1 Introduction

The applications of electric charge transport in materials have been known since before the discovery of the electron at the end of the nineteenth century. However, its special applicability in what is known as “electronics” took place in the twentieth century. Electronics, defined as that part of physics that studies the controlled transport of electric charges by means of the laws of electromagnetism, was first associated with vacuum physics, giving rise to electronic devices based on tubes with diode and triode functionality, among others, with which oscillators, amplifiers and modulators were produced.

Around the 1930s, the semiconducting properties of Si and Ge became known, giving rise to p-n junctions and the creation of the first transistor in the 1940s. From that moment on, advances in solid-state physics modified electronics as it was previously known, drastically changing the applications of semiconductor materials, creating new, more efficient and functional devices, and promoting successive miniaturization processes that turned it into microelectronics in the 1950s. Thus, individual components were replaced by the integrated circuit or chip: a Si crystal of small dimensions, containing tiny capacitors, diodes, resistors and transistors appropriately interconnected. These processes were the direct promoters of the transformation between analog and digital processing, with an unusual growth in the 60’s and 70’s that led to the evolution of microprocessors used in personal computers and following the so-called G.E. Moore's law [1]. At the beginning of this century, the design of processors with nanometer technology (90 nm) gave rise to the so-called nanoelectronics, which gave rise to a race among large companies producing devices for information processing and storage, including USB memories, portable hard disks and innovations in cell phones, among others [2].

Up to this point, the functionalities in electronic devices were obtained by controlling the electron charge transport, aided by semiconductor materials in processing and logic operation [3]. On the other hand, the storage of information, as well as the processes of reading, writing and erasing it, was introduced through the use of magnetic materials [4]. Recently, a new word related to the intrinsic properties of the electron has made the old electronics reach unsuspected levels of applicability and multifunctionality: spintronics. Spin transport electronics (spintronics) is that in which the manipulation of the electron spin and the resulting magnetic moment allows the modification of the functionalities of materials, facilitating the design of more versatile and faster devices than their predecessors and giving rise to a technology in which, in addition to the control of the fundamental charge, the intrinsic spin and the magnetic moment of the electron are controlled [5]. The generation of spin currents can be performed by injection of magnetic spins from a ferromagnetic material, the application of electric and magnetic fields, the introduction of electromagnetic waves, the inclusion of elements that promote Zeeman splitting, the application of thermal gradients and mechanical rotations or geometric modifications [6]. These spin current generation methods applied to the design of spintronic devices are intrinsically related to exotic properties such as giant [7] and colossal [8] magnetoresistance, half-metallicity [9], exchange-bias [10] multiferroicity [11] and coexistence of ferromagnetism and semiconductivity [12], among others. The design of new magnetic semiconductor materials that allow the control of logic operations as well as the reading and recording of information in a single material could substantially simplify the development of spintronic devices. Efforts to reconcile the two properties, semiconductivity and magnetism, in a single functional material started with the epitaxial growth of tetrahedrically coordinated semiconductor films containing magnetic transition metals [13], up to the so-called dilute magnetic semiconductors, in which spin-dependent magneto-electro-optical properties can be observed [14, 15]. The main obstacle of applicability in spintronic devices at room temperature of the former lies in the difficulty to increase the densities of the included transition metals and hollow carriers [14], while in the latter the major difficulty has to do with the spin-dependent coupling between the semiconductor bands and localized states, which occurs because many of the magnetic particles are not part of the structure of the semiconductor material and remain interstitial or form secondary crystallographic phases [16].

In the present chapter, two ceramic material systems are presented that show semiconducting and ferromagnetic behaviors simultaneously in a single crystalline structure, which may represent an excellent perspective in the generation of new multifunctional materials for applications in spintronic devices. These materials belong to a particular family of oxides known as perovskites [17], which, due to their particular crystallographic characteristics, allow the inclusion of alkaline earth cations, rare earths and transition metals, so that the physical properties can be tuned by modifications in the stoichiometric composition and structural variations of the material [18]. An advantage of perovskites is that their degree of structural complexity depends on the type of substitutions that can be made from their basic formula, which, in turn, allows modifying and even predicting the occurrence of physical properties in new materials [19].

One family of versatile materials that represents a large percentage of the systems currently being investigated in solid physics and chemistry is the so-called perovskites. In general, perovskites are represented by the ideal formula ABX3, where A is an alkali earth element, a rare earth, or a metal or semimetal of large ionic radius, B represents a transition metal or lanthanide element, and X, usually, is oxygen or a halogen [20]. Modifications of the atomic radii of A and B introduce structural distortions and new crystalline phases, while inclusions of rare earth elements give the possibility to produce materials with exotic electrical and magnetic properties [21]. Partial substitutions of the A and B cations give rise to complex materials such as double perovskites with generic formula A2BB'O6 [22] and even more complex perovskites AA'B2O6 [23] and the well-known triple perovskite A2A'B2B'O9 [24]. Their chemical configuration offers multiple possibilities to combine different elements, as well as the possibility to synthesize new materials with a wide variety of physical properties. Depending on the magnetic and electrical characteristics of A, A’, B and B’ it is relatively easy to create new perovskite systems with promising prospects in the new field of spintronic technology [25]. In the A2BB'O6 double perovskites it is possible to obtain materials evidencing B-B’ cation ordering along the crystallographic axes, forming a salt-rock type structure [26], which is not the case for other types of complex perovskite [27]. These structural characteristics limit the crystalline symmetries for the different families of complex perovskites, suggesting the occurrence of structural distortions that also correspond to octahedral inclinations and rotations that influence the physical properties as a result of the intrinsic characteristics of the constituents of the material [28].

