Fully-Compensated Ferrimagnetic Spin Filter Materials within the CrMNAl Equiatomic Quaternary Heusler Alloys

Equiatomic quaternary Heusler alloys (EQHA’s) provide a relatively unexplored rich phase space which may harbor novel electronic, magnetic, topological, and correlated phases of matter. Recently, EQHA’s have attracted attention due to their variety of low energy spin-polarized transport and tunneling properties. This makes them ideal candidate materials for the next generation of spintronic and quantum information processing applications,Bainsla and Suresh (2016); Xu et al. (2013); Gao et al. (2013) including direct use in magnetoresistive random access memory (MRAM), spin diodes and transistors, and magnetic tunnel junction (MTJ) devices.Hirohata and Takanashi (2014)

MTJ’s are the standard generator of spin-polarized currents and therefore the fundamental building blocks of MRAM. They are fabricated by inserting a thin barrier in between two electrodes in order to suppress the transmission rate for electrons of one spin over the other. Therefore, when a potential difference is applied between the electrodes a spin polarized current flows. Over the years, developments in functional configurations of magnetic and electronic properties for the electrodes and the barrierWorldedge and T.H. (2000); Moodera et al. (1988, 2010) have pushed spin-polarization very close to 100% at varying temperatures.Miao et al. (2011) However, room temperature MTJ’s without net magnetism and high polarization are still elusive.

Fully-compensated ferrimagnets,Özdoğan et al. (2015) exhibiting an asymmetric gap between the two spin channels, provide a clear route to discriminate between electronic spin states by yielding a highly asymmetric spin-dependent tunneling rate. Since the spin polarization is generated by intrinsic exchange splitting with no net moment, this class of materials can be used in nano scale devices without the need to correct for the deleterious effects of stray coercive fields. Since MTJ’s are one of the most critical components for spintronic applications, the accurate prediction of new candidate barrier materials is of paramount importance.

Many half-metals,de Groot et al. (1983) spin gapless semiconductors (SGS),Wang (2008) and spin filter materials (SFM)Galanakis et al. (2016) have been predicted, but only a few have been synthesized.Stephen et al. (2016, 2019a, 2019b); Tsuchiya et al. (2019) CrVTiAl, a member of the XX’YZ EQHA family, has been of particular interest due to its prediction to be a fully-compensated ferrimagnetic SFM with exceptional properties.Özdoğan et al. (2015); Stephen et al. (2016); Cao et al. (2019) However, when synthesized in bulk form only a small component of an ordered L2₁ or Y phase was present. Similarly, the thin film was found in an SGS phase due to atomic swapping between magnetic sub-lattices,Stephen et al. (2019b) illustrating the key role the arrangement of the atomic species across the four magnetic sub-lattices plays in the electronic and magnetic properties of the quaternary Heusler alloys. Therefore, further study of the Cr-based EQHA’s is needed to find other atomic combinations which are immune to disorder effects.

In this article, we predict 27 new spin-gapless semiconductors, half metals, and spin-filter material candidates within the equiatomic quaternary Heusler alloy family. Specifically, the electronic and magnetic properties of Cr$MN$Al compounds with $M$ and $N$ selected from Group IVB and Group VB elements, respectively, is presented. Most compounds were predicted to be fully-compensated ferrimagnets, following the variant of the Slater-Pauling ruleÖzdoğan et al. (2013) for $18$ valence electrons, with only four materials displaying a finite net magnetic moment. In particular, CrVZrAl, CrVHfAl, CrTiNbAl, and CrTiTaAl are identified as robust spin filter candidates, the latter two exhibiting $\sim 0.10$ eV band gaps, and all four with an exchange splitting of $\sim 0.20$ eV. Additionally, their properties are maintained in two of the three atomic arrangements, making them partially immune to atomic site disorder. Finally, we find the CrAlVM subfamily of compounds to be exceptionally sensitive to hydrostatic stress, yielding transitions between all spin-dependent electronic phases.

