Next Article in Journal
Organocatalytic Asymmetric Halocyclization of Allylic Amides to Chiral Oxazolines Using DTBM-SEGPHOS—Mechanistic Implications from Hammett Plots
Next Article in Special Issue
MoO3 Interlayer Modification on the Electronic Structure of Co/BP Interface
Previous Article in Journal
Development and Application of Non-Destructive Testing Instrument for Wall Impermeability Based on a Water Drenching Method
Previous Article in Special Issue
Behavior of Vortex-Like Inhomogeneities Originating in Magnetic Films with Modulated Uniaxial Anisotropy in a Planar Magnetic Field
 
 
Search for Articles:
Title / Keyword
Author / Affiliation / Email
Journal
Article Type
 
 
Advanced Search
 
Section
Special Issue
Volume
Issue
Number
Page
 
Journals
Symmetry
Volume 14
Issue 5
10.3390/sym14050988
symmetry-logo
Submit to this Journal Review for this Journal Propose a Special Issue
► Article Menu

Article Menu

share Share announcement Help format_quote Cite question_answer Discuss in SciProfiles thumb_up
...
Endorse textsms
...
Comment
first_page
settings
Order Article Reprints
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

Influence of Symmetry from Crystal Structure and Chemical Environments of Magnetic Ions on the Fully Compensated Ferrimagnetism of Full Heusler Cr2YZ and Mn2YZ Alloys

by
Zhigang Wu
1,
Yajiu Zhang
1,*,
Zhuhong Liu
2 and
Xingqiao Ma
2
1
School of Civil Engineering, Guangzhou University, Guangzhou 510006, China
2
Department of Physics, University of Science and Technology Beijing, Beijing 100083, China
*
Author to whom correspondence should be addressed.
Symmetry 2022, 14(5), 988; https://doi.org/10.3390/sym14050988
Submission received: 7 April 2022 / Revised: 4 May 2022 / Accepted: 9 May 2022 / Published: 12 May 2022
(This article belongs to the Special Issue Chirality Applied in Spintronics, Ferroics and Light–Matter Interactions)
Browse Figures
Versions Notes

Abstract

:
Fully compensated ferrimagnets do not create any magnetic stray field and allow for a completely polarized current of charges. As a result, these alloys show promising prospects for applications as spintronic devices. In this paper, we investigated the phase stability, the site preference, the tetragonal distortion and the influence of symmetry from the crystal structure and chemical environments of magnetic ions on the magnetic properties of Cr2YZ and Mn2YZ (Y = void, Ni, Cu, and Zn; Z = Ga, Ge, and As) full Heusler alloys by first-principles calculations. We found that the selected Cr2-based alloys, except for Cr2NiGa and Cr2NiGe, prefer to crystallize in the centrosymmetric L21-type structure, while the selected Mn2-based alloys, except for Mn2CuAs, Mn2ZnGe and Mn2ZnAs, tend to crystallize in the non-centrosymmetric XA-type structure. Due to the symmetry, the antiferromagnetism of the selected L21-type alloys is very stable, and no spin-polarized density of states could be generated. In contrast, the magnetic moment of the selected XA-type alloys depends heavily on the number of valence electrons and tetragonal distortion, and spin-polarized density of states is generated. Therefore, the selected alloys with L21-type structures and their tetragonal-distorted structure are potential candidates for conventional antiferromagnets, while those with XA-type structure and their tetragonal-distorted structure are promising candidates for (fully) compensated ferrimagnets.