Double perovskite-type materials will be presented in this chapter, which are generated from the general formula A2BB'O6. As can be inferred from the title of the chapter, experimental results suggesting macroscopic behaviors applicable in spintronics will be presented. Results of structural properties and electrical, magnetic and optical responses for materials belonging to these families of perovskite-type materials will be duly correlated with predictions from calculation of density of electronic states. These will be performed by means of Density Functional Theory (DFT), which has become a potentially powerful tool for predicting physical properties in perovskite-type materials [29]. Additionally, considering that the macroscopic thermodynamic properties are strongly correlated with the microscopic dynamics of the atoms of the material, and assuming that the collective vibrations of the crystal lattices in these solid materials take place via phonons, it is possible to study the fundamental excitations that are associated with these thermodynamic properties. The most representative function of phonons takes place in insulators and semiconductors, where they make direct contributions to properties such as specific heat and thermal expansion, indicating that they are temperature-dependent properties. In relation to phonons, vibrations in perovskite-type crystals will be considered to have a harmonic character, which will be valid for temperature values below the Debye temperature of the solid. In this way, theoretical methods can constitute a complementary tool for the study of atomic dynamics at relatively high temperatures, through approximations such as the quasi-harmonic Debye model [30]. Thus, the results of the density of electronic states and the behavior of specific heat, Debye temperature, entropy, thermal expansion and the Grüneisen parameter as functions of pressure and temperature will be presented.

The single perovskite-type material LaFeO3, known as lanthanum orthoferrite, has been extensively investigated for more than 60 years because of its G-type antiferromagnetic character [31], as well as for simultaneously exhibiting weak ferromagnetic [32] and ferroelectric [33] responses, so it can be classified as a multiferroic material [34]. Experimentally, the lanthanum orthoferrite LaFeO3 adopts an orthorhombic crystal structure, belonging to the space group Pnmb (# 62), with tolerance factor t = 0.951 and octahedral distortions given by a+b−b− [35]. On the other hand, the so-called lanthanum cobaltite LaCoO3 has been studied since 1953, showing a total effective magnetic moment greater than the value expected from Hund's rules and Co3+ electronic spin splittings [36]. Initially, the suggested explanations for this effect were based on contributions due to the appearance of Co3+ ions in the structure in octahedrally coordinated sites under strong crystal field intensity [37] and spin–orbit type effects, considering a cubic perovskite type structure with trigonal distortions [38]. The exotic electrical and magnetic properties, as well as the different possible interactions, including the coexistence of low and high spin configurations of the Co cation have been analyzed for many years, including theoretical studies of electronic structure with variations in possible spin configurations and proposals for the occurrence of a transition between a low spin configuration and an intermediate spin state [39]. Despite the apparent similarity in the chemical formula of lanthanide cobaltite and lanthanide orthoferrite, LaCoO3 crystallizes in a rhombohedral structure, belonging to space group R-3c (# 167), with a tolerance factor t = 1.012 and out-of-phase distortions on the three octahedral sub-axes according to Glazer's a−a−a− notation, showing a mostly antiferromagnetic ordering, with incipient ferromagnetic coupling [40] and dielectric response [41]. The interesting properties exhibited by these two materials, as well as their substantial structural differences, in addition to the magnetic nature due to the crystalline field and the types of bonding of the Co3+ and Fe3+ cations with oxygen in the Co-O6 and Fe-O6 octahedra, have direct inference on the exotic physical properties they exhibit. These are the reasons why it is of particular interest to study a complex perovskite-type material from the union of these two simple perovskites to produce the double perovskite La2CoFeO6 [42]. On the other hand, from the theoretical point of view, the correct determination of the electronic structure of materials containing 3d elements such as Co and Fe in the structure is not easy because of the complex splitting of low spin and high spin electrons due to the crystal field and their interactions with the oxygens in the tetrahedral coordination. Although some calculations have been reported [43], the Hubbard potential applied may not have been exactly the best fit to the experiment, so it is important to revisit the applied concepts, as well as the correction to the exchange potential and correlation considered in the calculations. When La is substituted by rare earths in orthoferrite, its structure remains [44] but its magnetic response depends on the specific cation, showing antiferromagnetism for R = Er, Dy [45], ferrimagnetism for R = Sm [46] and ferromagnetism for R = Tb [47]. On the other hand, the rare earth cobaltite RCoO3 crystallizes in an orthorhombic structure (space group Pbnm, #62), evidencing different magnetic responses depending on the cation R [48], with non-collinear antiferromagntism-type character for R = Tb [49]. Due to the ferromagnetic character of the orthoferrite TbFeO3 and antiferromagnetic character of the cobaltite TbCoO3, the study of the double perovskite Tb2FeCoO6 is particularly interesting [50].

