A Model of the Electronic Structure of a FeRh Alloy Undergoing an Antferromagnetic–Ferromagnetic Phase Transition

Interest in studying the FeRh alloy has not weakened for several decades due to the presence of a first-order phase transition between antiferromagnetic (AFM) and ferromagnetic (FM) ordering at a critical temperature T_tr ~ 320–370 K [1‐5]. The AFM–FM phase transition in FeRh is accompanied not only by an abrupt change in the magnetization, \(\Delta M,\) but also by an increase in the size of the unit cell, \({{{{\Delta }}a} \mathord{\left/ {\vphantom {{{{\Delta }}a} a}} \right. \kern-0em} a} \approx 0.01\) [6], and significant changes in the electrical resistivity [1, 5]. In the vicinity of the AFM–FM transition, anomalously large values of magnetostriction [7, 8], magnetoresistance [9, 10], and elastocaloric [11] and magnetocaloric effects [12, 13] are observed. Studies have shown that the temperature intervals for the existence of AFM and FM states and the critical temperature (T_tr) in FeRh-based alloys can be changed over a wide range by applying external pressure [6, 14], as well as by small substitutions of iron or rhodium by atoms of other d-metals [10, 15‐17]. In connection with the prospects for the practical use of FeRh-based alloys not only for magnetic cooling, but also in magnetic recording devices, in recent years, intensive studies of the dynamics of the nature of the AFM–FM transition have been carried out by various methods. The main results on the magnetic and magnetothermal properties of FeRh alloys are presented in a recently published review [5].

Our interest in FeRh alloys is related to a specificity of the mechanism of the AFM–FM metamagnetic transition. In the early 1960s, a discussion arose about the effect of the lattice parameter \({{{{\Delta }}a} \mathord{\left/ {\vphantom {{{{\Delta }}a} a}} \right. \kern-0em} a}\) on the AFM–FM phase transition in FeRh alloys after Kittel’s suggestion that the increase in the lattice parameter upon the phase transition is related to the inversion of the exchange energy at a certain value of this parameter. The authors of [18, 19] believe that the observed increase in the lattice parameter, \({{{{\Delta }}a} \mathord{\left/ {\vphantom {{{{\Delta }}a} a}} \right. \kern-0em} a} \approx 0.01\) [6], is quite sufficient for such changes. However, there are publications in which such an exchange-striction mechanism of the AFM–FM transition in FeRh is questioned. As an example, the authors of [20, 21] reported the results of studying these alloys by femtosecond optics. It turned out that the changes in their spin structure upon the AFM–FM transition outrun the changes in the atomic structure. If the atomic structural transition were the cause of the metamagnetic transition, then one would expect a reverse sequence of these changes. This means that many problems remain in the theory of the AFM–FM transition in FeRh alloys and any new approach to their resolution may be useful.

Such an approach, which is different from the generally accepted ones, is proposed in our work. It is based on qualitative assumptions about the properties of 3d and 4d electrons in FeRh alloys, which, at first, can explain two anomalies in their magnetism. One of them is the absence of magnetic moments of 4d electrons in the low-temperature AFM phase and their appearance in the high-temperature FM phase. The second is a radical rearrangement of the spin structure of 3d and 4d electrons, which causes the change in the lattice parameter \({{{{\Delta }}a} \mathord{\left/ {\vphantom {{{{\Delta }}a} a}} \right. \kern-0em} a} \approx 0.01\) upon the AFM–FM transition. Usually, the assumptions are substantiated using first-principles calculations. Since the known data of such calculations [22‐25] was insufficient for this, we used a different approach. It consists in predicting new properties of FeRh alloys, which may be a consequence of the above assumptions, and verifying the validity of these predictions experimentally. Experimental confirmation of the predictions of a theory has always been considered the main indicator of its quality.

