Superspin glass phase and hierarchy of interactions in multiferroic PbFe_1/2Sb_1/2O₃: an analog of ferroelectric relaxors?

Figure 1 presents the temperature dependence of the dielectric permittivity and ac magnetic susceptibilitiy measured at different frequencies. Dielectric permittivity shows one distinct peak at 188–190 K along with a second smaller feature at higher temperature (figure 1(a)). The temperature position of the former peak is slightly frequency-dependent and is related to a paraelectric–ferroelectric phase transition. The ferroelectric origin of this peak is confirmed by dielectric hysteresis loops exhibiting typical ferroelectric behavior with saturation polarization up to 5–6 μC cm⁻² (see inset in figure 1(b)). Remanent polarization appears below the dielectric peak temperature. Note that, as already mentioned above, x-ray diffraction studies of PFS showed that this material experiences a structural transformation to the tetragonal I4/m phase in the range 210–220 K [32]. However, the latter result contradicts our present data as I4/m is a non-polar phase, whereas we found that the low-temperature phase is polar. At the same time, the dielectric anomaly that we observed in PFS can be preliminarily explained by a cubic-to-tetragonal phase transition, while the actual symmetry of this tetragonal phase still needs to be determined.

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The data in figures 1(c) and (d) show huge peaks in both real, ${{\chi }^{\prime }}$, and imaginary, $\chi$", parts of the ac magnetic susceptibility at ${{T}_{{\rm m}}}\;\approx$ 150–160 K, which strongly shifts to lower temperatures as the frequency decreases. This behavior is similar to that of ac PMN dielectric permittivity [28]. Such unusually large high-temperature magnetic dispersion requires detailed discussion.

Besides the main peak, the magnetic susceptibility has a second magnetic anomaly at 30–32 K (see figures 1(c) and (d)), and we found it to be field- and frequency-independent. Therefore, this peak in the magnetic susceptibility seems to correspond to a normal phase transition. Below, we will show that it is of AFM nature. This means that PFS belongs to the multiferroic family. Although the multiferroic behavior of PFS is also very interesting, hereafter we will focus only on the magnetic relaxation properties.

The unusually strong frequency dispersion of the magnetic susceptibility discovered experimentally in PFS suggests that its spins are involved in thermally activated relaxation processes. The main characteristic of this relaxation, the mean relaxation time $\tau$, can be obtained from the standard relation between $\tau$, the temperature of magnetic susceptibility maximum T_m and the glassy phase transition temperature T_g [33]:

$$\begin{eqnarray}&&\tau ={{\tau }_{0}}{{\delta }^{-z\nu }},\end{eqnarray} \tag{ 1 }$$

where $\delta ={{T}_{{\rm m}}}/{{T}_{{\rm g}}}-1$ is the reduced temperature, $\nu$ is the critical exponent of the correlation length and z is the dynamic exponent. From the fit of equation (1) to the experimentally measured relaxation time (see inset in figure 1(c)) we obtained physically reasonable values: ${{\tau }_{0}}=2\cdot {{10}^{-12}}$ s, T_g = 140 K and the critical exponent $z\nu =10$. A tentative fit of the Arrhenius law for $\tau ({{T}_{{\rm m}}})$ to the experiment (also known as the Neel–Brown law obtained for weakly interacting SS magnetic moments [5, 11]) led to unphysical values of ${{\tau }_{0}}\sim {{10}^{-54}}\;{\rm s}$ and ${{E}_{{\rm act}}}\sim 18000\;{\rm K}$. However, we obtain reasonable agreement with the experiment by employing the Vogel–Fulcher law [34] which is common for the description of spin glass dynamics:

$$\begin{eqnarray}&&\tau ={{\tau }_{0}}{\rm exp} \left[ \frac{{{E}_{{\rm act}}}}{{{T}_{{\rm m}}}-{{T}_{0}}} \right].\end{eqnarray} \tag{ 2 }$$

Here T₀ is a characteristic temperature, which is close to the paramagnet–glass phase transition temperature determined as the temperature of the ZFC dc susceptibility maximum. Our experimental data were satisfactorily fitted by equation (2) with the following parameters: ${{T}_{0}}=130\;{\rm K}$, ${{\tau }_{0}}=2.8\cdot {{10}^{-10}}\;{\rm s}$ and ${{E}_{{\rm act}}}=467\;{\rm K}$.