Therefore, the investigation of these materials, whose magnetic and electrical transport properties may have interesting implications for the prospects of application in spintronics technology, is of special interest.

2 Experimental Techniques

(La,Tb)2FeCoO6 samples were produced via the solid reaction method starting from powdered oxides of (La,Tb)2O3, Fe2O3 and Co2O3 (Sigma-Aldrich, > 99.99% purity). The precursor oxides were dried at a temperature T = 120 °C, then weighed in stoichiometric proportions, using an analytical balance of 0.1 mg accuracy. The mixture obtained was crushed in an agate mortar for 3 h. The resulting homogeneous powders were heated at 600 °C for 24 h in a Lindberg-Blue tube furnace, after which it was remixed for 30 min and pressed under the application of 4.5 kN in a 9 mm diameter die to form cylindrical tablet-shaped samples. The samples were then subjected to thermal sintering processes at 800 °C, 1000 °C and 1200 °C for 24 h each to ensure the grain growth process and densification of the samples. The structural analysis of the samples was performed by means of a PANalytical X’pert-Pro X-ray diffractometer (λCuKα = 1.540598 Å), in Bragg–Brentano configuration, with step of 0.001° in 2θ and time of 10 s. The structural analysis was performed by Rietveld type refinement to determine crystallographic properties of the material. For the surface morphological study, images were taken by scanning electron microscopy (SEM) in a Vega 3 TESCAN equipment. A semi-quantitative electron energy dispersive X-ray energy dispersive spectroscopy (EDS) analysis was performed to determine the composition of the samples. Complex impedance measurement as a function of temperature was carried out using an Agilent-4194A phase gain and impedance analyser, a Janis Research cryogenic system (VPF-475 model) and a Lake Shore 332 temperature controller. These measurements were performed at a temperature variation rate of 1.7 K/min and in a frequency range between 102 and 107 Hz. The I-V curve at room temperature was elaborated from the data measured in a Keithley-6517A DC electrometer and a sample holder with gold-plated silver contacts designed by the authors, which was adapted to the cryogenic system and the temperature controller. The band gap and energy excitation regimes at room temperature were examined by using a VARIAN Cary 5000 UV–Vis–NIR diffuse reflectance spectrophotometer (DRS), which has an integration sphere with a PMT/Pbs detector. The magnetic response was evaluated by means of magnetization measurements as a function of temperature, following the Zero Field Cooling (ZFC) and Field Cooled (FC) procedures under the application of varying magnetic field strengths. Likewise, magnetic hysteresis measurements were performed, varying the applied field intensity in isothermal curves at different temperatures. The equipment used to measure the magnetic response was a VSM-PPMS Quantum Design. Various temperature and applied field regimes were considered for the different measured samples.

3 Calculation Methods

As mentioned above, the calculations were performed by means of DFT, since it has been proven that in perovskite-type materials its results agree with those obtained experimentally [51,52,53,54]. For the study of the structural, cohesive, electronic and thermodynamic properties of complex perovskites, the VASP code was used, which is based on DFT and the Augmented Wave Projector (PAW) [55,56,57,58,59,60]. Since the Generalized Gradient Approximation (GGA) does not fully describe the system containing 3d-Fe, 3d-Co and 5d-La orbitals, the exchange and correlation energy was evaluated using the GGA with the inclusion of corrections to the Hubbard U potential (GGA + U) [50,51,52,53,54, 61]. Due to the ferromagnetic characteristic of the material, the value of U was obtained following the method described for this type of magnetic ordering [62], finding optimal potentials UFe = 5.3 eV for Fe and UCo = 3.32 eV for Co. The ion–electron interactions were described through the PAW [57,58,59] and all calculations were performed considering spin-polarized configurations. The kinetic energy cutoff for the plane wave expansion of the electronic wave function was 520 eV. Gratings of k-points defined according to the Monkhorst–Pack method [63] were used, where the convergence of the gratings was verified until an energetic accuracy higher than 1 meV/atom was obtained. The Methfessel-Paxton technique [64] with a mixing factor of 0.1 eV was adopted for filling the electronic levels. For these configurations, grids of 9 × 9 × 7 k-points were considered. These grids represent 284 k-points in the irreducible Brillouin zone. The conjugate gradient method was used to find stable ionic positions, with an energy value of 0.1 meV used as the convergence criterion for the total self-consistent energy calculations. The lattice parameters and internal coordinates of the unit cell were fully optimized by maintaining the spatial symmetry group of the crystal structure, until the forces were less than 30 meV/Å and the energy due to ionic relaxation was less than 1.0 meV/atom. These calculations were performed considering a monoclinic crystalline structure with space group P21/n (#14), in agreement with experimental results for (La,Tb)2FeCoO6 [42, 59]. To determine the Density of States (DOS) parameters, the total energy (E) and external pressure (P) were calculated for different volumes (V) varying around the equilibrium volume (V0) up to ± 5%, allowing relaxation of the internal coordinates. The bulk modulus (B0) and its pressure derivative (B’) were obtained by fitting the pressure curves as a function of volume from the Murnaghan equation of state [65].

