induced by emission from the thermally populated Stark levels of the ⁴I_13/2 state (or lattice thermal vibrations). The intensities at 1533 nm in Fig. 2(c) also decrease with decreasing temperature, indicating that the transition between the ⁴I_13/2 and ⁴I_15/2 states at 1533 nm results not only from the 0-0 transition, but also from the emission of the thermally populated Stark levels of the ⁴I_13/2 state. These observations indicate that the transition mechanism at 1533 nm depends on the concentration of Er³⁺.

Figures 2(e) - 2(l) show the 5 K (below the Curie temperature) and 80 K PL spectra of the ⁴I_13/2→⁴I_15/2 transition as a function of magnetic field, which are all in high crystal-field symmetry. The 0-0 phonon lines (the highest peak) for all Gd₃Ga₄FeO₁₂:Yb³⁺, Er³⁺ samples split into two weaker peaks when a high external magnetic field (≥ 20 T at 5 K, and ≥ 25 T at 80K, respectively) is applied. It is also clear that these splitting peaks are redshifted with increasing field, indicating that the peaks of the ⁴I_13/2→⁴I_15/2 transition are split into Zeeman doublets in high external magnetic fields both for ferromagnetic and for paramagnetic Gd₃Ga₄FeO₁₂: Yb³⁺, Er³⁺. Er³⁺ has the electronic configuration [Xe] 4f¹¹6s², and therefore the electronic state must be at least doubly degenerate in the absence of a magnetic field. All these doublets can be split in the presence of an external magnetic field. ⁴I_13/2 and ⁴I_15/2 split into 14 and 16 M_J sublevels, respectively, in an external magnetic field.

The small peaks at the left (1515 nm) and right (1610 nm) are only shifting and the highest middle peaks (1544 nm) are splitting into 2 weaker peaks for all Gd₃Ga₄FeO₁₂: Yb³⁺, Er³⁺, which has a very similar behavior with Zeeman effect in photoluminescence of Er³⁺ ion imbedded in crystalline silicon [35]. Moreover, with increasing field (30-38 T), the highest middle peaks intensities of the Zeeman splitting at 1544 nm become higher at 80 K but weaker at 5 K. This indicates that these peaks at 1544 nm may be influenced by the large Er-Er bond length, which results in lattice thermal vibrations [20].

It is noteworthy that the intensities of the 0-0 phonon lines of the ⁴I_13/2→⁴I_15/2 transition (the highest peak and the highest middle peak) decrease with increasing magnetic field (≥ 10 T). This indicates the intensities of ⁴I_13/2→⁴I_15/2 transition depend on magnetic field, which is in conflict with the J-O theory [38]. While, the smaller peak (1515 nm) on the left suddenly disappear in an external magnetic field for Gd₃Ga₄FeO₁₂: 0.042Yb³⁺, 0.042Er³⁺ (Fig. 2(f)). There is a redshift with increasing magnetic field for the 0-0 zero phonon line of the ⁴I_13/2→⁴I_15/2 transition. Moreover, the Zeeman splittings of this transition (the highest peak at 1533 nm) for different Er concentrations depend on magnetic field as shown in Figs. 3(a) and 3(b). It can be seen that the Zeeman splittings increase with increasing magnetic field and that Gd₃Ga₄FeO₁₂: 0.042Yb³⁺, 0.042Er³⁺ exhibits the maximum Zeeman splitting. This indicates that a particular choice of Er concentration can change the Zeeman splitting of the 0-0 phonon line of the ⁴I_13/2→⁴I_15/2 transition, which agrees with the results for the magnetic properties of Gd₃Ga₄FeO₁₂: Yb³⁺, Er³⁺. The magnetic field dependence of the redshift of the 0-0 phonon line of the ⁴I_13/2→⁴I_15/2 transition is shown in the insets in Figs. 3(a) and 3(b). The dependence changes to a linear one at high magnetic fields, with crossovers at about 25 and 20 T at 5 and 80 K, respectively. This linear dependence at high fields is typical of magnetic confinement (compression) [14] of electron wave functions in a variety of materials. The dE/dB ratio about 0.32 cm⁻¹/T, 1.57 cm⁻¹/T, 1.43 cm⁻¹/T, and 0.996 cm⁻¹/T for the highest peaks in Gd₃Ga₄FeO₁₂: 0.042Yb³⁺, 0.021Er³⁺, Gd₃Ga₄FeO₁₂: 0.042Yb³⁺, 0.042Er³⁺, Gd₃Ga₄FeO₁₂: 0.042Yb³⁺, 0.084Er³⁺, and Gd₃Ga₄FeO₁₂: 0.042Yb³⁺, 0.126Er³⁺, respectively. There is the biggest dE/dB ratio in Gd₃Ga₄FeO₁₂: 0.042Yb³⁺, 0.042Er³⁺, indicating there is the biggest Zeeman splitting in Gd₃Ga₄FeO₁₂: 0.042Yb³⁺, 0.042Er³⁺. Comparing with the Er³⁺ in other crystal, such as Er³⁺-doped nano-glass-ceramics (dE/dB~0.66 cm⁻¹/T) [14], the dE/dB ratio is also bigger in Gd₃Ga₄FeO₁₂: Yb³⁺, Er³⁺ (except Gd₃Ga₄FeO₁₂: 0.042Yb³⁺, 0.021Er³⁺), which indicates these materials are better for magnetic sensors. Besides, both Zeeman splitting and redshifting of the 0-0 phonon line need a lower magnetic field at 5K than at 80 K, indicating that splitting in the presence of a magnetic field is easier for Gd₃Ga₄FeO₁₂: Yb³⁺, Er³⁺ at lower temperature. In addition, Table 2 lists the Lande g factors estimated from the highest peaks positions of the Zeeman components for Gd₃Ga₄FeO₁₂: Yb³⁺, Er³⁺ at 5 K, which is closed to theoretical value (6/5) very well. The magnetic moments μ are calculated from g factors (Table 2), which is also in agreement with the data given in Table 1 and Eq. (2). The Lande gJ-tensor about 19 for the Er³⁺ ion in orthorhombic garnet type structure [37].

