Magnetic Properties of FeBO₃ in the Low-Spin State

Changes in the magnetic properties of FeBO₃ single crystals in the course of spin crossover occurring with increasing pressure up to 63 GPa are studied both experimentally and theoretically. Simultaneous measurements of the nuclear diffraction and nuclear forward scattering spectra make it possible to detect the antiferromagnetic high-spin state at low pressures up to 48 GPa and to reveal the antiferromagnetic state within the range characterized by the coexistence of Fe³⁺ ions in the high-spin and low-spin states, which occurs at pressures from 48 to 54 GPa, where the hysteresis takes place. Above 58 GPa, only the low-spin state is observed, whereas the magnetic order is absent down to 9 K. An analysis of changes in the exchange interactions manifesting themselves at spin crossover suggests that there exist competing ferromagnetic and antiferromagnetic contributions nearly compensating each other. A possible Néel temperature in the low-spin state does not exceed 7 K; thus, an ordered state cannot be observed in our experiments.

1 INTRODUCTION

With an increase in the pressure in the range of P_c = 50–70 GPa, a large number of iron oxides with Fe³⁺ and Fe²⁺ ions exhibit a spin crossover between the high-spin (HS) and low-spin (LS) states [1–8]. At atmospheric pressure, most of these oxides are Mott insulators with the antiferromagnetic (AFM) order in the HS state [9]. Above P_c, the crystals with Fe²⁺ ions have spin S = 0 and are nonmagnetic. The crystals with Fe³⁺ ions are characterized by S = 1/2 in the LS state above P_c and could be magnetically ordered. Magnetic properties at high pressures are usually measured by laboratory Mössbauer spectroscopy or using synchrotron radiation (nuclear forward scattering, NFS). For the HS state of FeBO₃, NFS spectra demonstrate an increase in the Néel temperature from 350 K at atmospheric pressure up to 600 K at P = 47 GPa [10].

Magnetic collapse and a sharp decrease in the energy of the optical absorption edge were observed at the crossover point P_c = 47 GPa [11]. In the LS state above P_c, the temperature dependence of NFS spectra were interpreted in [10] as a manifestation of the AFM order below 50 K. Note that the symmetry of the FeBO₃ crystal does not change in the case of spin crossover, but the unit cell volume and lattice parameters change abruptly owing to the first order isostructural phase transition. Such stepwise changes are caused by a noticeable (up to 10%) difference in the ionic radii of the HS and LS ions. Recent careful measurements of the X-ray diffraction and Mössbauer spectra also confirmed an isostructural nature of spin crossover in FeBO₃ [12] with the initial crystal structure retained at least up to 105 GPa. A transition from the $\bar {R}3c$ structure to ${\kern 1pt} C2{\text{/}}c{\kern 1pt}$ was detected at 106 GPa [12]. The magnetic properties of the LS state remain unknown. Two points in the phase diagram [10] corresponding to T_N = 50 K belong to the hysteresis range where the HS and LS states coexist; therefore, the magnetic order can be induced by the presence of HS states. In [12], the coexistence of Fe³⁺ ions in the HS and LS states was revealed up to 140 GPa in the crystals with T_N = 60 K. Moreover, in the theoretical work [13], the ferromagnetic (FM) exchange interactions possibly arising in the low-spin state were discussed. Thus, a change in the type of magnetic order is possible.

In this paper, we describe the results of the magnetic structure measurements for FeBO₃ performed at the DESY synchrotron using the original technique proposed by I. Sergeev. Details of the experiment can be found in [14]. The time-of-flight NFS and nuclear diffraction spectra for the (111) and (222) reflections were measured using natural (for NFS) and isotopically enriched (for nuclear diffraction) ⁵⁷FeBO₃ crystals in the pressure range of 0–70 GPa at temperatures of 9–300 K. The samples were placed within a high-pressure diamond anvil cell alongside a ruby chip and Ne as the pressure-transmitting medium, using beryllium gaskets. The high-pressure cell was placed in a cryostat. A low (∼50 Oe) applied magnetic field was used to magnetize the sample along the direction of the X-ray beam. Under such conditions, the time-of-flight NFS and nuclear diffraction spectra differ qualitatively for FM and AFM orderings [15]. While the (222) reflection exists for any magnetic structures, the (111) reflection is characteristic only of the AFM state. The recorded spectra were analyzed using the CONUSS software package, which takes into account the chosen geometry of the nuclear diffraction experiment [16]. Thus, the proposed method allows reliable and unambiguous conclusions.