For the materials for which thermodynamic property calculations were performed, these were carried out following the Debye quasi-harmonic model [66, 67]. In this model, macroscopically measurable thermophysical properties are directly associated with the microscopic dynamics of the atoms within the material when subjected to external changes in pressure and temperature. Considering that the collective vibrations of the crystalline cells of solids occur in the form of phonons, the fundamental excitations associated with these thermodynamic properties can be comfortably calculated. In materials evidencing relatively high electrical resistivities, phonons play a particularly important role, giving relevant contributions to some properties such as specific heat and thermal expansion as a function of temperature. In this way, the crystal lattice vibrations can be expected to exhibit a harmonic character for temperature values below the Debye temperature of the material. In this work, the Debye quasi-harmonic model is applied as part of the study of atomic dynamics in the appropriate temperature regime.

4 Semiconductor Ferromagnetic Features and Thermophysical Properties of Superstructurated (La,Tb)2CoFeO6 Double Perovskites

4.1 Structural Characteristics

The diffraction pattern shown in Fig. 1 was obtained under the same instrumental conditions specified in Sect. 2. Close observation of the diffractograms reveals practically the same reflection peaks for the materials La2FeCoO6 and Tb2FeCoO6, corresponding to perovskite-type monoclinic structure belonging to the P21/n (#14) space group.

Fig. 1
The graphs depict diffraction patterns obtained for two materials La2FeCoO6 and Tb2FeCoO6. The y-axis depicts Intensity and the x-axis depicts degrees. The red line depicting the calculated pattern has the highest values in both graphs.

Diffraction patterns obtained for materials a La2FeCoO6 and b Tb2FeCoO6

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Although the structure is the same, for R = Tb a shift of the diffraction peaks to the right with respect to R = La is observed, which is related to the change in lattice parameters, as presented in Table 1 and Fig. 2, where the atomic distribution is exemplified from the results of the Rietveld analysis. This shift is related to a contraction of about 3% in the crystallographic cell of the material for R = Tb with respect to R = La. The structure depicted in Fig. 2 corresponds to a monoclinic perovskite, belonging to space group P21/n, which characterizes a primitive cell with rotational symmetry of two orders of 180° around a screw-shaped axis, a subsequent translation of half a cell parameter in the [010] direction and a sliding reflection plane perpendicular to the [010] direction.

Table 1 Lattice parameters, crystal distances and bond angles obtained from the Rietveld refinement
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Fig. 2
Structural diagram depicts the crystallographic structure of R2FeCoO6. The atomic distribution is depicted obtained from Rietveld's analysis.

Crystallographic structure of R2FeCoO6 (R = La,Tb) established from Rietveld refinement of XRD experimental data

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The differences between the ionic radii of the constituent atoms of the material, together with the bond characteristics and electronic correlations, as well as the crystal field effects due to the presence of 3d orbitals in the octahedral coordinations, cause rotations and tilts of the octahedra that influence the magnetic and electric transport responses of the material. In Glazer's notation for octahedral distortions [68], this structure corresponds to the system denoted a−b+a−, where the superscripts (+) indicate in-phase tilt and (−) out-of-phase tilt. Therefore, in the R2FeCoO6 material the octahedra rotate out of phase along the a and c crystallographic axes, while they rotate in phase along the b axis. This is corroborated by the appearance of indexed reflections such as (004), (204) and (404) observed in Fig. 1, which are directly related to the in-phase octahedral tilt (ood) along the direction of the b cell parameter.

Table 1 shows evident differences in the Fe-O and Co-O interatomic distances for the three sub-axes, as well as in the φi bond angles, both for R = La and R = Tb. These differences, which can be clearly seen in Fig. 3, have an impact on the so-called tolerance factor [69, 70]. The obtained values τLa = 0.978 and τTb = 0.921 are far from the ideal value (τ = 1) corresponding to the cubic perovskite structure, evidencing the degree of structural distortion of the materials analyzed. Meanwhile, the presence of the incipient peaks (111) and (313) in the diffraction pattern suggest reflections due to the occurrence of cation ordering due to the formation of a superstructure with the Fe and Co cations alternating along the crystallographic axes [71, 72].

Fig. 3
Structural diagram depicts the bond angle between Fe-O1-Co, Fe-O2-Co, and Fe-O3-Co and Fe-O6 and Co-O6.

