The Electronic Structure, Thermoelectric, and Optical Properties of Heusler Alloys Mn₂MeAl (Me = Ti, V, Cr)

At present, semiconductor compounds, such as tellurides and selenides of bismuth and antimony, and their solid solutions are used as thermoelectric materials. Metallic compounds have not received much attention in this respect due to their low Seebeck coefficient. However, Heusler alloys (HAs) X₂MeZ (where X and Me are transition metals, and Z is an element of III–V groups) have come into the focus of interest of researchers in recent years. Currently, materials with unique physical properties, such as half-metallic high-temperature ferri- and ferromagnets, multiferroics, shape memory alloys, and tunable topological dielectrics, are among these kinds of alloys with high potential for numerous applications [1]. It has been theoretically and experimentally shown that it is possible to substantially change the properties of HAs by replacing constituent elements, nanostructuring, heat treatment, etc.

Theoretical studies show that HAs classified as half-metallic ferromagnets or spin gapless semiconductors can exhibit a relatively high Seebeck coefficient [2‐12].

In many studies, emphasis is placed on the relationship between crystal structure and thermoelectric properties through the electronic structure (see, for example, [3] and references therein). If the calculations are performed for an ideal crystal structure, then the calculated values do not reflect, as a rule, the experimental dependence of thermoelectric properties. This is due to the fact that real alloy samples cannot be in a completely ordered structure, for example, the L2₁ structure for Heusler alloys Co₂MeZ (Me = Ti, Cr, Mn, Fe; Z = Al, Si, Sn [3, 4]). Most of them crystallize in B2 and/or A2 structures. The presence of defects and/or disorder has an effect on the temperature dependence, as well as the sign of the Seebeck coefficient.

Furthermore, the equation for calculating the Seebeck coefficient includes parameters that are functions of relaxation time τ(k, T), and the calculations are performed under the assumption that τ is the same for electrons with spins up (↑) and down (↓) [3].

The electronic structure of the material is primarily determined by the type of crystal structure. Numerous studies have shown that manganese-based Heusler alloys can have a β-Mn structure (space group P4₁32) that can be either the inverse X structure (Hg₂CuTi type, space group F-43m, 216) or the L2₁ structure (Cu₂MnAl type, space group Fm-3m, 225) [3, 6, 13‐17]. Chemical or structural disorder leads to partially disordered structure B2 or A2.

Experimental studies of electrical, magnetic, and galvanomagnetic properties and calculations of the electronic structure of Heusler alloys Mn₂YAl (Y = Ti, V, Cr, Mn, Fe, Co, and Ni) have shown that the states of ferro- and antiferromagnets, a compensated ferrimagnet, and a frustrated magnet can be achieved in them. Phase transitions occur with a change in the magnetic structure [8, 15, 16]. Some alloys show an anomalous behavior of electrical resistance for metals, namely, there are regions with a positive, negative, or zero temperature coefficient of resistivity (TCR) in different temperature ranges. The presence of a negative TCR may indicate the proximity to the state of a spin gapless semiconductor with a vanishingly small energy gap.

The aim of this study was to obtain information about the electronic structure, and the thermoelectric and optical properties of Heusler alloy Mn₂MeAl (Me = Ti, V, Cr) from the theoretical calculations and experimental research.

Polycrystalline samples of Mn₂TiAl, Mn₂VAl, and Mn₂CrAl alloys were prepared in an induction furnace in a purified argon atmosphere. Next, they were annealed for 72 h at T = 650°C in an argon atmosphere with subsequent cooling to room temperature at a rate of 100 °C/h.

To measure the electrical resistance and thermoelectric voltage (Seebeck coefficient), samples with a size of about 1 × 1 × 10 mm³ were prepared. The electrical resistance was measured according to the conventional four-contact method with a direct current by switching the current through the sample. The thermal EMF (Seebeck coefficient) was measured at room temperature as described in [18].

Mirror surfaces for optical studies were obtained by grinding samples on boron carbide micropowders of different grits and by polishing with chromium oxide. The frequency dependences of the real and imaginary parts of the permittivity ε₁(ω) and ε₂(ω), respectively; ω is the cyclic frequency of the light wave) were studied by the Beattie ellipsometric method at room temperature in air in the spectral range of 0.07–5 eV (λ = 0.25–13 µm). The measurement accuracy was 2–5% in the visible, ultraviolet, and infrared regions of the spectrum. Optical conductivity is calculated by the formula σ(ω) = ε₂ω/4π.

The results of determining the structural state of alloys from X-ray diffraction data (Fig. 1) are given in Table 1.