 

Fig. 3 Zeeman splittings of 0-0 zero phonon line of the ⁴I_13/2→⁴I_15/2 transition (the highest peak at 1533 nm) for different Er concentrations depend on magnetic field at (a) 5 K and (b) 80 K. The insets are the magnetic field dependence of the redshift of the 0-0 phonon line of the ⁴I_13/2→⁴I_15/2 transition at 5 K and 80 K.

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Table 2. Lande g factors and μ estimated from the highest peaks positions of the Zeeman components for Gd_3-xGa₄FeO₁₂: Yb³⁺, xEr³⁺ [x = 0.021 (I), 0.042 (II), 0.084 (III), and 0.126 (IV)] at 5 K.

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The Zeeman splittings of the highest middle peak at 1544 nm for different Er concentrations depend on magnetic field as shown in Figs. 4(a) and 4(b). It can be seen that the Zeeman splittings decrease with increasing magnetic field and that Gd₃Ga₄FeO₁₂: 0.042Yb³⁺, 0.042Er³⁺ exhibits the maximum Zeeman splitting. This indicates that a particular choice of Er concentration can change the Zeeman splitting of the 0-0 phonon line of the ⁴I_13/2→⁴I_15/2 transition, which agrees with the results for the magnetic properties of Gd₃Ga₄FeO₁₂: Yb³⁺, Er³⁺. In addition, we obtained the Zeeman components of the highest peak and middle highest peak as function of magnetic field for Gd₃Ga₄FeO₁₂: 0.042Yb³⁺, 0.042Er³⁺ as shown in Fig. 5 There is a nonlinear dependency of the Zeeman components on the magnetic field, which resulted from the Er f-f spin-orbit coupling and the Jahn-Teller displacement of the impurity Er 4f orbit [36]. The others Gd₃Ga₄FeO₁₂: Yb³⁺, Er³⁺ system have similar behavious to Gd₃Ga₄FeO₁₂: 0.042Yb³⁺, 0.042Er³⁺. Therefore, to explain the f-f spin-orbit coupling, these relationships between the Zeeman splitting of the 0-0 phonon line and the Er concentration and between the emission intensity and the magnetic field, the electronic structures of Gd₃Ga₄FeO₁₂: Yb³⁺, Er³⁺ are now analyzed.

 

Fig. 4 Zeeman splittings of the highest middle peak at 1544 nm for different Er concentrations depend on magnetic field at (a) 5 K and (b) 80 K.

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Fig. 5 Zeeman components of the highest peak and middle highest peak as function of magnetic field for Gd₃Ga₄FeO₁₂: 0.042Yb³⁺, 0.042Er³⁺ at 5 K and 80 K.