In Section 2, we analyze the assumption that the 4d electron band of Rh submerges below the Fermi level to a depth that ensures its complete filling at a temperature T = 0 K. This assumption provides compensation for the spins of 4d electrons in the low-temperature AFM phase of the FeRh alloy and the appearance of uncompensated 4d spins in its high-temperature FM phase. A significant part of Section 2 discusses the magnetic properties of the FeRh alloy following from this assumption. The results of the experimental verification of the predicted properties are given in Section 4. Section 3 analyzes the assumption that there are two spatial distributions of 3d and 4d electrons compatible with the same atomic structure of the FeRh alloy. For brevity, these distributions are called in the text uniform and nonuniform. The nonuniform distribution corresponds to the low-temperature AFM phase of the FeRh alloy, while the uniform distribution corresponds to its FM phase. The compatibility of these two electron distributions with the same atomic structure makes it possible to explain the weak change in the crystal parameter, \({{{{\Delta }}a} \mathord{\left/ {\vphantom {{{{\Delta }}a} a}} \right. \kern-0em} a} \approx 0.01\), upon the AFM–FM transition. An analysis of other properties of the nonuniform and uniform electron distributions made it possible to predict the existence of local magnetic moments of 3d electrons, \({{\mu }_{{{\text{loc}}}}}.\) Their indirect interaction via paramagnetic spins can provide AFM ordering in the low-temperature FeRh phase. Experimental results confirming the existence of the moments μ_loc are given in Section 4. The results we obtained are discussed in Section 5.

It is believed that the magnetic properties of crystalline substances are determined by the spin moments of electrons [26, 27]. Their exchange ordering, as a rule, can be described in a simple model of localized electrons, in which interstitial transitions are neglected. The spectrum of such electrons must consist of discrete levels, as in isolated atoms. The equivalence of the lattice sites makes this spectrum degenerate in the site coordinates r_j, which are the quantum numbers of such localized states. The ordering of spins is determined by their exchange interaction V_ex. Negative values of the exchange energy provide the FM ordering. The ordering of the AFM type is realized if terms with positive exchange parameters dominate. This means that the AFM–FM phase transition in the model of localized electrons requires changes in the atomic structure of the matter.

We propose an alternative approach based on the model of delocalized electrons to describe the AFM–FM transition in FeRh. This model takes into account the transitions of electrons between the sites of the crystal lattice with different \({{{\mathbf{r}}}_{j}},\) which transforms the quasi-atomic levels into Bloch bands with a continuous spectrum \(\varepsilon \left( {\mathbf{q}} \right)\) [28] (q is the wave vector in the Brillouin zone).

Figure 1a shows the assumed densities of states of 3d and 4s electrons in Fe for spin projections \(\sigma = \pm \frac{1}{2}.\) The 3d electron band of Fe is split into two subbands for electrons with different spin directions. The 4s electron band of Fe is the same for spins up and down.

Figure 1b shows the densities of states of the 4d and 5s electrons in Rh for the spin projections \(\sigma = \pm \frac{1}{2}.\) The picture for the 4d band is assumed to be qualitatively different and corresponds to the “submersion” of the 4d band of width \(\Delta \) to a depth \(\delta \) below the Fermi level \({{\varepsilon }_{{\text{F}}}}.\) This spectrum is realized with a weak exchange splitting \(\Delta {{E}_{{{\text{ex}}}}}\) in the spin quantum number \(\sigma = \pm \frac{1}{2},\) satisfying the condition

The opposite condition \(\Delta {{E}_{{{\text{ex}}}}} > \delta \) corresponds to the exchange splitting, in which at least part of the 4d band with spin \(\sigma = - \frac{1}{2}\) is higher than \({{\varepsilon }_{{\text{F}}}},\) as shown in Fig. 1a.

Thermal smearing of the Fermi level ensures the appearance of electrons with energy \(\varepsilon \left( {\mathbf{q}} \right) > {{\varepsilon }_{{\text{F}}}}\) and free states (holes) with energy \(\varepsilon \left( {\mathbf{q}} \right) < {{\varepsilon }_{{\text{F}}}}.\) The concentration of such holes in the 4d band (\({{c}_{{\text{h}}}}\)) reaches appreciable values at temperatureswhere \({{k}_{{\text{B}}}}\) is the Boltzmann constant.