A powder X-ray phase analysis showed that Mn₂VAl is a single-phase alloy and has a bcc crystal structure. The distribution of atoms over lattice sites are as follows: position 1a is occupied by 0.97 Mn and 0.03 Al; position 1b is occupied by 0.54 V and 0.46 Al.

An analysis of the X-ray diffraction pattern of Mn₂CrAl showed that the alloy crystallizes in a two-phase state in which the bcc structure is also revealed in addition to the cubic phase of the β-Mn type. It is not possible to distinguish which of the Mn and Cr atoms occupies a certain position from the X-ray diffraction data since they have similar X-ray scattering amplitudes. Therefore, the distribution of atoms in the β-Mn structure is denoted as follows: position 8c is occupied by 0.9 Mn(Cr) and 0.1 Al; position 12d is occupied by 0.4 Mn(Cr), 0.43 Al, and 0.17 Mn(Cr). The distribution of atoms in the bcc structure is as follows: position 2a is occupied by 0.12 Al and 0.88 Mn(Cr).

The two-phase state is also detected in the Mn₂TiAl alloy. The fraction of β-Mn in the structure is about 29% and the distribution of atoms over positions is as follows: position 8c is occupied by 0.88 Mn and 0.09 Al; position 12d is occupied by 0.36 Mn and 0.64 Al. The fraction of the D6h structure is about 71%, and the distribution of atoms over positions is as follows: position 2a is occupied by 0.72 Mn and 0.28 Al; position 6h is occupied by 0.7 Mn and 0.3 Al; and position 4f is occupied by 0.98 Ti.

The electronic structure was calculated using the VASP software package [19, 20] with the electron–ion interaction pseudopotential based on the projected augmented wave method (PAW [21]) and the exchange-correlation functional was chosen in the GGA PBE form [22]. The cutoff energy for plane waves was 500 eV. Integration in reciprocal space was carried out over a grid of k-points 8 × 8 × 8. Preliminarily, the geometric optimization of the crystal structure of each of the compounds was performed.

The pattern of the density of electronic states is shown in Fig. 2. For Mn₂VAl, it can be seen that the Fermi level is in the gap for the electrons with “majority” spin projection; the density of states at the Fermi level is nonzero for the opposite direction of the spin, which makes it possible to attribute this alloy to half-metallic ferromagnets (Fig. 2a). The calculated magnetic moments are 1.31 μ_B/atom on the manganese atoms, 0.76 μ_B/atom on the vanadium atom, and 0.03 μ_B/atom on the aluminum atom. The total magnetic moment per formula unit is 1.82 μ_B, which is in good agreement with the experimental value equal to 1.98 μ_B/formula unit [16].

As can be seen from Table 1, the phase composition of Mn₂TiAl consists of 71% of the D6h type structure (space group P63/mmc) and 29% of the β-Mn structure (space group P4₁32). The results of calculations of the densities of electronic states for both phases are shown in Fig. 2b. As can be seen from Fig. 2b, both phases have a high density of states at the Fermi level and the same density of states for electrons with spin up and spin down. The latter is confirmed by the values of the magnetic moments, which are close to zero: 0.05 μ_B/formula unit and 0.03 μ_B/formula unit for the D6h and β-Mn structures, respectively. These values are in good agreement with the experimental value equal to 0.05 μ_B/formula unit [16].

Similar results were obtained for both crystalline modifications of Mn₂CrAl (Fig. 2c). In this case, there is a noticeably stronger difference in the density of states of the ordered bcc phase compared to the phase with the β-Mn structure. As in the case of Mn₂TiAl, both types of structures exhibit an equally high density of states at the Fermi level for both spin directions. For the bcc structure, a nonmagnetic ground state with zero magnetic moments of atoms is observed; for the β-Mn structure, ferrimagnetic ordering with low values of moments at individual atoms and a resulting moment of 0.12 μ_B/formula unit is obtained. Thus, Mn₂CrAl is a nonmagnetic alloy, which agrees with the published experimental results [16], according to which the magnetization of this compound is zero.

Based on the calculated electronic structure, the temperature dependences of the Seebeck coefficient of the systems under study were calculated using the Boltztrap2 software package [23], which makes it possible to numerically solve the linearized Boltzmann transport equation (BTE). This software package provides various ways to set the relaxation time τ from the constant relaxation time approximation (CRTA) to setting the τ(k, T) values obtained in the calculations of the electron–phonon interaction, which require significant computational costs though. On the other hand, the model approximation of type τ⁻¹(E) = cg(E), where g(E) is the density of electronic states, is sufficiently accurate, as was demonstrated in [23, 24]. In this study, we use this approximation to estimate τ. It should be noted that calculations of thermoelectric properties require information about the electronic spectrum with a high resolution in k. Therefore, calculations were additionally performed with a 24 × 24 × 24 k-grid for all structures except for β-Mn, for which a 16 × 16 × 16 grid was used due to a larger unit cell.