Fe-O1-Co, Fe-O2-Co and Fe-O3-Co bond angles between Fe-O6 and Co-O6 octahedra in the R2FeCoO6 double perovskites

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4.2 Ferromagnetic Behavior

A first part in the process of studying the magnetic response is shown in Fig. 4, where graph 4a presents the magnetic susceptibility R = La, in a temperature range between 50 and 330 K, under the application of magnetic fields of 0.5, 2.0 and 10 kOe, while graph 4b shows the magnetic susceptibility for R = Tb, in the temperature range 5 K < H < 340 K and in external fields of 2.0, 5.0, 10 and 20 kOe. In both cases the thermal procedures applied were the well-known Zero Field Cooling (ZFC) and Field Cooled (FC). For both R = La and R = Tb, the difference in the magnetic field trajectory as a function of temperature between the ZFC and FC recipes suggests the occurrence of some kind of magnetic disorder giving rise to irreversibility effects, with typical features of spin glasses [73]. In double perovskite-type materials, this behavior is attributed to the cationic disorder of the B and B’ ions in the A2BB'O6 structure [54]. Meanwhile, for R = La it is not possible to establish the value of the irreversibility temperature, while for R = Tb it can not only be determined but it also evidences a shift towards the lower temperature regime with increasing applied field. This result reveals that the material with R = Tb contains a higher cationic ordering of Fe and Co ions than the compound with R = La. In this ordering, the Fe and Co cations tend to intercalate along the crystallographic axes, ensuring a higher correlation between their magnetic moments in the unit cell and along the whole structure, while the lack of correlation between the magnetic moments in the R = La is due to the low tendency of the crystalline arrangement to form a salt-rock type superstructure [74]. Due to the non-formation of a superstructure in which the FeO6 and CoO6 octahedra adopt an alternating distribution along the structure causes the breakdown of the correlation between magnetic domains, which is characteristic of ordered ferromagnetic materials, so that the ferromagnetism remains in the material but the lack of correlation between domains gives rise to the difference between the ZFC and FC responses in magnetic susceptibility. Evidence for this interpretation is in the presence of the peak (111) in the diffractogram corresponding to R = Tb in Fig. 1b that is not observed in Fig. 1a for R = La, and which is the fingerprint of cation-ordered superstructure formation in double perovskites [75].

Fig. 4
Graph depicts the magnetic susceptibility measured at low temperatures under the application of various magnetic field intensities for La2FeCoO6 and Tb2FeCoO6. It is plotted between temperature in kelvin and emu/mol.ue

Magnetic susceptibility measured at low temperatures under the application of various magnetic field intensities for La2FeCoO6 (a) and Tb2FeCoO6 (b) double perovskites

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The irreversible character could also be associated with effects due to octahedral distortions such as the tilts and rotations of FeO6 and CoO6, which take place in phase along the b-direction of the subcell of the octahedrons and out of phase in the a and c directions, introducing blocking of the magnetic domains at low temperatures and making it difficult to align the moments with the applied field during the ZFC procedure. The occurrence of irreversibility only takes place in the ferromagnetic state, which allows classifying this material and can be corroborated through the finite value of magnetization observed in the susceptibility curves for the two materials under study. This magnetically ordered behavior occurs throughout the temperature region under study. The experimental confirmation of the ferromagnetic response in the two compounds was carried out by measurements of magnetization as a function of the applied magnetic field in isothermal curves at 50, 100, 200 and 300 K for R = La (Fig. 5a) and 5, 25, 50, 100, 200 and 300 K for R = Tb (Fig. 5b). The applied external field regime was −40 kOe < H < 40 kOe.

Fig. 5
Graph depicts the Hysteretic magnetization curves as a function of applied magnetic field for rare earth ferrocobatltite with R=La and R=Tb using graphs. The x-axis depicts Applied Field while the y-axis depicts Magnetization.

Hysteretic magnetization curves as a function of applied magnetic field for rare earth ferrocobaltite with R = La (a) and R = TB (b)

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As predicted from the magnetic susceptibility curves, Fig. 5 evidences a clearly hysteretic behavior characteristic of a ferromagnetic type response for all cases studied at room temperature and below for both R = La (Fig. 5a) and R = Tb (Fig. 5b). Similarly, it is observed that with increasing temperature, there is a decrease in the number of aligned magnetic domains because the increase in entropy decreases the effective exchange energy that characterizes the ferromagnetic feature. For this reason, with increasing temperature, not only the magnetization decreases, as previously observed in the susceptibility curves, but also the magnetic energy of the system decreases as evidenced in the area enclosed by the hysteresis curve. Therefore, the characteristic magnetic parameters are proportionally affected by the change in temperature, observing a systematically decreasing behavior of the coercive field (CF) and remnant magnetization (RM) values. The behavior of the coercive field and remanent magnetization with change in temperature is shown in Fig. 6.

Fig. 6
The x-axis depicts temperature while the y-axis depicts RM and CF. The graph concludes that the remnant magnetization and coercive field values decrease with an increase in temperature.