2 EXPERIMENTAL RESULTS

The NFS and nuclear diffraction spectra of FeBO₃ at room temperature and atmospheric pressure are shown in Fig. 1. Their form suggests the presence of a long-range AFM order. The parameters of the hyperfine structure deduced from the NFS and nuclear diffraction data are close to each other. The found hyperfine field of 43.5 T agrees with the earlier data [17]. In the diffraction spectra, we can see that the quantum beatings for the (222) and (111) directions practically coincide. This is possible owing to the period of the magnetic lattice [18]. Since the (222) crystallographic direction includes all quantum beatings, only the data corresponding to this particular direction are considered below.

Fig. 1.

(Color online) (a) Nuclear forward scattering spectra and (b) nuclear diffraction spectra for the FeBO₃ single crystal at room temperature and atmospheric pressure for the (111) and (222) reflections. Solid lines illustrate the results of fitting.

Changes in the shape of the spectra at room temperature caused by an increase in the applied hydrostatic pressure are illustrated in Fig. 2. They are rather trivial up to a pressure of 48 GPa; in this pressure range, the hyperfine field increases monotonically from 34.5 to 48.6 T. These values are close to those obtained earlier for the HS state [10]. The monotonic increase in the hyperfine field is related to an increase in the overlap of the electron wave functions. When the critical pressure P_c = 48 GPa is achieved (the critical pressure reported in [10] is 46.5 GPa; small differences are obviously associated with an error in measuring the pressure in diamond anvil cells at about 3 GPa), we observe a sharp change in the form of the spectra. At P > P_c, only the low-frequency beatings caused by quadrupole nuclear transitions remain. Such changes imply spin crossover between the HS and LS states of Fe³⁺ ions and the suppression of the magnetic order at room temperature.

Fig. 2.

(Color online) (a) Nuclear forward scattering spectra and (b) nuclear diffraction spectra for FeBO₃ at room temperature and different pressures. Solid lines illustrate the results of fitting of the spectra.

Note that the cause of crossover between the configurations, |⁶A₁〉 → |²T₂〉, is an increase in the cubic component of the energy of the crystal field 10Dq in accordance with the Tanabe–Sugano diagrams for d ⁵ ions. Since the ion spin in the low-spin state is S = 1/2, magnetic ordering is also possible above P_c. Under the simplest assumption that the interatomic exchange interaction does not change with pressure, the temperature T_N should decrease proportionally to a factor of S(S + 1), which is equal to 35/4 and 3/4 for the HS and LS states, respectively; i.e., the AFM ordering can be expected in the low-spin state below 30 K. An interesting situation was observed at a pressure of 51.6 GPa corresponding to a two-phase region within the hysteresis loop. Before cooling at a pressure of 47.9 GPa, the sample exhibited magnetic beats. Upon cooling down to 18 K, the magnetic beats disappear at the center of the sample, but remain at its edge. These two points are visible in the case of the (222) reflection. One point corresponds to the HS phase, and the other one, to the LS phase; the distance between them is about 3 mm. At pressures above 53.9 GPa, the time-of-flight spectra exhibit only quadrupole beats corresponding to the low-spin state (as seen from the diffraction data). Typical time-dependent NFS and nuclear diffraction spectra at a pressure of 60.4 GPa and a temperature of 9 K are shown in Fig. 3. We can see that there is no magnetic order in the low-spin state above 9 K.

Fig. 3.

(Color online) (a) Nuclear diffraction spectra and (b) nuclear forward scattering spectra at T = 9 K and P = 60.4 GPa.