The outline of this paper is as follows. In Sec. II the computational details are summarized. In Sec. III the crystal structure is introduced along with a comparison of the various atomic orderings. The ground state electronic and magnetic structure for each compound studied is broken down into classes and presented in subsections A-C. Section IV delineates the effects of hydrostatic stress on the CrAlVM subfamily of compounds. Finally, Sec. V is devoted to the conclusions.

Ab initio calculations were carried out by using the pseudopotential projector-augmented wave methodKresse and Joubert (1999) implemented in the Vienna ab initio simulation package (VASP) Kresse and Furthmüller (1996); Kresse and Hafner (1993) with an energy cutoff of $400$ eV for the plane-wave basis set. Exchange-correlation effects were treated using the strongly constrained and appropriately normed (SCAN) meta-GGA scheme.Sun et al. (2015) An 18 $\times$ 18 $\times$ 18 $\Gamma$-centered k-point mesh was used to sample the Brillouin zone. All atomic sites in the unit cell along with the cell dimensions were relaxed using a conjugate gradient algorithm to minimize the energy with an atomic force tolerance of 0.008 eV/Å. When the energy difference between subsequent electronic steps was less than 1 $\mu$eV self-consistency was determined to be achieved. Pulay stress in VASP was used to simulate hydrostatic stress on the lattice, with relaxations being performed after each increase in stress. The Wigner-Seitz magnetic moment is the total moment of the unit cell, as defined by the Bravais lattice vectors, while the magnetic moments on atoms are the moments on spheres defined by the Wigner-Seitz radius in the pseudopotential. Crystal structure images were rendered using VESTA.Momma and Izumi (2011)

Figure 1 shows the crystal structure of a prototypical EQHA of spacegroup $\text{F}\bar{4}3m$ (216) with Wyckoff positions $a$, $b$, $c$, and $d$ occupied. For four distinct atomic species X, X’ , Y, and Z arranged across the four sites, 4! unique configurations are possible. We can further reduce this by noticing that cyclic permutations of the Wyckoff positions along the diagonal must form an equivalence relation between permutations, resulting in six equivalence classes. Lastly, half the remaining arrangements are related by inversion symmetry. Therefore, there are only three unique configurations of atomic species along the [111] body diagonal in the cubic Bravais lattice Stephen et al. (2019b). To consistently label the various compounds we use the naming convention where XX’YZ corresponds to the 1st, 3rd, 2nd, and 4th site as shown in Fig. 1. This notation is distinct from that used by Galanakis et al. in Ref. Galanakis et al., 2016. Each of the three unique arrangements along the [111] diagonal were studied for all combinations of Cr and Al with Group IVB (M = Ti, Zr, Hf) and Group VB elements (N = V, Nb, Ta), for a total of 27 unique quatenary Heusler compounds.

To classify the electronic phase of each compound we examine the relative energies of each spin population at valence and conduction band edges. The six distinct classifying phases are illustrated in Fig. 2. A half-metal (Fig. 2a) is defined as any compound that is gapped in one spin channel and metallic for the opposite spin, yielding mobile spin polarized charge carriers. The formation of a zero-energy gap in a given spin channel can appear in two different ways. A Type I SGS (Fig. 2b) has the valence band maximum of one spin channel coincide with its conduction band minimum, whereas a Type II SGS (Fig. 2c) has one spin channel’s valence band maximum coincide with the conduction band minimum of the opposite spin channel. In a fully gapped system, we define three unique cases. In a Type I SFM (Fig. 2e) the valence band maximum and the conduction band minimum belong to the same spin channel. A Type II SFM (Fig. 2f), on the other hand, has the valence band maximum and the conduction band minimum belong to different spin channels. Finally a symmetric semiconductor (SSC) (Fig. 2d) exhibits no exchange splitting, or spin polarization at the Fermi level. In MTJ’s employing magnetic insulators the properties of the occupied states are not believed to strongly influence the tunneling, rendering the distinction between Type I and II SFM’s moot, but in situations where gating voltages are necessary, the two classes give rise to physically different situations.