We now turn to dc measurements. The results of our measurements of the ZFC and FC dc magnetic susceptibilities of PFS are reported in figure 2(a). One can see that the magnetic susceptibility strongly depends on the temperature and applied field. Moreover, on cooling, it exhibits an abrupt increase at approximately 250 K and then, in the temperature region between 100 and 150 K, the ZFC magnetic susceptibility (dashed lines) shows a pronounced maximum. Below this maximum, we observe that FC and ZFC magnetic susceptibilities are very different, which indicates the appearance of the irreversibility, a peculiar feature of glasses; see, e.g. [33]. Finally, another smaller maximum is observed at 30–32 K in both ZFC magnetic susceptibility, ${{\chi }_{{\rm ZFC}}}(T)$, and FC magnetic susceptibility, ${{\chi }_{{\rm FC}}}(T)$. We notice a dramatic difference in our results for PFS from those obtained for PFN (see the lowest group of curves in figure 2(a)). One can see that all curves corresponding to different fields, 20 $\lt \;H\;\lt$ 5000 Oe, for PFN, merge into one master curve which means that its susceptibility is almost field-independent, except for the 10–12 K region at the spin–glass phase transition (see also [19]), and the magnitude of this susceptibility is rather low. Contrary to this, the magnetic susceptibility of PFS, at low fields, is large and has significant magnetic field dependence.

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All these data show the existence in PFS of thermally activated frustrated SSs, inherent to superparamagnets. Note that the superparamagnetic phase can actually be considered as the so-called Griffiths or paraglass phase, which can emerge in disordered magnets [35] or ferroelectrics [36]. In this phase, the weakly interacting SSs (which appear in finite magnetic regions as clusters) generate effects akin to those in ferroelectric relaxors (like strong frequency dispersion of magnetic susceptibility).

Let us discuss the field dependence of PFS magnetization presented in figure 2(b) for several selected temperatures. To understand these data, we approximated them by the sum of two Langevin functions L(x):

$$\begin{eqnarray}\begin{array}{rcl} M & = & M_{0}^{({\rm p})}L\left( \frac{{{\mu }_{{\rm fe}}}H}{{{k}_{{\rm B}}}T} \right)+M_{0}^{({\rm ss})}L\left( \frac{mH}{{{k}_{{\rm B}}}T} \right), \\ L(x) & = & {\rm coth} x-\frac{1}{x}. \\ \end{array}\end{eqnarray} \tag{ 3 }$$

The first term in equation (3) describes the paramagnetic contribution of Fe³⁺ ions with the magnetic moment ${{\mu }_{{\rm fe}}}\approx 5.9\;{{\mu }_{{\rm B}}}$. The second term is the contribution of SSs, which we believe to be responsible for the activation of the strong relaxation processes (see below). Here, m is the SS magnetic moment. The weights $M_{0}^{({\rm p})}$ and $M_{0}^{({\rm ss})}$ of the individual and SS contributions are temperature-independent in the considered temperature region. We fitted function (3) to the experimental data shown in figure 2(b) and obtained, at 150–175 K, the SS magnetic moment $m\approx {{10}^{4}}\;{{\mu }_{{\rm B}}}$. Thus, the SSs are extremely large, while, as we observed, their weight does not exceed 5.5 emu/mol at 175 K. This feature is a difference between magnetic and ferroelectric relaxors. Indeed, only a large magnetic moment m of SSs can have the product mH of the order of thermal energy k_BT that makes this SS particularly distinct from single Fe magnetic moments, for which μ_FeH is much smaller than k_BT. Below, we will describe some microscopic mechanisms supporting the emergence of such SSs in PFS due to special local chemical ordering of Fe ions.

The SS magnetic moment m was found to decrease on heating from 175 to 250 K, but it does not vanish completely even at room temperature. This magnetic moment also varies from sample to sample due to the influence of synthesis conditions on the chemical ordering of Fe and Sb ions.