The analysis of these spectra demonstrates the existence of two states of iron in the paramagnetic state (in contrast to usual conditions) with the quadrupole splitting ΔE_Q = 1.85 and 2.25 mm/s. Note the atomic fractions of these Fe³⁺ LS states are the same. The two found states of iron agree with the observations reported in [12]. Although the atomic fractions of these states in [12] are in a ratio of about 1 : 2, we can unambiguously argue that two different characteristic local distortions on iron nuclei in the crystal are present. This is due to the presence of an uncompensated electron in the Fe³⁺ shell in the LS state. At the same time, our experiment does not demonstrate any signs of iron in the high-spin state, in contrast to the data of [12]. Some difference in the values of ΔE_Q, in our opinion, cannot be related to the coexistence of the LS and HS phases since the corresponding values are quite close to each other.

3 THEORETICAL RESULTS

For metals and alloys, the parameters of the effective Heisenberg Hamiltonian can be calculated using the density functional theory (DFT) [19]. For FeBO₃ and other Mott insulators, this approach is inapplicable because it does not take into account strong electron correlations. To generalize the DFT approach for strongly correlated systems, an effective s–d model [20] was recently proposed, but it is inapplicable to FeBO₃. Therefore, we use the many-electron approach [21], which, for the case of spin crossovers, must be generalized taking into account not only the ground state but also the excited states, which change places as a result of crossover. Such a generalization of the calculation of the superexchange interaction in Mott insulators taking into account various multielectron terms of a cation was performed in our works [13, 22]. In Fig. 4, we show the local environment of the central cation belonging the A sublattice. It is seen that the largest contribution to the Fe(A)–Fe(B) superexchange is due to the hopping via the oxygen anion from the BO₃ group having the bond angle close to 120°. The 180°-type interaction along the vertical axis in Fig. 4 arises for the next-nearest neighbor cations.

Fig. 4.

(Color online) Local environment of the central cation from the A sublattice (brown circle) and six nearest neighbors from the B sublattice (blue circles). Oxygen t-riangles in BO₃ groups are shown in blue. A large contribution to the formation of a chemical bond comes from BO₃ groups. Boron ions are shown by empty circles.

In the framework of the Hubbard model, the superexchange interaction arises in the second order of the perturbation theory with respect to the parameter t/U ≪ 1 [23, 24]. This mechanism does not contradict the traditional picture of cation–ligand–cation hoppings, since the parameter t of the effective Hubbard model is expressed in terms of the p−d hopping parameter t_pd. The exchange interaction arises as a consequence of the creation and subsequent annihilation of virtual electron‒hole pairs. At the initial time, there are two neighboring cations with spin S = 1/2 in the d ¹ configuration. The virtual hopping of an electron from atom 1 to atom 2 leads to the formation of a hole in atom 1 in the d ⁰ configuration and an electron in atom 2 in the d ² configuration. The reverse hopping destroys the virtual electron‒hole pair and restores the original S = 1/2 configurations d ¹, coupled by the exchange integral J = 4t ²/U. This well-known mechanism for inducing the superexchange interaction in the Hubbard model is presented here to explain many-electron calculations of the exchange interaction taking into account the excited terms [13, 22]. In the many-electron case, we consider two ions in electrically neutral configurations d ⁿ (d ⁵ for Fe³⁺), taking into account all the necessary terms (in our case, HS and LS). The interatomic hopping leads to the formation of hole states in the d ⁿ⁻¹ configuration (here, d ⁴) and electron states corresponding to the d ⁿ⁺¹ configuration (here, d ⁶). Using the projection properties of the Hubbard operators constructed on the basis of many-electron terms of the electrically neutral, hole, and electron configurations, we generalize the method for calculating the superexchange interaction for the Hubbard model [25] to the case of arbitrary many-electron configurations [13]. This approach allowed us to write the total exchange interaction as a sum of partial contributions from various terms, not only from the main one but also from excited ones. Thus, we can distinguish the contributions from the HS and LS terms and trace the changes in the exchange interaction in the course of spin crossover [22].