Decrease in remnant magnetization (left scale) and coercive field (right scale) values with increasing temperature for double perovskite-type lanthanide ferrocobaltites

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The response obtained for R2FeCoO6 decays exponentially, following an empirical law, where, MoLa = 1.12 emu/g and MoTb = 2.73 emu/g represent the spontaneous remanent magnetization for R = La and R = Tb at T = 0 K, aLa = 2.15 emu/g and aTb = 15.86 emu/g are phenomenological constants and kBToLa = 1.14 × 10–22 J and kBToTb = 5.52 × 10–22 J are the magnetic interaction energy corresponding to the exchange potential between the ferromagnetic moments. The obtained value is lower than the energy values calculated for the interaction between ferromagnetic spins in Fe (12.1 × 10–22 J) and Co (51.5 × 10–22 J) [76], showing that the ferromagnetic interaction in Tb2FeCoO6 requires a lower energy cost than in monatomic systems of Fe or Co. Likewise, the coercive fields vary slightly with increasing temperature but decays roughly following Kneller’s law [77, 78] given by, where HoLa = 108.16 Oe and HoTb = 492.64 Oe represent the coercive fields independent of temperature, TBLa = 0.0011 K and TBTb = 0.0035 K, are the blocking temperatures due to the granular feature of the samples, and a La»0.35 and aTb»0.5 are the Bloch’s exponent, which has been observed in ferromagnetic nanoparticles randomly oriented [79], multilayers [80] and bulk spinels [81], suggesting that the variation of the coercive field is sensitive to thermal activation mechanisms, with contributions due to the occurrence of magnetic multidomain states due to the microstructural anisotropy of crystallites and grains in the material [77].

4.3 Electric Response

I-V response measurements were experimentally elaborated with the objective of analyzing the behavior of the electrical response of the materials at room temperature, under application of voltage ranges between −400 V and 400 V for R = La and between −200 V and 200 V for R = Tb. The curves obtained are shown in Fig. 7, which clearly shows that the change in applied voltage causes a nonlinear response of the current in the material. The trend followed by the curve has the characteristics observed in varistor type semiconductors [53]. As the voltage increases, the current increases following a power law of the type. Initially, increasing the voltage induces small currents due to the occurrence of polarization effects and mixing between dielectric responses and granular boundary relaxation effects. Subsequently, electrons promoted across the gap from the semiconductor valence band conduct, tending to linearize the behavior of the I-V curve for high applied voltages. The fit with the potential spike for the two materials yields the following results: IoLa = 0.19 mA, IoTb = 0.27 mA and b» 1.53, b Tb» 1.43. This behavior implicitly contains an additional thermal response that contributes to deviating the electrical response from Ohm's law. Thus, the power dissipation is expected to increase rapidly as the applied voltage increases. The dissipation, then, has contributions due to the appearance of Schottky barriers formed by the grain boundaries, which give rise to the nonlinear characteristic of the I-V curve [53]. These barriers are formed by micro-junctions in which a pair of grains can be seen as two consecutive Zener diodes, such that the intergranular boundaries along the material adopt the behavior of resistors that generate currents like those expected in a varistor diode. For high voltage values, the resistance follows the expected ohmic trend, but in this regime the resistivity decreases because the intragranular contributions are more relevant than the intergranular ones. Therefore, the electrical transport due to microstructural defects can be modeled through equivalent electrical circuits that are related to the intra- and intergranular transport currents within the material [82]. According to the model, the I-V figure of merit of the varistor has the form of a power law, as mentioned above, where Io is a free constant corresponding to the nonlinear coefficient, whose value close to 3/2 is predicted by the quasi-hydrodynamic equations of semiconductors and was attributed to regions of the sample with evanescent carriers [83, 84].

Fig. 7
The graph depicts the I-V characteristic curves measured La2FeCoO6 and Tb2FeCoO6. The x-axis depicts voltage while the y-axis depicts current. The graph depicts that the current increases with an increase in voltage for both.

I-V characteristic curves measured on La2FeCoO6 (a) and Tb2FeCoO6 (b) samples

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4.4 Electronic Structures

It is well known that the macroscopically observable properties of perovskite-type materials can be explained through the study of their electronic behavior [85, 86]. For this reason, the density of electronic states of the compounds R2FeCoO6 following the procedure detailed in Sect. 3. The base magnetic state of the material was considered through ferromagnetic coupling between Fe-O-Fe, Fe-O-Co and Co-O-Co atomic bonds, in addition to possible antiferromagnetic ordering configurations between the aforementioned atoms. In order to explore different configurations of antiparallel spin distribution in the [001], [110] and [111] directions for the antiferromagnetic coupling cases, a 2 × 2 × 2 supercell of the double perovskite was used. For the ferromagnetic case, all the spins due to the magnetic moments of Fe and Co atoms were considered to be oriented in the direction of application of the external magnetic field. By minimizing the energy for all the mentioned cases, the result reveals that the lowest energy distribution corresponds to a ferromagnetic type of exchange interaction. Since the GGA does not satisfactorily describe the system for the 3d-Fe, 3d-Co, 5d-La and 4f-Tb orbitals, it became necessary to calculate the exchange and correlation energy by means of the generalized gradient approximation including the Hubbard U-correction (GGA + U) [87]. From the ferromagnetic nature of the material, evidenced both in experimental results and by the energy minimization process, the U correction in the potential was established following the method reported by Liechtenstein et al. [62]. The DOS results as a function of energy are presented in Fig. 8 for (a) La2FeCoO6 and (b) Tb2FeCoO6 compounds.