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3.3 Electronic structure

To obtain the optimal electronic structures of Gd₃Ga₄FeO₁₂: Yb³⁺, Er³⁺, the various effective Hubbard U parameters are used for the Fe d orbitals and Gd f orbitals for Gd₃Fe₅O₁₂. Figure 6 shows the spin-up and spin-down band structures of Gd₃Fe₅O₁₂ along the high-symmetry direction of the Brillouin zone modified by adding the U values of the d orbitals of Fe (U_Fe = 2, 4, 6, 8 eV) and the f orbitals of Gd (U_Gd = 2, 4, 6, 8 eV). The valence-band maximum (VBM) and conduction-band minimum (CBM) occur at the G point (one of the high-symmetry points). Gd₃Fe₅O₁₂ is predicted to be a direct band gap material with energy gaps of 1.819, 1.995, 2.172, 2.287, 1.996, 1.974, and 1.984 eV in Figs. 6(a) - 6(g), respectively. It is found that for Gd₃Fe₅O₁₂, an energy gap of 1.996 eV (U_Fe = 4 eV and U_Gd = 2 eV) gives the best agreement between theory and optical experiments [Fig. 6(h)].

 

Fig. 6 (a)-(g) Band structures of Gd₃Fe₅O₁₂ along the high-symmetry direction of the Brillouin zone modified by adding the U values of the d orbitals of Fe (U_Fe = 2, 4, 6, 8 eV) and the f orbitals of Gd (U_Gd = 2, 4, 6, 8 eV). (h) Calculated energy gap of Gd₃Fe₅O₁₂ from optical absorption experiments.

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In Fig. 7, we display the projected spin-up and spin-down densities of states (DOS) of Gd₃Ga₄FeO₁₂: Yb³⁺, Er³⁺. All of the Fermi energies are superimposed on the conduction band and valence band, and the band edges of all samples are lower than E_f. It is possible that these complexes can act as donors. While the VBM and CBM of GGG did not cross the Fermi energy in our previous study [20], the conduction band edge of Gd₃Ga₄FeO₁₂ [Fig. 7(a)] is lower than that of GGG, indicating that Fe acts as a donor. The conduction band edge energy above E_f of Gd₃Ga₄FeO₁₂: 0.042Er³⁺ [Fig. 7(b)] is lower than that of Gd₃Ga₄FeO₁₂, indicating that Er acts as a donor, while the conduction band edge energy of Gd₃Ga₄FeO₁₂: 0.042Yb³⁺, 0.042Er³⁺ [Fig. 7(d)] above E_f is higher than that of Gd₃Ga₄FeO₁₂: 0.042Er³⁺, indicating that Yb acts as a acceptor. However, when the concentration of Er is lower or higher than 0.042, the conduction band edge energy [Figs. 7(c), 7(e), and 7(f)] is lower than that of Gd₃Ga₄FeO₁₂: 0.042Yb³⁺, 0.042Er³⁺, indicating that Er acts as a acceptor at low concentrations (below 0.042) but as a donor at high concentrations. The gap properties can also be tuned by doping with Fe, Er, and Yb, which is in agreement with what is found for doping of semiconductor nanocrystals. The replacement of Ga by the magnetic Fe induces s-p hybridization between the occupied Fe 3d orbitals and the valence states of Gd₃Ga₄FeO₁₂. In addition, there is s-p hybridization between the occupied Er 4f orbitals and the valence states of Gd₃Ga₄FeO₁₂: Yb³⁺, Er³⁺, which results from the addition of the magnetic Er. These hybridizations can enhance the Zeeman splitting in the presence of an external magnetic field. In addition, the Fe 3d, Gd 5d, and Gd 4f orbitals mix with the Er 4f orbitals for all Yb³⁺ and Gd₃Ga₄FeO₁₂: Yb³⁺, Er³⁺, as a consequence of which the Er 4f become sensitive to an external magnetic field owing to the presence of the magnetic Fe 3d and Gd 5d.

 

Fig. 7 Projected spin-up and spin-down DOS Gd₃Ga₄FeO₁₂ with a complex formed by substitutional Yb and Er at the dodecahedral sites: (a) Gd₃Ga₄FeO₁₂; (b) Gd3Ga4FeO12 Gd₃Ga₄FeO₁₂: 0.042Er³⁺; (c) Gd₃Ga₄FeO₁₂: 0.042Yb³⁺, 0.021Er³⁺; (d) Gd₃Ga₄FeO₁₂: 0.042Yb³⁺, 0.042Er³⁺; (e) Gd₃Ga₄FeO₁₂: 0.042Yb³⁺, 0.084Er³⁺; (f) Gd₃Ga₄FeO₁₂: 0.042Yb³⁺, 0.126Er³⁺. The zero of the energy (the vertical yellow line) denotes the Fermi energy E_f.