The results of calculating the Seebeck coefficient S(T) together with experimentally found values at room temperature are shown in Fig. 3. For Mn₂VAl, one can see that the Seebeck coefficient is positive in the studied temperature range from 100 to 800 K. Moreover, a good agreement with the experimental value at 300 K should be noted (Fig. 3a).

According to the measurements of the crystal structure (see Table 1), Mn₂TiAl and Mn₂CrAl are two-phase systems, so the calculation of the Seebeck coefficient in these cases was performed for both types of crystal structures (Figs. 3b and 3c). Better agreement with the experimental data is observed for the β‑Mn phase in the case of Mn₂TiAl and for the A2 bcc phase in the case of Mn₂CrAl. The observed patterns may be due to the fact that the used approximation for the relaxation time is not quite correct for heterophase samples and direct calculations of τ(k, T) taking into account the electron–phonon interaction are required.

The general form of the temperature dependence of the electrical resistivity of the studied alloys is shown in Fig. 4. The Mn₂CrAl and Mn₂TiAl alloys have anomalous temperature dependences, i.e., a weak negative TCR and high ρ(T) values of about 300 μΩ cm (Fig. 4). Only Mn₂VAl has a weak positive TCR.

According to the resistivity and the room temperature Seebeck coefficient, we estimate the thermoelectric Q factor as S²/ρ = (0.13–0.84) × 10^–4 W/(K² m). According to the published data, the highest values of the thermoelectric Q factor in the Heusler alloys were observed in Co₂MnSi; they are 2.9 × 10^–3 W/(K² m) at 550 K and 1.7 × 10^–3 W/(K² m) at room temperature [4].

Optical conductivity σ(ω) is the most informative function for revealing the features of optical absorption. In the IR region of the spectrum, metals exhibit Drude’s rise on the σ(ω) curve due to the contribution from the absorption of the energy of the incident wave by free electrons (intraband absorption) [25]. As the frequency of the incident light increases, the mechanism of quantum excitation of electrons is turned on; then, the mechanism of quantum excitation of electrons begins to dominate. A contribution from interband absorption appears in the optical conductivity, which gives information about the electronic energy spectrum.

For all investigated alloys, interband transitions play a dominant role in the formation of optical properties and formation of complex spectral dependence σ(ω) in the entire studied region of the spectrum (Fig. 5). The main band of intense interband absorption forms at energies of E > 0.23 eV in the Mn₂TiAl alloy, E > 0.13 eV in the Mn₂VAl alloy, and E > 0.14 eV in the Mn₂CrAl alloy. In the visible and UV regions, the intensity of interband absorption gradually decreases to the level of σ(ω) ≈ 22 × 10¹⁴ s^–1.

The dispersion of the optical conductivity is generally the same in the samples of Mn₂TiAl and Mn₂CrAl, in which two phases were revealed, and differs only in intensity. The center of the main absorption band is in the energy region near 1.3 eV.

For the Mn₂VAl alloy with the B2 structure, the optical conductivity in the entire studied region is substantially lower than that for Mn₂TiAl and Mn₂CrAl. The minimum on the σ(ω) curve at an energy of 0.13 eV, a peak at an energy of 0.5 eV, and shoulders in the vicinity of 1 eV and in the region of 2–2.5 eV should be singled out. The intensity of interband absorption gradually decreases with an increase in the energy of the incident light.

For all alloys, a slight increase in the optical conductivity is observed in the IR region of the spectrum, which should be associated with the onset of the Drude rise. However, there are issues that prevent one from drawing such a conclusion.

In the limit of ω → 0, the optical conductivity reaches a static value, which can be obtained from electrical resistivity measurements. The static conductivity σ_st at room temperature is 30 × 10¹⁴ s^–1 for Mn₂TiAl, 53 × 10¹⁴ s^–1 for Mn₂VAl, and 31.6 × 10¹⁴ s^–1 for Mn₂CrAl. Therefore, the optical conductivity should increase only for Mn₂VAl in the limit of ω → 0; in this alloy the rise of the optical conductivity curve can be associated with the beginning of the Drude rise. For the Mn₂TiAl and Mn₂CrAl alloys, the increase of σ(ω) should be associated with the emergence of new interband absorption peaks.