Fig. 8
Graph depicts the total and partial DOS calculated for La2feCoO6 and Tb2FeCoO6. The x-axis represents Energy while the y-axis depicts DOS. The DOS increase between -0.4 to 0 electron volt.

Total and partial DOS calculated for a La2FeCoO6 and b Tb2FeCoO6 double perovskites

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In Fig. 8 the zero-energy value is labeled as Fermi energy EF. The correction applied to the exchange potential and correlation was U = 5.3 eV as it was the most in agreement with the experimental results [61]. The first feature observed in Fig. 8 for the total DOS in both materials is the asymmetric character of their electronic states between the up and down spin orientations. For spin up polarization the band gaps clearly correspond to an insulator with values EgLa­ > 3.6 eV and EgTb­ > 3.8 eV, while for spin down polarization the two materials exhibit semiconducting behavior with EgLa¯ = 1.10 eV and EgTb¯ = 1.19 eV. The obtained valence and conduction band configurations for the up and down spin orientations of the compounds with R = La and R = Tb are summarized in Fig. 9.

Fig. 9
Diagram depicts the schematics of the valence and conduction bands having spin polarization for La2FeCoO6 and Tb2FeCoO6 ferrocobalties. Both depict parabolic shapes.

Schematics of the valence and conduction bands with spin polarization for La2FeCoO6 and Tb2FeCoO6 ferrocobaltites

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The generation of a spin-polarized charge transport channel has been theoretically predicted and experimentally measured in ferromagnetic semiconductors [88, 89]. This type of insulator-semiconductor distribution, which here we will call half-semiconductor in analogy with half-metallic behavior, is unusual in double perovskites based on essentially metallic cations such as Fe and Co. Meanwhile, the occurrence of ferromagnetism and half-semiconductivity in the same crystalline structure is really interesting because it differs substantially from the case of dilute magnetic semiconductors, where there could be no guarantee of control in the positioning of magnetic moments in specific sites of the structural cells of the compound, besides the risk of having interstitial magnetic atoms outside the crystalline cells of the semiconductor material acting as matrix. These results allow inferring possible applications in devices that require magnetic control for the transport of both electric charge and spin moments, what we defined before as spintronics.

Analysis of the hybridizations between the outer electron shells of the atoms in the octahedrally coordinated structural octahedra with the oxygen anions suggest that the microscopic mechanism associated with the ferromagnetic behavior is of the super-exchange type between the high spin electrons of Fe2+3d (eg) and the low spin electrons of Co4+3d (t2g) mediated by the O2−2p orbitals, as shown in the diagram in Fig. 10.

Fig. 10
Structural diagram depicts the formation of a weak ferromagnetic response in (La, Tb)2FeCoO6 due to super-exchange interaction between eg moments of Fe and t2g moments of Co.

Super-exchange interaction between eg-moments of Fe and t2g-moments of Co, giving rise to a weak ferromagnetic response in (La,Tb)2FeCoO6 complex perovskites

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In this scenario, the interaction through the pi-bonds Fe-O-Co between the FeO6 and CoO6 octahedra, favours a total antiferromagnetic ordering, while through the sigma-bonds it can be observed that the bond angle 90° < Phi < 180° promotes ferromagnetic dominance through a Goodenough-Kanamori-Anderson type interaction [90].

4.5 Thermophysical Properties

An almost unexplored feature in double perovskite-type materials is the influence of the change in temperature and external pressure on their physical properties. These behaviors could provide relevant information in the study of the macroscopic responses that characterize this family of materials, so their modeling is of particular interest. Considering that during the electronic structure calculation process the equation of state was obtained, it was used in Debye's quasi-harmonic approximation formalism [91], facilitating the calculation of thermophysical properties for (La,Tb)2FeCoO6 materials in the 0 K < T < 900 K temperature regime, obtaining isobaric curves for applied pressures up to 5 GPa. The results of the specific heat at constant volume, CV(T,P), are shown in Fig. 11, where it can be observed that for T > 870 K the Dulong-Petit limit is reached, where the trend of the specific heat becomes independent of the temperature for values CDPLa = 491 J/mol.K (Fig. 11a) and CDPTb = 493 J/mol.K (Fig. 11b). It is evident from the insets of Figs. 11a,b that the specific heat at constant pressure, Cp(T,P), has a behavior similar to that of Cv(T,P) at T < 450 K but does not reach the Dulong-Petit limit in the temperature regime considered in the present work.

Fig. 11
Graph depicts the curves of CV for La2FeCoO6 and Tb2FeCoO6. The x-axis represents Temperature while the y-axis represents C subscript v. It can be concluded that the values of C subscript v increase with the increase in temperature.