Since states with uncompensated spin correspond to holes, their appearance must affect the magnetic properties of 4d electrons. This influence depends on the distance \({{\rho }_{{\text{h}}}}\) between them:

At low temperatures corresponding to the condition

the exchange interaction \(({{V}_{{{\text{ex}}}}})\) with a radius \({{\rho }_{{{\text{ex}}}}}\) cannot affect the magnetic properties of 4d electrons. This means that the state of 4d electrons under condition (4) corresponds to Pauli paramagnetism with temperature-independent magnetic susceptibility \({{\chi }_{{\text{p}}}}\) [28]. The condition \({{\chi }_{{\text{p}}}} = {\text{const}}\) is a consequence of the compensation of the rise of temperature \({{c}_{{\text{h}}}}\left( T \right)\sim {{k}_{{\text{B}}}}T\) by the decrease in \({{\chi }_{{\text{p}}}}\) according to the Curie law:

A decrease in \({{\rho }_{{\text{h}}}}\left( T \right)\) with increasing temperature imposes restrictions on the condition for the applicability of inequality (4). The opposite inequality

Inequality (7) means that the exchange ordering of the hole spins in the 4d band is weaker than the thermal disordering; therefore, the 4d spins remain paramagnetic at any temperature. However, the temperature dependence \({{\chi }_{{\text{p}}}}\left( T \right)\) (5) for Rh can change, since the Curie law under the conditions of an FM-type exchange interaction is replaced by the Curie–Weiss law [26]:where \({{T}_{{\text{C}}}}\left( {{{c}_{{\text{h}}}}\left( T \right)} \right)\) is the Curie temperature of 4d spins, which depends on the hole concentration in the 4d band.

The substantiation of the applicability of the Curie–Weiss law for the itinerant electron model, as well as the explicit calculation of the dependence \({{T}_{{\text{C}}}}\left( {{{c}_{{\text{h}}}}\left( T \right)} \right),\) is not the goal of this work and is accepted purely phenomenologically. Inequality (7) also provides the condition for the paramagnetism of 4d spins, \({{T}_{{\text{C}}}}\left( {{{c}_{{\text{h}}}}\left( T \right)} \right) < T.\) Experimental data on the temperature dependence \({{\chi }_{{\text{p}}}}\left( T \right)\) of pure Rh are given in [29]. The possibilities for the formation of FM ordering of 4d spins of Rh in FeRh alloys are discussed in the next section.

The schemes of the density of states of d and s electrons in Fig. 1 make it possible to qualitatively describe the differences in the magnetic properties of pure Fe and Rh. In this case, it is important to take the exchange interaction between their spins and the thermal excitation of holes with uncompensated spin in the 4d band into account.

The existence of the AFM–FM phase transition in FeRh alloys at \(T = {{T}_{{{\text{tr}}}}} \approx 370\,{\text{K}}\) leads to additional complications in the mathematical description of the magnetic properties of the 3d and 4d electron spins. In the simplest version of this description, we took into account the exchange interaction between 3d and 4d spins in a thermodynamic potential \({{\Phi ,}}\) which depends on the magnetizations \({{{\mathbf{M}}}_{{{\text{Fe}}}}}\) of the 3d spins of iron and \({{{\mathbf{M}}}_{{{\text{Rh}}}}}\) of 4d spins of rhodium:

Formula (9) takes the terms quadratic in \({{{\mathbf{M}}}_{{{\text{Fe}}}}}\) and \({{{\mathbf{M}}}_{{{\text{Rh}}}}}\) into account in the expansion of \({{\Phi }}\) in the AFM phase of the FeRh alloy. The first term in (9) describes the dependence of \({{\Phi }}\) on the AFM vector \({{{\mathbf{L}}}_{{{\text{Fe}}}}} = {{{\mathbf{M}}}_{1}} - {{{\mathbf{M}}}_{2}},\) where \({{{\mathbf{M}}}_{1}}\) and \({{{\mathbf{M}}}_{2}}\) are the magnetizations of the spin sublattices [26, 27]. We will only need the dependence of \({{\Phi }}\) on the vector \({{{\mathbf{M}}}_{{{\text{Fe}}}}} = {{{\mathbf{M}}}_{1}} + {{{\mathbf{M}}}_{2}},\) thus the explicit form of \({{\Phi }}\left( {{{{\mathbf{L}}}_{{{\text{Fe}}}}}} \right)\) is not important for us. The second term in (9) is responsible for the increase in the exchange energy of the 3d spins of iron upon distortions of the AFM order. Its minimum at \({{J}_{{33}}} > 0\) corresponds to \({{{\mathbf{M}}}_{{{\text{Fe}}}}} = 0.\) The third term in (9) is the exchange energy of the 4d spins of rhodium. It is determined by holes with uncompensated spins in the 4d band, associated with thermal excitation of 4d electrons; therefore, the parameter \({{J}_{{44}}}\) depends on T. The condition \({{J}_{{44}}}\left( T \right) > 0\) means that the minimum of this energy corresponds to the paramagnetic state of 4d spins for all T. The inequality \({{J}_{{44}}}\left( T \right) > 0\) is equivalent to the conditionwhere \({{T}_{{\text{C}}}}\left( {{{c}_{{\text{h}}}}\left( T \right)} \right)\) is the Curie temperature in formula (8).