All possible electron‒hole virtual excitations from the HS and LS configurations of Fe³⁺ ions are shown in Fig. 5. Electrically neutral terms are denoted by the symbol n₀, the terms corresponding to the formation of an additional electron are specified by the symbol e, and the terms with the removal of an electron and the formation of a hole are marked by the symbol h. These terms differ for the case of high-spin and low-spin configurations of an electrically neutral ion.

Fig. 5.

(Color online) Diagram illustrating the formation of the Fe−O−Fe superexchange: (a) the cross marks the filled HS ground state at $P < {{P}_{{\text{c}}}}$. The electron and hole excitations characterized by nonzero matrix elements are shown in blue and green, respectively. These excitations form the AFM exchange between the nearest neighbors $J_{{^{5}{{E}^{5}}{{T}_{2}}}}^{{{\text{AFM}}}}$ (120°-type exchange) and between the next-nearest neighbors $J_{{^{5}{{E}^{5}}E}}^{{{\text{AFM}}}}$ (180°-type exchange). The dashed and solid lines denote ${{t}_{{2g}}}$ and ${{e}_{g}}$ virtual electrons and holes. (b) The cross marks the filled LS state at high pressures $P > {{P}_{{\text{c}}}}$. The FM contributions $J_{{^{3}{{T}_{1}}^{1}{{T}_{1}}}}^{{{\text{FM}}}}{\kern 1pt} \left( {120^\circ } \right)$ and $J_{{^{1}{{T}_{2}}^{3}{{T}_{1}}}}^{{{\text{FM}}}}{\kern 1pt} \left( {120^\circ } \right)$ are indicated in red and yellow, respectively; the AFM contributions $J_{{^{3}{{T}_{1}}^{3}{{T}_{2}}}}^{{{\text{AFM}}}}{\kern 1pt} \left( {120^\circ } \right)$ and $J_{{^{1}{{T}_{2}}^{1}{{T}_{1}}}}^{{{\text{AFM}}}}{\kern 1pt} \left( {120^\circ } \right)$ are indicated in blue and green, respectively.

The superexchange Hamiltonian can be written in terms of the Heisenberg model

$${{\hat {H}}_{s}} = - \sum\limits_{i \ne j} J_{{ij}}^{{{\text{tot}}}}\left( {{{{\hat {S}}}_{{i{{n}_{0}}}}}{{{\hat {S}}}_{{j{{n}_{0}}}}} - \frac{1}{4}\hat {n}_{{i{{n}_{0}}}}^{{\left( h \right)}}\hat {n}_{{j{{n}_{0}}}}^{{\left( e \right)}}} \right)$$

(1)

with the exchange integral

$$J_{{ij}}^{{{\text{tot}}}} = \sum\limits_{he} \frac{{{{J}_{{ij}}}\left( {{{n}_{0}}h,{{n}_{0}}e} \right)}}{{\left( {2{{S}_{h}} + 1} \right)\left( {2{{S}_{{{{n}_{0}}}}} + 1} \right)}}.$$

(2)

Expression (2) involves the sum of all possible virtual electron−hole processes of creation and annihilation (called exchange loops in [13]). The sign of each partial contribution is easily determined by the following rule: if the spins of the hole and electron terms coincide S_h = S_e, the AFM coupling occurs. If the spins of the electron and hole terms differ, S_h = S_e ± 1, the coupling is FM. The full exchange energy is the sum of the AFM and FM contributions. The same rule for determining the sign of superexchange was obtained in [21] for the exchange interaction of ions in the ground state. The magnitude of each contribution has the standard form corresponding to the second order of perturbation theory, i.e., ∼t²/U_eff, where the effective Hubbard parameter for each exchange loop is given by the energies of the involved terms as E(d ⁿ⁻¹) + E(d ⁿ⁺¹) − 2E(d ⁿ). For example, the energies of the low-spin terms involved in the formation of the AFM contribution $J_{{^{3}{{T}_{1}}^{3}{{T}_{2}}}}^{{{\text{AFM}}}}(120^\circ )$ can be written as

$$E({{d}^{4}}{{,}^{3}}{{T}_{1}}) = {{E}_{{\text{C}}}}({{d}^{4}}) - 3{{J}_{{\text{H}}}} - 16Dq,$$