Note also that the SSʼs contribution to the magnetization depends on temperature at low fields only. This is because the high field completely aligns the magnetic moments of SSs, making their thermal motion impossible. This is seen in figure 2(b) (main panel), which exhibits a nonlinearity of magnetization owing to the SSs at low fields only. Consequently, magnetic susceptibility $\chi =M/H$ has a strong low-field contribution of the SSs. Below, we will illustrate this fact theoretically.

To elucidate the question of how the atomic spins in the above magnetic relaxor form the SSs, we need to investigate the microscopic structure of PFS. Having this goal in mind, we measured the EPR spectrum. These measurements were performed at 9.407 GHz and in a temperature range from 4 to 300 K. We obtain that, at T $\gt$ 250 K, the EPR spectrum of PFS contains only one resonance line, which has a Lorenzian shape (figure 3). We attribute this to the paramagnetic resonance of individual Fe³⁺ ($3{{d}^{5}},S=5/2,L=0$) ions. As the temperature decreases below 250 K, another line shows up at low fields. This line shifts towards lower resonance fields (in accordance with ⁸ ). Fortunately, its intensity, as a function of temperature, correlates well with the magnitude of the low-field dc ZFC susceptibility (see the inset in figure 3). Thus, it is reasonable to ascribe this low-field line to the magnetic resonance of SSs, which starts to increase in size and number below 250 K.

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It is interesting that the first EPR line, obtained at 3.34 kOe, which we attributed to the Fe³⁺ individual ions, broadens critically and disappears completely on cooling at ${{T}_{{\rm N}}}=30$ K. This fact can be explained by a spontaneous antiparallel alignment of the Fe³⁺ spins below this temperature. This suggestion agrees with the independence of the magnetic susceptibilities peak at 30 K on the magnetic field magnitude and frequency (figures 1 and 2).

One can see from figure 2(a) that both the peak temperature T_m(H) of the ZFC susceptibility and the irreversibility onset temperature (i.e. the temperature around the ZFC susceptibility maximum T_m(H), where FC and ZFC susceptibilities begin to be significantly different) strongly decrease on the field increase. Surprisingly, our data can be well fitted with a so-called de Almeida–Thouless (AT)-type formula [37] of the H-T phase diagram: $H({{T}_{{\rm SG}}})=A{{\left[ 1-\frac{{{T}_{{\rm SG}}}(H)}{{{T}_{{\rm SG}}}(0)} \right]}^{3/2}}$, where we assumed that T_SG(H) is equal to the SS glass transition temperature defined by the onset of strong irreversibility in the temperature variation of magnetization as suggested, for example, in [33]. Initially [37], the AT line was introduced theoretically for systems such as ergodic spin glass, as the line of stability of the replica symmetric solution, in the framework of the Sherrington–Kirkpatric model with an infinite range of random interactions between the constituting spins in a spin glass. Further Monte Carlo simulations [38] showed that this line also occurs in vector spin glasses, which is the case for our system.

In order to eliminate possible ambiguity in the determination of T_SG(H), we analyze the temperature derivative of the magnetization, ${\rm d}M(T)/{\rm d}T$, for both ZFC and FC data. T_SG(H) was determined as the temperature where the derivative ${\rm d}{{M}_{{\rm ZFC}}}(T)/{\rm d}T$ starts to deviate markedly from that of M_FC(T). The fit of the above AT-type formula to our experimental data is shown in figure 4 by the solid line. One can see that this fit is nearly perfect and signifies the stability of the superparamagnetic phase above the phase boundary line, which, at low fields, has some features of ferroelectric relaxors and can thus be referred to as the magnetic relaxor phase. This essentially differs from that in PFN, where a similar AT-type formula was used to describe the stability of the AFM phase against the spin-glass state, which appears as a re-entrant phase at $T\ll {{T}_{{\rm N}}}$ [19].

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The H–T phase diagram presented in figure 4 shows a remarkable transformation, on cooling, of the superparamagnetic phase into the SS glass phase. On further cooling, the majority of the Fe³⁺ spins, uncoupled into SSs, undergo AFM long-range ordering with the Neel temperature T_N = 30 K. Obviously, the state at $T\lt {{T}_{{\rm N}}}$ can be considered as a mixed AFM–SS glass state.