1. Introduction

Spintronic devices based on antiferromagnets (AFMs) have attracted great interest in recent years [1,2]. The most important characteristic of AFM materials is net-zero magnetic moment; thus, they do not carry any macroscopic magnetic field [3]. The spintronic devices based on AFM materials are not easily interfered with by external magnetic fields and do not generate stray fields to disturb adjacent devices [4]. Recent studies also show that AFM materials display ultrafast dynamics and generate large magneto-transport effects [5]. There are two different kinds of AFMs. One is the conventional AFM, such as metals Mn and Cr, which has a zero net spin magnetic moment and zero spin polarization (Figure 1a). The other is called half-metallic AFM (HMAFM), which was proposed by Van Leuken and Groot in the 1990s [6]. HMAFM is a special kind of ferrimagnet, which has reduced symmetry, half-metallic band structure and zero net magnetic moment at the same time. Therefore, a more precise term for HMAFM is half-metallic fully compensated ferrimagnets (HMFCFs), proposed by later researchers [7,8,9,10]. The following discussion will use the term HMFCF to represent this type of material. As shown in Figure 1b, the electronic structure of HMFCFs is fully spin polarized near the Fermi level. HMFCFs have the following properties: (1) create no magnetic stray field, (2) contribute a 100% spin-polarized charge, and (3) the magnetic susceptibility is zero.
Based on these features, HMFCFs could be even more useful than half-metallic ferromagnets (HMFs) and conventional AFM. The high spin polarization provides the possibility of reaching high magnetoresistance and low magnetic damping, while the zero net magnetic moment can lead to high storage density due to the absence of a stray field. In contrast to a conventional AFM, compensated ferrimagnets have reduced symmetry and allow for easier magnetic reading and writing. These unique properties make compensated ferrimagnets promising for fast, energy-efficient spintronic devices, such as spin-polarized scanning tunneling microscopy (SP-STM) tips, anchor layer in a spin valve and spin-orbit torque [9,11]. Experimental evidence of HMFCFs was recently observed in Heusler-type Mn2Ru0.5Ga [12]. To date, many alloys (e.g., Ti2YZ (Y = V, Cr, Mn; Z = Al, Ga, In, Ge, P, As, Sb) [13,14,15,16,17,18], Cr2MnZ [19,20], Co-doped Mn2VZ [21], Cr2YZ [22], Cr2CoGa [23], Mn2RuxGa [24], Mn2Z (Z = Si, Ge, Sn) [25,26,27], CrVTiAl [28,29], FeMnGa [30], and Sr2OsMoO6 [31]) have been predicted to be HMFCFs by electronic structure calculations. Most of them are Heusler alloys, which are a kind of intermetallic compound and an important material family in condensed matter physics and material science [32,33].
In Heusler alloys, there are several key points for the formation of HMFCFs. One of them is the number of valence electrons (NV) per unit cell of the alloys. The total spin magnetic moment (Mt) of a half-metal is quantized, and the relationship between Mt and NV follows the famous Slater–Pauling (SP) rule [34]. There are three modes of SP rule in Heusler alloys, which are Mt = NV − 18, Mt = NV − 24 and Mt = NV − 28 [35,36]. Therefore, only Heusler alloys with NV = 18, 24, or 28 will get a zero net magnetic moment. Generally, the number of magnetic ions in the alloys is also a key point to obtain HMFCF. An even number of magnetic ions may lead to conventional AFM because of the inversion symmetry of the magnetic structure. Three or more magnetic ions are usually necessary to break the inversion symmetry to get the FCF property [29]. However, the recently proposed C1b-type HMFCF Mn2Z (Z = Si, Ge and Sn) alloys do not meet the condition, since they have only two magnetic ions per unit cell but exhibit a typical FCF state [16,27].
Mn2YZ (Y = transition metals; Z = main group elements) full Heusler alloys, due to their diverse magnetic and unique electronic properties, are promising candidates for HMFs [37], HMFCFs [12,21], thermoelectrics [38], magnetic skymions [39], exchange bias, [40,41] and spin gapless semiconductors [42]. Most of them have higher Curie temperatures (TC) than room temperature, which makes them beneficial for practical applications. Cr2YZ (Y = transition metal; Z = main group elements) full Heusler alloys have been recently proposed, with the same crystal structure as Mn2YZ, to show similar potential applications in spintronics [19,20,22,23]. Antiparallel aligned magnetic moments, which are formed between Mn (Cr) atoms of different sublattices, are important for the formation of compensated ferrimagnetic state. Mn2YZ and Cr2YZ alloys have two inequivalent structure configurations, the non-centrosymmetric XA-type structure (F-43m, space group no. 216) and the centrosymmetric L21-type structure (Fm-3m, space group no. 225). The difference in the symmetry of the crystal structure might have a significant effect on the magnetic properties and density of states distribution, which, in turn, affects the formation of HMFCF.
The current research focuses mainly on alloys with a low number of valence electrons (NV) of Y, such as Y = Ti, V, Cr, Mn, Fe, and Co [36,43]. Few studies focus on alloys with high NV of Y, such as Y = Cu and Zn. Cu and Zn tend not to contribute local magnetic moments in Heusler alloys. Ferromagnetic Ni atoms also hardly contribute significant magnetic moments in Heusler alloys. Therefore, it is advantageous to study the influence of crystal structure symmetry on the magnetic and electronic density of states distribution in Mn2YZ and Cr2YZ (Y = Ni, Cu and Zn; Z = main group element) alloys, since Ni, Cu, an Zn do not contribute to the magnetic properties at all, and the number of magnetic ions is even.
Mn2YZ systems with a tetragonal structure or a spontaneous structural transition from cubic phase to tetragonal phase have important properties for applications, such as the perpendicular magnetic anisotropy (PMA) for spin-transfer torque magnetic random-access memory (STT MRAM) [44], and the ferromagnetic shape memory alloys (FSMAs) [45]. As alloys similar to Mn2YZ, the tetragonal distortion of Cr2YZ has been less studied. The tetragonal phase (I4/mmm, space group no. 139) distorted from the L21-type cubic phase remains centrosymmetric, while the tetragonal (I-4m2, space group no. 119) distorted from the XA-type cubic phase is noncentrosymmetric. The difference in the symmetry of these two tetragonal structures will also lead to the difference in the magnetic and electronic properties, which require further study.
In this paper, we investigated the phase stability, the site preference, tetragonal distortion, and the influence of symmetry from crystal structure and chemical environments of magnetic ions on the electronic and magnetic properties of Cr2YZ and Mn2YZ (Y = void, Ni, Cu, and Zn; Z = Ga, Ge, and As) full Heusler alloys by first-principles calculations. Void means that no atoms were selected for Y, such as Cr2Z and Mn2Z. It should be noted that the C1b-type and C1-type structures have the same symmetry as the L21-type and XA-type structures, respectively. Void can be regarded as a different atom from Mn or Cr, so Mn2 (void) Z and Cr2 (void) Z can be regarded as special full-Heusler alloys. All these alloys, except for Cr2NiZ and Mn2NiZ, have two magnetic ions per unit cell. Our results indicate that the selected alloys with centrosymmetric L21 types and their tetragonally distorted structure are conventional AFMs, while those with non-centrosymmetric XA types and their tetragonally distorted structure are candidates for (fully) compensated ferrimagnets.