(3)

$$E({{d}^{5}}{{,}^{2}}{{T}_{2}}) = {{E}_{{\text{C}}}}({{d}^{5}}) - 4{{J}_{{\text{H}}}} - 20Dq,$$

(4)

$$E({{d}^{6}}{{,}^{3}}{{T}_{2}}) = {{E}_{{\text{C}}}}({{d}^{6}}) - 7{{J}_{{\text{H}}}} - 14Dq.$$

(5)

Here, ${{E}_{{\text{C}}}}({{d}^{n}})$ is the spin-independent part of the ionic Coulomb energy and $10Dq$ is the crystal field energy corresponding to the cubic field. The Hund’s coupling constant J_H reduces the energy of each pair of electrons with parallel spins. The Hubbard parameter is $U = {{E}_{{\text{C}}}}({{d}^{4}}) + {{E}_{{\text{C}}}}({{d}^{6}}) - 2{{E}_{{\text{C}}}}({{d}^{5}})$. At low pressures in the HS state, both contributions ${{J}_{{^{5}{{E}^{5}}E}}}(180^\circ )$ and ${{J}_{{^{5}{{E}^{5}}{{T}_{2}}}}}(120^\circ )$ have the AFM sign, since ${{S}_{e}} = {{S}_{h}} = {\kern 1pt} 3{\kern 1pt} /{\kern 1pt} 2{\kern 1pt}$. The crystal field increases with the pressure and reaches the value 10Dq = 3J_H at the critical pressure P_c, at which spin crossover occurs and the HS ground state term ${{{\text{|}}}^{6}}{{A}_{1}}\rangle$ of Fe³⁺ ions is changed by the ${{{\text{|}}}^{2}}{{T}_{2}}\rangle$ LS term. In the LS state, the 180° e_g bond disappears, and the 120° exchange with six nearest neighbors also undergoes some changes. In Fig 5b, we illustrate four  contributions to the exchange interaction: $J_{{^{{^{3}{{T}_{1}}^{3}{{T}_{2}}}}}}^{{{\text{AFM}}}}(120^\circ )$, $J_{{^{{^{3}{{T}_{1}}^{1}{{T}_{1}}}}}}^{{{\text{FM}}}}(120^\circ )$, $J_{{^{{^{2}{{T}_{1}}^{3}{{T}_{2}}}}}}^{{{\text{FM}}}}(120^\circ )$, and $J_{{^{1}{{T}_{2}}^{1}{{T}_{1}}}}^{{{\text{AFM}}}}(120^\circ )$, two of which have the FM sign and the other two are antiferromagnetic. All these contributions are formed via the same overlap of ${{t}_{{2g}}}$ and ${{e}_{g}}$ c-orresponding to the hopping integral $t$. The total exchange integral in the LS state is ${{J}_{{{\text{tot}}}}}\left( {120^\circ } \right) = \left( {J_{{^{{^{3}{{T}_{1}}^{3}{{T}_{2}}}}}}^{{{\text{AFM}}}} - J_{{^{3}{{T}_{1}}^{1}{{T}_{1}}}}^{{{\text{FM}}}}} \right) + \left( {J_{{^{{^{1}{{T}_{2}}^{1}{{T}_{1}}}}}}^{{{\text{AFM}}}} - J_{{^{{^{1}{{T}_{2}}^{3}{{T}_{2}}}}}}^{{{\text{FM}}}}} \right)$. The partial contributions can be written as

$$\begin{gathered} J_{{^{{^{3}{{T}_{1}}^{3}{{T}_{2}}}}}}^{{{\text{AFM}}}} = 4{{t}^{2}}{\kern 1pt} {\text{/}}\Delta \left( {^{3}{{T}_{1}}^{3}{{T}_{2}}} \right), \\ {\text{where}}\quad \Delta \left( {^{3}{{T}_{1}}^{3}{{T}_{2}}} \right) = U + 10Dq - 2{{J}_{{\text{H}}}}, \\ \end{gathered} $$