The key to predicting strongly polarizing spin filters is the exchange splitting $2\Delta E_{ex}$. Here, we define it as the energy difference between the conduction band minimum in each spin channel. Additionally, the overall size of the spin-dependent gaps, total band gap and effective masses are important in designing an effective spin filter.Müller et al. (2000) Intrinsic semiconductors with smaller band gaps typically have higher conductance, which is ideal for the advancement of spintronic devicesDeiny et al. (2020) while the differences in electron effective mass are rarely taken into account.Lukashev et al. (2012)

Since the CrVAlM subfamily of compounds, along with their possible arrangements, display a diverse array of electronic phases, we gauge their stability and tunablity to external perturbations by applying hydrostatic compressive and tensile stress. Compressive stress was applied in increments of 5 kbar, while tensile stress was applied in increments of 2 kbar to mitigate numerical instabilities. The cell shape and volume along with the internal atomic degrees of freedom were relaxed for each step in pressure. Here, negative (positive) hydrostatic stress corresponds to tensile (compressive) stress. The applied stresses are readily experimentally accessible by induced substrate lattice mismatch during epitaxial growth or through the use of diamond anvil cells.Suwardi et al. (2016)

Figure 3 shows a phase diagram of the all the CrAlVM subfamily compounds as a function of pressure. Strikingly, the compounds containing V display the most varied behavior. The electronic phase is quite tunable with applied stress, exhibiting all electronic phases over a mild stress range. Starting at zero stress the majority spin band gaps in the SFM ground state of CrAlVZr and CrAlVHf quickly decrease with stress, precipitating a phase transition to an SGS state upon their closure. Subsequently for larger compressive (tensile) stress, a ferrimagnetic half-metal phase appears in CrAlVZr while CrAlVHf becomes a ferrimagnetic normal metal. CrAlVTi, on the other hand is a metal in the ground state and by applying tensile stress a transition opening a gap with small exchange splitting is induced. This suggests that systems synthesized with slight atomic site disorder may be tuned by a small applied stress into a SFM. Moreover, stress may be used to switch the electronic state from SFM to metal or half-metal allowing for greater control over spin polarized currents. The main effect of the pressure is to influence the magnitude of the magnetic moment of each atom. This behavior is most clearly seen for CrAlVTi. The phase change from metal to SFM in CrAlVTi can be attributed to the strengthening of the antiferromagnetic correlations of V and Ti due to the increase of their magnetic moments.

Figure 4 compares the atomic site projected density of states (DOS) for CrAlVHf under $0$, $30$ and $86$ kbar of applied tensile stress. The various colored lines denote the DOS for the different atomic species. By comparing Figures 4(a) and 4(b) the effect of $30$ kbar of tensile stress is readily seen. The relative energies of the Cr and V character states in the majority-spin channel are shifted to slightly higher energies, marking an increase in the exchange splitting. Figures 4(b) and 4(c) show the DOS for $30$ kbar and $86$ kbar of stress, respectively. Here a clear switch in the electronic phase is seen with pressure. For increasing tensile stress the band gap energy shrinks in both spin channels converting CrAlVHf into a Type II SGS and then for larger stress a metallic phase appears. Coincident with the transition from type II SGS to metal, a magnetic transition occurs at approximately 90 kbar and the cell is no longer fully compensated.

By calculating the electronic and magnetic properties of 27 new EQHA’s we identify several new possible spin filter materials, along with other SGS and half-metal phases. Importantly, CrVZrAl, CrVHfAl, CrTiNbAl, and CrTiTaAl are partly immune to atomic site disorder important to synthesizing single crystals and thin films. Overall, we find the ground state electronic phases intimately connected to the competition between AFM and FM correlations. This should serve as an accurate descriptor to search for other new SFM.