The presence of absorption peaks in the IR region of the spectrum indicates the existence of low-energy gaps in the band spectrum of the alloys. For the Mn₂TiAl and Mn₂CrAl alloys, in which the Fermi level is located in the region of high density of states for both spin subsystems, interband transitions can be expected starting from almost zero energy.

For the Mn₂VAl alloy, interband transitions of electrons with practically zero energy are possible in a system of bands with spins opposite to the magnetization direction. In the other system, interband transitions of electrons are possible at energies above the gap in which the Fermi level is located. A noticeable decrease in the intensity of interband absorption in the visible and UV regions of the spectrum indicates a weakening of the hybridization of deep states.

The dispersion curves of the real and imaginary parts (ε₁(ω) and ε₂(ω), respectively) of the permittivity of the Mn₂TiAl, Mn₂VAl, and Mn₂CrAl alloys are shown in Fig. 6. We observe a monotonic increase of ε₂(ω) with an increase in the wavelength of the incident light and low negative or positive values of ε₁(ω) up to the boundary of the studied interval. From the dependence (1 – ε₁)^–1 = f(ω²) for Mn₂VAl and Mn₂CrAl, estimates of squared plasma frequency Ω² ≈ (0.6–0.7) × 10³⁰ s^–2 and effective concentration N_eff = Ω²m/4πe² ≈ 2 × 10²⁰ cm^–3 of free carriers (where e and m are the charge and mass of a free electron) are obtained. The N_eff values are about 2 orders of magnitude lower than those typical for good metals. The ε₁(ω) values for Mn₂TiAl remain positive up to the long-wave boundary of the studied interval, thus it is impossible to estimate Ω² and N_eff.

For cubic crystals, the squared plasma frequency Ω² is determined by the electron velocity on the Fermi surface, which is related, in turn, to the density of states at the Fermi level: \(\Omega _{{}}^{2} = \frac{{{{e}^{2}}}}{{3{{\pi }^{2}}\hbar }}\int {\upsilon \,d{\kern 1pt} {{S}_{{\text{F}}}}} ,\) \(N\left( {{{E}_{{\text{F}}}}} \right) = \frac{1}{{4{{\pi }^{3}}\hbar }}\int {\frac{{d{{S}_{{\text{F}}}}}}{\upsilon }} \) [25]. According to the band calculations, a high density of states is noted for Mn₂TiAl and Mn₂CrAl at the Fermi level, which is formed by the contributions of the d states of the Mn and Me atoms (Fig. 2). For Mn₂VAl, the Fermi level is in the pseudo-gap in one of the spin subsystems and at the peak of the density of states in the other. Therefore, it is natural to expect low values of the squared plasma frequency Ω² and effective concentration N_eff of free carriers for all alloys.

High values of the relaxation frequency, which includes all mechanisms of electron scattering, in particular, scattering due to structural disorder, may be another reason for the low absolute ε₁(ω) and ε₂(ω) values. However, we see that the ε₁(ω) and ε₂(ω) curves are close for all alloys, regardless of the structure or degree of disorder.

The electronic structure of Mn₂MeAl (Me = Ti, V, Cr) Heusler alloys has been comprehensively studied theoretically on the basis of band calculations and experimentally on the basis of optical spectroscopy data.

Taking the real crystal structure of the alloys into account, the patterns of the density of electronic states are obtained. In the Mn₂TiAl and Mn₂CrAl alloys, the Fermi level is in the region of high density of states. The Mn₂VAl alloy is a half-metallic ferromagnet.

The calculated values of the Seebeck coefficient are in good agreement with the experimental data in the case of the single-phase alloy Mn₂VAl, while agreement with the experimental values is observed only for one of the phases in the case of the Mn₂CrAl and Mn₂TiAl two-phase alloys.

The Mn₂CrAl and Mn₂TiAl alloys have a weak negative temperature coefficient and high values of residual electrical resistivity. The Mn₂VAl alloy shows electrical resistance of the metallic nature.

It has been experimentally demonstrated that the alloys have a low thermoelectric Q factor of (0.72–1.07) × 10^–4 W/(K² m).

An anomalous behavior of the optical properties of the alloys in the IR region of the spectrum has been revealed, which consists in the absence of a contribution from intraband absorption and the presence of intense interband absorption.

The obtained pattern of the band spectrum allows us to give a qualitative explanation of the features of the temperature dependence of the electrical resistivity and the permittivity dispersion.