(6)

$$\begin{gathered} J_{{^{3}{{T}_{1}}^{3}{{T}_{2}}}}^{{{\text{AFM}}}} = 4{{t}^{2}}{\text{/}}\Delta \left( {^{3}{{T}_{1}}^{3}{{T}_{2}}} \right), \\ {\text{where}}\quad \Delta \left( {^{3}{{T}_{1}}^{3}{{T}_{2}}} \right) = U + 10Dq - {{J}_{{\text{H}}}}, \\ \end{gathered} $$

(7)

$$\begin{gathered} J_{{^{1}{{T}_{2}}^{3}{{T}_{2}}}}^{{{\text{FM}}}} = 4{{t}^{2}}{\text{/}}\Delta \left( {^{1}{{T}_{2}}^{3}{{T}_{2}}} \right), \\ {\text{where}}\quad \Delta \left( {^{3}{{T}_{1}}^{3}{{T}_{2}}} \right) = U + 10Dq - {{J}_{{\text{H}}}}, \\ \end{gathered} $$

(8)

$$\begin{gathered} J_{{^{1}{{T}_{2}}^{1}{{T}_{1}}}}^{{{\text{AFM}}}} = 4{{t}^{2}}{\kern 1pt} {\text{/}}\Delta \left( {^{1}{{T}_{2}}^{1}{{T}_{1}}} \right), \\ {\text{where}}\quad \Delta \left( {^{3}{{T}_{1}}^{3}{{T}_{2}}} \right) = U + 10Dq. \\ \end{gathered} $$

(9)

The sum of contributions (6)–(9) is the total exchange integral in the low-spin state. At the crossover point, where 10Dq = 3J_H, this can be written in the compact form

$${{J}_{{{\text{LS}}}}} = \frac{{4{{t}^{2}}J_{{\text{H}}}^{2}}}{{(3(U + {{J}_{{\text{H}}}})(U + 2{{J}_{{\text{H}}}})(U + 3{{J}_{{\text{H}}}}))}}.$$

(10)

Similarly, we can describe the superexchange interaction for the high-spin state at the crossover point

$${{J}_{{{\text{HS}}}}} = \frac{{2{{t}^{2}}}}{{(15(U + {{J}_{{\text{H}}}}))}},$$

(11)

$$\frac{{{{J}_{{{\text{LS}}}}}}}{{{{J}_{{{\text{HS}}}}}}}\left( {P = {{P}_{{\text{c}}}}} \right) = \frac{{10J_{{\text{H}}}^{2}}}{{(U + 2{{J}_{{\text{H}}}})(U + 3{{J}_{{\text{H}}}})}}.$$

(12)

Typical parameters for FeBO₃ found from comparison with experimental data [10] are U = 4.2 eV and J_H = 0.7 eV. With these parameters, ratio (12) equals 0.14. In the mean field approximation, we can write T_N = JzS(S + 1)/3 with S = 5/2 for the HS state and S = 1/2 for the LS state; the ratio T_N(LS)/T_N(HS) at P = P_c is 0.012. The Néel temperature before crossover in the HS state is 600 K [10]. For the Néel temperature in the LS state, we find T_N(P > P_c) =7.2 K. As known, in the mean-field approximation, the Néel temperature is a factor of 2–3 higher than the experimental value; i.e., a more realistic estimate of the possible temperature of magnetic ordering in the low-spin state is 2–3 K.

4 CONCLUSIONS

The nuclear diffraction method used in this work allows one to distinguish the FM and AFM orderings, in contrast to Mössbauer and nuclear forward scattering measurements. The nuclear diffraction spectra measured for the (222) reflection take place both for the FM and for the AFM ordering. At the same time, the spectra for the (111) reflection are typical only of the AFM ordering. The observed coexistence of the high-spin and low-spin states in the pressure range of 48–54 GPa is due to the two-phase nature of the system within the hysteresis loop characteristic of a first-order phase transition. The above theoretical estimates explain the absence of magnetic ordering in the low-spin state at the measurements above 9 K. The sharp decrease in the exchange interaction above the crossover point is due to the significant compensation of the AFM and FM contributions. Note that, in the high-spin state, there are no FM contributions to the exchange interaction.