Here we calculate the complex magnetic susceptibility (in the sense of the Casimir–du Pré approximation [39]) ${{\chi }^{*}}=$ ${{M}^{*}}/{{H}_{0}}$, where M* is magnetization and H₀ is the external magnetic field, by means of deriving an original phenomenology valid for the new class of materials, magnetic relaxors. Note that, in its turn, the magnetization can be expressed via the reduced (normalized) dynamical magnetization ${{m}^{*}}\;=$ ${{M}^{*}}/{{M}_{0}}$, where M₀ is the saturated magnetization (see [40] and references therein). This magnetization is dynamical, i.e. dependent on the frequency $\omega$ of the external magnetic field, revealing the existence of relaxing constituents (here we consider the SSs arising from extended AS defects, which form chemically disordered regions in PFS, see below) with different relaxation times τ in the system. In fact, the distribution of the relaxation times is a consequence of the randomness of the positions and fields at different SSs [41]. This implies that, to describe our experiment adequately, we need to employ a random field approach.

The idea of the random field model [42] is to express the distribution function of the relaxation times τ via the random fields H acting among the SSs, due to the spatial (different sizes of SS clusters as well as presence or absence of a cluster at a certain spatial point) and thermal (different orientation of the magnetic moment in an SS cluster) disorder. In principle, the distribution function of these random fields, $f(H,{{m}^{*}})$, can be expressed via m* self-consistently in a manner that m* can be presented as the average, with this distribution function⁹ , of a single Ising-like SS magnetic moment μ (see [40] and references therein):

$$\begin{eqnarray}&&{{m}^{*}}=\int _{-\infty }^{\infty }{\rm tanh} \left[ \frac{\mu (H+{{H}_{0}})}{{{k}_{{\rm B}}}T} \right]\frac{f(H,{{m}^{*}})}{1+i\omega \tau (H)}{\rm d}H.\end{eqnarray} \tag{ 4 }$$

It can be shown [40] that the magnetic dipole interaction between SS clusters generates the Lorentzian distribution function of random magnetic fields

$$\begin{eqnarray}&&\begin{array}{l} f(H,{{m}^{*}})=\frac{\delta }{\pi }\frac{1}{{{\delta }^{2}}+{{(H-\alpha {{m}^{*}})}^{2}}}, \\ \qquad \delta \approx 5.065n{{\mu }_{{\rm B}}},\ \alpha \approx 2.1\delta , \\ \end{array}\end{eqnarray} \tag{ 5 }$$

where n = N/V is the SS concentration and $\tau (H)=\tau {\rm cosh} (2\beta \mu (H+{{H}_{0}}))/{\rm cosh} (\beta \mu (H+{{H}_{0}}))$, $\beta =1/({{k}_{{\rm B}}}T)$, τ is a single SS relaxation time, which has the Arrhenius form [41] (note that this Arhenius form for the individual relaxing constituents does not prevent the emergence of the cooperative Vogel–Fulcher effect in the set of the relaxors, as we will show below)

$$\begin{eqnarray}&&\tau ={{\tau }_{0}}{\rm exp} \left( \frac{{{E}_{{\rm act}}}}{{{k}_{{\rm B}}}T} \right)\end{eqnarray} \tag{ 6 }$$

with E_act being the activation energy or energy barrier between different SS cluster orientations. Substituting these equations into equation (4) and dividing the result by H₀, we obtain the desired complex susceptibility.