The last term in (9) is the energy of the exchange interaction of 3d and 4d spins, which can ensure the instability of the AFM state of spins, provided that \({{{\mathbf{M}}}_{{{\text{Fe}}}}} = 0\) and \({{{\mathbf{M}}}_{{{\text{Rh}}}}} = 0.\)

In formula (9), there is no part of Φ that describes the interaction of magnetic moments \({{{\mathbf{M}}}_{{{\text{Fe}}}}}\) and \({{{\mathbf{M}}}_{{{\text{Rh}}}}}\) with a magnetic field \({\mathbf{H}}.\) The magnetic properties of FeRh in the AFM phase in a field \({\mathbf{H}}\) are discussed in Section 4. However, one of these properties does not depend on \({\mathbf{H}}.\) This is the loss of stability of the AFM phase at the temperature \(T = {{T}_{{{\text{tr}}}}}.\) Using this condition to estimate parameters \({{J}_{{33}}},\) \({{J}_{{44}}}\), and \({{J}_{{34}}}\) is the subject this section.

The condition for the loss of stability of the AFM phase can be written explicitly using condition (9) for the minimum of \({{\Phi }}\) in \({{{\mathbf{M}}}_{{{\text{Rh}}}}},\) which allows us to relate the variables \({{{\mathbf{M}}}_{{{\text{Rh}}}}}\) and \({{{\mathbf{M}}}_{{{\text{Fe}}}}}\):

Relation (11) allows us to rewrite \({{\Phi }}\) (9) as a function of one variable \({{{\mathbf{M}}}_{{{\text{Fe}}}}}{\text{:}}\)where

At \(Q\left( T \right) > 0\), the minimum in (12) corresponds to the AFM state with equilibrium values \({{{\mathbf{M}}}_{{{\text{Fe}}}}} = 0\) and \({{{\mathbf{M}}}_{{{\text{Rh}}}}} = 0.\) The AFM state loses stability at a temperature \(T = {{T}_{{{\text{tr}}}}},\) which is the root of the equation

Formula (13) makes it possible to estimate the parameters \({{J}_{{33}}},\) \({{J}_{{44}}}\), and\({{J}_{{34}}}\) on the basis on the experimental data available for FeRh alloys and model assumptions about their electronic structure. We used two such assumptions. One of them corresponds to the spatial distribution of the density of 3d and 4d electrons that conserves the translational symmetry of the atomic structure of the FeRh alloy. For brevity, we will call it the uniform distribution. The second of the discussed electronic structures will be called nonuniform. The possibility of the existence of nonuniform structures in the distribution of 3d electrons of Fe and 4d electrons of Rh was discussed earlier in [30‐32].

The absence of the temperature dependence of the magnetic susceptibility \(\chi \left( T \right)\) at zero anisotropy energy is considered to be a distinctive feature of AFM ordering [26, 27]. The proposed model of the electronic structure of FeRh alloys allows the coexistence of two mechanisms of the temperature dependence \(\chi \left( T \right).\) One of them determines the dependence \({{\chi }_{{\text{p}}}}\left( T \right)\) (8) of paramagnetic 4d spins of Rh. This dependence is due to the exchange interaction between holes in the 4d band, excited by thermal motion.