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The main peaks of the Fe d states and Er f states appear at the valence band and the conduction band. Owing to the dodecahedral symmetry in GGG [29,30], the Fe and Er electrons are split. We are interested in the magnetic coupling between Er and GGG states equivalent to the band edge states. The projected DOS around valence and conduction edge states indicate that the conduction edge of the host compound hybridizes with the 4f orbitals of Er in all Gd₃Ga₄FeO₁₂: Yb³⁺, Er³⁺. Hence, the superexchange interaction arises from Coulomb repulsion, the Pauli exclusion principle, and spin splitting of these states. Meanwhile, the valence-band edge states of the host compound hybridize fully with the Er 4f orbitals except in the case of Gd₃Ga₄FeO₁₂: 0.042Er³⁺. This hybridization yields an effective exchange mechanism of interaction between Er atoms. Therefore, the f-f interaction for Gd_3-xGa₄FeO₁₂: 0.042Yb³⁺, xEr³⁺ (x = 0.021, 0.042, 0.084, and 0.126) leads not only to spin splitting of the valence-band edge, but also to spin splitting of the conduction-band edge. In this paper, to describe the f-f exchange mechanism of interaction between Er f orbits, we define the Er f-f spin-orbit coupling interaction as f-f superexchange interaction, which is used to study the Yb spin-orbit coupling interaction [17].

3.4 Dependence of Zeeman splitting on superexchange interaction

We calculated the f-f superexchange interaction parameters of Er using the DOS from Fig. 7, together with the bulk expressions, which are given in standard mean-field theory as [31–34]

Using Eqs. (3)–(4), we calculate $N_0(\alpha -\beta )$ as a function of the concentration of Er, with the results shown in Fig. 8(a). It can be seen that the superexchange interaction parameter is greatest when x = 0.042, because the band edge is located at the conduction band above E_f. Figure 8(b) shows the effect of $N_0(\alpha -\beta )$ on Zeeman splitting of the 0-0 phonon line of the ⁴I_13/2→⁴I_15/2 transition in the presence of a 38 T magnetic field at 5 K and 80 K. It is clear that the Zeeman splitting increases with increasing f-f superexchange interaction of Er. This explains the maximum Zeeman splittings in Gd₃Ga₄FeO₁₂: 0.042Yb³⁺, 0.042Er³⁺ in an external magnetic field. Furthermore, the f-f superexchange interaction parameters reveal the magnetic coupling between Er and the band edges of Gd₃Ga₄FeO₁₂, which enhances the sensitivity of the intensity of the ⁴I_13/2→⁴I_15/2 transition in a magnetic field. There is thus a very clear explanation for the dependence of the intensity of the 0-0 phonon line of the ⁴I_13/2→⁴I_15/2 transition on an external magnetic field, in contrast to the prediction of the J-O theory.

 

Fig. 8 (a) Calculated N₀(α − β) as a function of the concentration of Er. (b) Zeeman splitting of the highest peaks as functions of N₀(α − β) in the presence of a 38 T magnetic field at 5 K and 8 K.

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4. Conclusion

Zeeman splitting of the intra-4f-shell luminescence band (⁴I_13/2→⁴I_15/2) and f-f superexchange interactions of Er³⁺ have been studied in high magnetic fields up to 38 T. There is redshifting and Zeeman splitting of the ⁴I_13/2→⁴I_15/2 luminescence band of Er³⁺, and the Gd₃Ga₄FeO₁₂: Yb³⁺, Er³⁺ is easier splits at lower magnetic fields in the lower temperature. Additionally, the superexchange interaction contribution to Zeeman splitting and the magnetic-field-dependent intensity of the ⁴I_13/2→⁴I_15/2 luminescence band have been discussed based on results obtained using DFT. It is found that Zeeman splitting increases with increasing f-f superexchange interaction parameters, and the intensity of the ⁴I_13/2→⁴I_15/2 luminescence band depends on the magnetic field, in conflict with the predictions of J-O theory. This is a new phenomenon and results from spd-f hybridization between the occupied Er 4f orbital and the valence states of Gd₃Ga₄FeO₁₂: Yb³⁺, Er³⁺. The approach adopted here can provide further understanding of the f-f superexchange interaction and Zeeman splitting of the 4fⁿ configuration. In addition, the high sensitivity of intra-4f-shell PL to a magnetic field means that these materials are potentially applicable in remote magnetic sensing, magnetic field detection, and magnetic guidance systems.