In figure 5, we fit the high-temperature 'glassy' maximum of the real part of the theoretical susceptibility to the experimental curves taken from figure 1(c). The parameters of the best fit are: ${{E}_{{\rm act}}}/({{\mu }_{{\rm B}}}\delta )$ = 15 and ${{\tau }_{0}}=2.05\;\;{{10}^{-10}}\;{\rm s}$. Also, similar to equation (3), we take the magnetic moment $\mu \;\approx$ $8500\;{{\mu }_{{\rm B}}}$ for the high-temperature peak. This corresponds to the SS concentration $n\approx 9.46\;\;{{10}^{17}}\;{\rm c}{{{\rm m}}^{-3}}$. It can be seen that the theoretical curves coincide pretty well with the experimental ones for $T\gt 120\;{\rm K}$, i.e. near the maximum $T={{T}_{{\rm m}}}$ and at the high-temperature 'tail' at $T\gt {{T}_{{\rm m}}}$. The latter tail corresponds to the situation where the interaction between SSs is weaker than their thermal energy so that its details are unimportant. In that case, the decay of magnetic susceptibility is close to the Curie law (generally speaking, the so-called high-temperature expansion is valid; see below), which is well captured by our random field method. The good coincidence between theory and experiment at $120\;{\rm K}\lt T\lt {{T}_{{\rm m}}}$ is related to the fact that in this temperature range the strongest interaction between SS clusters is the magnetic dipole interaction, which we take into account. At $T\lt$ 120 K the theoretical curves decay rapidly as, most probably, SSs of smaller size come into play with stronger short-range exchange interaction between them. At lower temperatures this interaction finally overpowers the relativistic magnetic dipole interaction, giving rise to AFM order at $T\approx 30$ K.

The above means that to achieve the quantitative agreement in the entire temperature range, we need to consider the hierarchy of interactions: the short-range exchange interaction between spins in SSs along with the magnetic dipole and other relativistic interactions between SSs. The next step is to consider the vector (i.e. not Ising) spin model as well as to take into account the real symmetry of an elementary cell of the substance under consideration. Such a procedure requires extensive numerical calculations and will be published elsewhere.

We note here that the above random field method also permits us to describe the less-pronounced low-temperature AFM peak in the $\chi ^{\prime}$. This is because for a high concentration of spins (our estimations show that for the AFM case $\mu =5.9\;{{\mu }_{{\rm B}}}$ so that $n\approx 1.9\;\;{{10}^{21}}\;{\rm c}{{{\rm m}}^{-3}}$) the distribution function of random fields becomes much narrower, giving a $\delta$-function [41], so that our random field method only yields the well-known mean field result. We stress that the μ and n values were obtained from the fit of the formulas derived to the susceptibility maxima temperatures. Note that the large concentration value in the low-temperature ordered phase simply reflects the fact that the spins contributing to the AFM phase are not clusters (in contrast to the SSs) but rather atomic Fe spins. Consideration of the above hierarchy of interactions would permit automatic switching between the concentrations of single spins and SSs, thus describing the experiment quantitatively for all temperatures.

We emphasize again that the very large SS magnetic moment in PFS is a consequence of specialities of magnetic relaxors compared to ferroelectric ones. Indeed, it is hard to imagine such a large ferroelectric reorientable cluster as it would have comparatively large activation energy. In magnets, in spite of the large size of the SS, the potential barrier can still be small (compared to ferroelectric relaxors), as these barriers are due to relativistic interactions, which are much smaller than exchange interactions. This fact also facilitates the dynamics of large SSs, whereas smaller SSs will probably either be destroyed by thermal fluctuations, or be seen at much lower temperatures like those in PFN [19].

The static, $\omega =0$, or quasistatic, $\omega \to 0$, versions of our random field equation (4) permit us to describe the temperature variation of the M(H) curve. Indeed, at high temperatures, where ${{k}_{{\rm B}}}T\gt$ ${{\mu }_{{\rm B}}}\delta$, the slope of the magnetization curve M(H) is small and uniform; i.e. there is only one slope. However, at temperatures below ${{\mu }_{{\rm B}}}\delta$, the slope near H = 0 becomes steeper. The presence of this steep part near H = 0 is actually what has been observed and depicted in the inset in figure 2(b).

Another way to model the partial disorder in the distribution of Fe and Sb ions within the B-sublattice of the perovskite structure is to consider a periodic lattice with a large supercell having different ion distributions within it. This approach is suitable for numerical calculations, based on the n-th order expansions of the system thermodynamic characteristics (such as magnetic susceptibility and specific heat) in a small parameter proportional to the inverse temperature ${{T}^{-1}}$, named high-temperature expansions (HTE).