The second mechanism of the dependence \(\chi \left( T \right)\) is related to the magnetization of the 3d spins of Fe in the AFM phase by an external magnetic field \(H.\) Their magnetization \({{{\mathbf{M}}}_{{{\text{Fe}}}}}\left( T \right)\) is determined by the condition for the minimum of the thermodynamic potential:

In (27), the fact is taken into account that far from the AFM–FM phase transition the potential \({{\Phi }}\left( T \right)\) (9) can be restricted to the second term, which is responsible for the deformation of the AFM ordering of magnetic moments \({{\mu }_{{{\text{loc}}}}}\) (20). The minimum of \({{\Phi }}\left( {T,~H} \right)\) (27) is reached atwhere

The exchange parameter \({{J}_{{33}}}\left( T \right)\) determines the indirect interaction of the moments \({{M}_{{{\text{Fe}}}}}\left( T \right)\) via the paramagnetic spins of other electrons. At low temperatures, \(T \ll {\delta \mathord{\left/ {\vphantom {\delta {{{k}_{{\text{B}}}}}}} \right. \kern-0em} {{{k}_{{\text{B}}}}}}\), the 4d electron band is completely filled (Fig. 1a); therefore, 4d electrons cannot participate in the indirect interaction of the moments \({{\mu }_{{{\text{loc}}}}}.\) This possibility appears with an increase in temperature, which stimulates the appearance of mobile holes in the 4d band. This allows us to write the formula for \({{J}_{{33}}}\left( T \right)\) as

It follows from (8), (28), and (29) that the increase in the hole concentration \({{c}_{{\text{h}}}}\left( T \right)\) with increasing temperature determines both mechanisms of the temperature dependence of the magnetic susceptibility of FeRh alloys in the AFM phase \(\chi \left( T \right),\) predicted by the proposed model. In the linear approximation in \({{c}_{{\text{h}}}}\left( T \right)\), the formula for \(\chi \left( T \right)\) can be written as

To experimentally detect this dependence, magnetic measurements were carried out on a magnetic measuring complex MPMS-XL7 with a primary converter based on SQUID (Quantum Design) on a polycrystalline sample of the (Fe_0.9875Ni_0.0125)_0.49Rh_0.51 alloy (Fig. 2). The sample had the shape of a parallelepiped with linear dimensions \(2 \times 2 \times 6\) mm. The field was applied along the long side of the sample. The method of obtaining and certification of the sample was described in detail in [10].

The \({{T}_{{{\text{tr}}}}}\) of the sample, as determined from the first derivative \({{dM} \mathord{\left/ {\vphantom {{dM} {dT}}} \right. \kern-0em} {dT}}\), was ~283 K in a field of 3 kOe, which is in good agreement with the published data [10, 17]. The typical form of the field dependence of the magnetization M(H) of the sample below the AFM–FM transition temperature is shown in the lower inset in Fig. 2. The least-squares method makes it possible to describe this dependence by an analytical formula

with

It follows from (32) that \(M\left( {T,0} \right) = 0\) up to the measurement error. This means the absence of ferromagnetic inclusions in the sample of the alloy in the AFM phase, the possibility of which was discussed in [34].

The dashed line in Fig. 2 shows the result of fitting analytical curve (32) to the experimental data for the values

The data of magnetic measurements also make it possible to estimate \(\nu \) in formula (20). For this, it is convenient to write formula (28) for \({{M}_{{{\text{Fe}}}}}\left( {T,H} \right)\) aswhereHere, \({{H}_{{\text{E}}}}\) is the exchange field and \({{M}_{0}}\) is the magnetization of the sublattice of a two-sublattice antiferromagnet.

To estimate \({{M}_{0}}\), we used the formulawhere \({{M}_{S}}\) is the saturation magnetization of the FeRh alloy in the FM region at \(T = {{T}_{{{\text{tr}}}}}.\)

Equation (37) takes the fact into account that \({{\mu }_{{{\text{Fe}}}}} = 3{{\mu }_{{\text{B}}}}\) and \({{\mu }_{{{\text{Rh}}}}} = 1{{\mu }_{{\text{B}}}};\) therefore, the magnetization of 3d spins in the FM phase is related to \({{M}_{{\text{S}}}}\) as \({{M}_{{3d}}} = {3 \mathord{\left/ {\vphantom {3 {4{{M}_{{\text{S}}}}}}} \right. \kern-0em} {4{{M}_{{\text{S}}}}}}.\) The estimate \({{M}_{0}} = {{{{M}_{{3d}}}} \mathord{\left/ {\vphantom {{{{M}_{{3d}}}} 2}} \right. \kern-0em} 2}\) corresponds to the condition under which all 3d spins participate in the formation of the AFM ordering. Since \({{\mu }_{{{\text{loc}}}}}\) are determined only by a part of the Fe atoms, estimate (37) for \({{M}_{0}}\) should be considered overestimated. With this reservation, one should accept the estimate for the field