Curves of CV(T,P) for the La2FeCoO6 (a) and Tb2FeCoO6 (b) double perovskites. The insets represent the respective CP(T,P) behaviors

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A literature search shows that more symmetric perovskite-type cells, such as cubic ones, present low CDP values, with maximum values of 240 J/mol.K [30], which corresponds to 53% of the value obtained for monoclinic (pseudo-orthorhombic) structures of (La,Tb)2FeCoO6 materials. On the other hand, in perovskites with less symmetric structures the CDP value exceeds 475 J/mol.K [53], showing that at low crystalline symmetries effects appear that give rise to nonharmonic contributions to the thermodynamic functions. This anharmonicity seems to increase with structural complexity [92, 93]. The density of states in Fig. 8 shows a very incipient contribution of the La-5p, La-5d and La-6 s orbitals to the specific heat, while the majority effect corresponds to the O-2p orbitals. This suggests that at high temperatures the electronic contributions could be superimposed on the phonon contributions produced by the vibrations in the cation-oxygen bonds, as a product of the large absorption of thermal energy, significantly increasing the value of the specific heat.

The dependence of the coefficient of thermal expansion with temperature and pressure calculated for (La,Tb)2FeCoO6 materials is shown in Fig. 12. The result shows that for values T < 300 K, the α(T,P) curves present an abrupt growth with increasing thermal energy, which is expected because, in addition to the strongly anisotropic character originating from the elastic properties of the material, the dramatic octahedral distortions, as well as the elongation of the structural cell along the crystallographic axes, contribute to the appearance of rotations and inclinations of the FeO6 and CoO6 octahedra. This behavior occurs because the increase in the thermal energy supplied generates a substantial increase in entropy, randomizing the physical properties and intensifying the phononic effects coming from the vibrations in the intermolecular bonds [67]. At T > 300 K the curve slows down and the change with increasing temperature is less significant because the growth in cell volume tends to stabilize at its maximum value, below which the increase in the coefficient of thermal expansion becomes asymptotic. On the other hand, the effect of the applied pressures partially neutralizes the vibrational activity, so that the increase in the value of the external pressure produces a systematic decrease in entropy which, in turn, shifts the thermal expansion curves towards lower values.

Fig. 12
Graph depicts the coefficient of thermal expansion dependence with temperature and pressure for the R2FeCoO6 ferrocabaltites with R=La and R=Tb. The x-axis represents Temperature while the y-axis represents the coefficient of thermal expansion. From the graph, we can conclude that the coefficient of thermal increases with an increase in temperature.

Dependence of the coefficient of thermal expansion with temperature and pressure for the R2FeCoO6 ferrocobaltites with R = La (a) and R = Tb (b)

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Figure 13 shows the behavior of the Grüneisen parameter γ(T,P), which grows nonlinearly with increasing thermal energy and decreases with increasing applied pressure. This result takes place due to the variations caused in the vibrational frequency of the crystal lattice, increasing the elastic waves of the system and increasing the Debye temperature and making the contribution of phonons large due to the low structural symmetry of the system [85]. On the other hand, as the pressure increases, the vibrational frequency of the cation-oxygen bonds throughout the structure also increases, while, simultaneously, the effect of the temperature increase expands the crystalline cells, causing an increase in the wavelength of these vibrations and promoting a decrease in the vibrational frequencies that shifts the Grüneisen parameter curves towards lower values.

Fig. 13
The graph is between the Gruneisen parameter depicted on the y-axis and temperature on the x-axis for La2FeCoO6 and Tb2FeCoO6. The Gruneisen parameter increase with the increase in temperature.

Grüneisen parameter γ(T,P) for the La2FeCoO6 (a) and Tb2FeCoO6 (b) materials

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5 Conclusions

Rietveld analysis of experimental XRD data of the solid-reacted (La, Tb)2FeCoO6 material reveals its crystallization in a monoclinic superstructure of the double perovskite type belonging to space group P21/n (#14). The unit cell contains distortions due to differences in the sizes and orientation of the FeO6 and CoO6 octahedra, as well as the angles of the O-Fe-O and O-Co-O bonds along the basal and apical octahedral axes. Magnetic susceptibility reveals magnetic domain disorder effects, which cause ferromagnetic long-range order breaking, leading to irreversibility between curves measured through the ZFC and FC procedures, similar to those characterizing spin glasses. Meanwhile, the magnetization curves exhibit a hysteretic behavior characteristic of weak ferromagnetic materials, which is maintained up to room temperature, with values of the coercive field following Kneller’s law in its dependence on temperature. On the other hand, the I-V curves obtained are semiconductor type with strong varistor type tendency, dramatically influenced by the granular nature of the materials.

The results of density of states calculations suggest that the material adopts an antiferromagnetic structure, in which the crystal field effects, and the type of chemical bonding promote a weak ferromagnetic response through a superexchange type mechanism such as that due to a Goodenough-Kanamori-Anderson type interaction. The energy bands evidence an asymmetry between the spin up and down channels, creating a lower band-gap for one of the spin orientations than for the other, which gives rise to polarized spin currents in the semiconductor ferromagnetic material, suggesting its applicability in spintronic devices such as the so-called spin transistors.