Six possible types of chemical order PFB0 ... PFB5 exist for a $2\times 2\times 2$ supercell; see figure 3 of [43]. For PFS, the ideal rock-salt structure (PFB0, see figure 6) has the minimal energy. Two other structures are close in energy: the layered PFB5 structure (alternating Fe and Sb planes), and the ferrimagnetic PFB2 structure. Here, we also consider a system with a connected sublattice of anti-site (AS) defects; i.e. an extended defect such as a stacking fault or dislocation. A $2\times 2\times 4$ supercell for the AS structure is shown in figure 6.

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Using the program package presented in [44] for the tenth-order HTE, we have calculated the temperature dependence of the magnetic susceptibility $\chi (T)$ for the spin-S Heisenberg models corresponding to PFB0, PFB2, PFB5, and to the doubled-in-size AS structure described in the right panel of figure 6 ($S=5/2$). The program calculates the exact coefficients of the HTE and Padé approximants (ratios of two polynomials of mth and nth order, P_m(T) and P_n(T), respectively) $\chi (T)\approx [m,n]={{P}_{m}}(T)/{{P}_{n}}(T)$. These approximants allow us to extend the region of validity of the HTE down to $T\sim 0.5{{J}_{m}}S(S+1)/{{k}_{{\rm B}}}$, where J_m is the maximal exchange value in the system [44]. We take into account the nearest and next-to-nearest neighbor exchange interactions J₁ and J₂, and, besides this, ${{J}_{4}}={{J}_{2}}$ (interaction between Fe ions in different planes of the PFB5 structure). The results presented in the left panel of figure 6 show the calculated temperature dependence of the magnetic susceptibility for the configurations considered above, within the approach developed in [44]. One can see that the magnetic response of the layered PFB5 structure is strongly suppressed by the large nearest-neighbor AFM interaction, but the same interaction J₁ leads to the huge and even divergent magnetic response in the idealized ferrimagnetic PFB2 configuration, and the latter is also true in the extended AS structure (see also [45, 46]).

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The ordering temperature T_fe may be found from the condition $1/\chi ({{T}_{{\rm fe}}})=0$, and we have obtained that, if ${{J}_{2}}=0.1{{J}_{1}}$ then ${{k}_{{\rm B}}}{{T}_{{\rm fe}}}/{{J}_{1}}\;\approx$ 4 and 2.5, for PBF2 and AS structures, respectively. The frustrating interaction J₂ diminishes the ordering temperature of the PFB2 structure with respect to the non-frustrated value ${{k}_{{\rm B}}}{{T}_{{\rm fe}}}/{{J}_{1}}\approx 5.6\;{\rm K}$ for ${{J}_{2}}=0$ (see [46] for more details of this approach). Below, we estimate the exchange constants J₁ and J₂ from the first-principles calculations.

To study possible chemical ordering in PFS, we performed first-principles calculations in the framework of density-functional (DFT) theory [47] within the LDA+U method [48] by employing the Vienna ab initio simulation package (VASP) [49]. The on-site repulsion parameter $U=$ 4 eV was employed for Fe ions (the influence of the variation of this parameter was studied earlier [43]). In the present study we use the projected augmented wave (PAW) method with four types of functionals: GGA, LDA, PBE and PBEsol, as it has been implemented in VASP [49]. Forty-ion $2\times 2\times 2$ supercells without any symmetry restriction and with translations in the [200], [020] and [002] directions were used. The k-grid was an automatic $4\times 4\times 4$ Monkhorst–Pack mesh. We used an energy cutoff at 500 eV and looked for the ground state collinear magnetic structure by energy minimization until the precision ${{10}^{-5}}\;{\rm eV}$ was achieved.

Among the possible different atomic configurations, here we consider three structures with the lowest energy, as shown in the top and right panel of figure 6). The first one has rock-salt order of antimony and iron, PFB0 (hereafter we employ the abbreviations from [43]); the second, PFB2, has a ferrimagnetic structure; and the last one, PFB5, has a planar arrangement of iron and antimony ions. Only the ferrimagnetic PFB2 structure has a non-zero ground-state total spin moment of the unit cell. The other two configurations do not possess a finite magnetic moment: configuration PFB0 has I-type AFM, and configuration PFB5 possesses G-type AFM in the iron plane.