The convenience of estimate (38) is that the quantities \({{M}_{{\text{S}}}}\) and \({{M}_{{{\text{Fe}}}}}\left( {T,H} \right)\) are measured experimentally using the same method. In our experiments, \({{{{M}_{{{\text{Fe}}}}}} \mathord{\left/ {\vphantom {{{{M}_{{{\text{Fe}}}}}} {{{M}_{{\text{S}}}}}}} \right. \kern-0em} {{{M}_{{\text{S}}}}}} \cong 0.003\) at \(H = 3\) kOe (inset in Fig. 2), which gives for \({{H}_{{\text{E}}}}\) the estimate

This allows us to write the exchange energy of the moment \({{\mu }_{{{\text{loc}}}}}\) (20) in the AFM phase as

that contains only one variable parameter \(\nu .\) Since the energy \(\left| {{{V}_{{{\text{ex}}}}}} \right|\) must exceed the thermal energy at the temperature of the AFM–FM transition \({{T}_{{{\text{tr}}}}} = 283\) K and be lower than the thermal energy at the Curie temperature of the FeRh (\({{T}_{{\text{C}}}} \approx 600\) K) alloy, we obtain the following estimate for \(\nu \):

which was used for estimating the exchange parameters \({{J}_{{33}}}\) (21) and \({{J}_{{34}}}\) (26).

The description of the magnetic properties of FeRh alloys proposed in this article is based on two assumptions about the properties of 3d electrons of Fe and 4d electron of Rh:

(I) the assumption that the 4d band is submerged under the Fermi level to a depth that ensures its complete filling at a temperature T = 0 K;

(II) the assumption of the existence of two types of satial distributions of 3d and 4d electrons: uniform, which coincides in structure with the atomic distribution in the FeRh alloy, and nonuniform, with 3d and 4d superstructures with respect to the atomic structure of the FeRh alloy.

If these superstructures, which have the same wavelength, are in antiphase, then the spatial distribution of the total electric charge will be close to atomic, i.e., to a uniform distribution.

We hope that an analysis of other possible properties of FeRh will make it possible to identify those that are implications of assumptions (I) or (II). Experimental detection of the predicted properties of FeRh alloys would be the most reliable substantiation of these assumptions.

In this work, in order to substantiate assumption (I), it was proposed to use the temperature dependence of the magnetic susceptibility \({{\chi }_{{{\text{AF}}}}}\) of the AFM phase of FeRh due to thermal excitations of electrons from the 4d band. The reason for this decision was the absence of temperature dependence of the Pauli paramagnetic susceptibility and the transverse magnetic susceptibility of conventional antiferromagnets [26]. Our experimental data confirmed the dependence \({{\chi }_{{{\text{AF}}}}}\left( T \right),\) but only near the AFM–FM transition temperature. At these temperatures, \({{\chi }_{{{\text{AF}}}}}\left( T \right)\) can be strongly affected by the processes responsible for the destruction of superstructures of 3d and 4d electrons. The lack of data on the mechanisms of these processes did not allow us to isolate their contribution to the observed dependence \({{\chi }_{{{\text{AF}}}}}\left( T \right)\) and estimate the position of the 4d band relative to the Fermi level.

The use of the temperature dependence of the magnetic susceptibility on the magnetic field H was more successful. It enabled us to estimate the magnitude of the exchange field \({{H}_{{\text{E}}}}\) (39), responsible for the antiparallel orientation of the spin sublattices. In turn, the knowledge of \({{H}_{{\text{E}}}}\) made it possible to estimate the local magnetic moments of the 3d electrons of Fe, \({{\mu }_{{{\text{loc}}}}}\) (20), which determine the AFM ordering in FeRh. The inequality \({{\mu }_{{{\text{loc}}}}} > 3{{\mu }_{{\text{B}}}},\) which follows from relations (40) and (41), is an experimental proof of the existence of a superstructure in FeRh. The substantiation of assumption (II), which made it possible to predict the existence of \({{\mu }_{{{\text{loc}}}}},\) is the main result of our work.