The energies of the PFB2 and PFB5 configurations, calculated with the use of several DFT functionals, are reported in table 1, with respect to the energy of the PFB0 configuration. One can see that in PFS (in contrast to PFN [43], where the planar structure dominates) the rock-salt structure is the ground state, independently of the DFT functional form. However, in PFS, the ferrimagnetic PFB2 and planar PFB5 structures compete with each other as first, low-energy, structurally excited states. Thus, according to these results, both such chemical configurations can be observed as extended defects, like a stacking fault or a dislocation, on the background of the rock-salt structure, but this type of order cannot be the ground state of the crystal.

Similar calculations have been performed for the two times larger $2\times 2\times 4$ supercell using the PAW-LDA functional. Figure 7 presents a sketch of the configuration considered, obtained from the ground-state rock-salt PFB0 structure by swapping the nearest-neighbor Fe and Sb ions in different atomic chains directed along one of the main crystallographic axes. Below, we will call this structure an extended AS defect (figure 6). The energy of this structure, obtained with respect to the energy of the PFB0 configuration, is positive and higher than the energies given in table 1 (it equals 67 meV per five-ion primitive unit cell). Thus, the calculations performed on smaller supercells provide the excited (defect) states with smaller energies.

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We believe that the ferrimagnetic extended defect structure of the PFB2 configuration of PFS is responsible for the unusual high-temperature magnetic relaxor properties observed in PFS. According to our experimental results, such a structure can contain a maximum of 10 000 Bohr magnetons, while the weight of these SSs was found to be rather small. This is in line with the fact that such chemical configurations do not constitute the ground state of PFS although they contribute significantly to the magnetic response.

We show here that the magnetic structure of PFS can be successfully modeled by the Heisenberg model containing only two exchange interaction parameters: J₁ is the first-nearest-neighbor and J₂ is the second-to-nearest-neighbor interaction parameter [45]. In the present study, we have calculated these magnitudes for the PFB2 configuration of PFS in the framework of the full-potential local-orbital (FPLO) code [50] and by employing the atomic limit LSDA+U functional, as implemented in FPLO [51]. For that we use the experimental value [32] of the lattice parameter, a = 3.991 81 Å.

We derive the following analytical expressions (see table 2 of [45]):

$$\begin{eqnarray}&&{{J}_{1}}=\frac{{{E}_{2,{\rm fm}}}-{{E}_{2,{\rm fe}}}}{12{{S}^{2}}},\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&{{J}_{2}}=\frac{{{J}_{1}}}{4}-\frac{{{E}_{2,{\rm a}}}-{{E}_{2,{\rm fe}}}}{16{{S}^{2}}},\end{eqnarray} \tag{ 8 }$$

where ${{E}_{2,{\rm fe}}}$ is the total energy (per supercell) of the ferrimagnetic ground state, ${{E}_{2,{\rm a}}}$ is the energy of a state that is obtained by one spin-flip of the two-fold coordinated Fe ion, ${{E}_{2,{\rm fm}}}$ is the energy of the ferromagnetic state and $S=5/2$ is the spin of the Fe³⁺ ion.

Our calculations give ${{J}_{1}}/{{k}_{{\rm B}}}\;\approx$ 73 K and ${{J}_{2}}/{{k}_{{\rm B}}}\;\approx$ 13 K, for U = 6 eV.¹⁰ It is interesting that the magnitude of the J₁ parameter obtained for PFS is close to the value obtained previously for PFN: ${{J}_{1}}/{{k}_{{\rm B}}}\approx 75\;{\rm K}$ [45]. In contrast to this fact, the second-to-nearest-neighbor interaction parameter, J₂, is more affected by the type of non-magnetic cation: in PFS, it is substantially larger than in PFN (in PFN, ${{J}_{2}}/{{k}_{{\rm B}}}\;\approx 2\;{\rm K}$ [45]). This large difference might be caused by the substantially different electronic structures of Sb⁵⁺, which is not a d-metal ion, and Nb⁵⁺ cations, which are $3d$-metal ions.