2. Calculation Method

The first-principles calculations based on density of functional theory (DFT) were performed with Vienna ab-initio simulation package (VASP) [46]. The generalized-gradient approximation (GGA) in the Perdew–Burke–Ernzerhof (PBE) was employed to deal with the electronic exchange and correlation. Previous calculation indicates that the results obtained by GGA are in good agreement with the experiment results in 3d Heusler alloys [42]. The projector-augmented wave (PAW) was selected to describe the electron–ion interaction [47,48]. To achieve good convergence, the cut-off energy was set to 500 eV for all of the calculations, and 15 × 15 × 15 k-points were used for reciprocal spatial integration. The convergence criteria for the calculation were chosen as the total energy tolerance within 10−6 eV and the atomic force tolerance within 0.01 eV/Å. According to Mavropoulos et al. [49], the effect of spin–orbit coupling on the band gap of Heusler alloys is negligible for 3d transition elements. As such, the spin–orbit coupling was not considered in this work. A full-potential method (FP-LAPW implemented in the WIEN2k code [50], GGA + PBE, and appropriate parameters, such as RMT × Kmax = 9, 10,000 k-points in the Brillouin zone and the cut off energy of −6 Ry, have been chosen) was also used to repeat some calculations to confirm the validity of the results obtained by pseudopotential method of VASP.

3. Results and Discussion

We first took Mn2Si as an example to start the discussion. The C1b-type Mn2Si was proposed as a potential candidate for FCF with spin gapless semiconducting (SGS) properties [16]. The crystal structure of C1b-type Mn2Si has three fcc sublattices, with Mn (A), Mn (B), and Si occupying Wyckoff coordinates A (0, 0, 0), B (0.25, 0.25, 0.25) and D (0.75, 0.75, 0.75), respectively. Figure 2a,b show the band structures of C1b-type Mn2Si, calculated by the pseudopotential method and the full-potential method, respectively. Both the calculation methods give the same band structure. This result indicates the validity of the pseudopotential method for 3d transition Heusler alloys. Figure 2c displays the corresponding density of states (DOS) of C1b-type Mn2Si. The DOS of the spin-down channel clearly exhibits a band gap near the Fermi energy (EF), while the DOS of the spin-up channel shows a vanishingly small gap. As shown by Wang et al. [51], the DOS indicates a typical characteristic of a spin gapless semiconductor (SGS) and, thus, allows for tunable spin transport. Figure 2d gives the total and atom-resolved magnetic moments as a function of the lattice constant. The zero net total magnetic moment indicates the FCF property. Similar behavior was also observed in many other HMFCFs.
HMFCF behaviors have also been identified in many Heusler alloys, such as C1b-type CrMnSb [52], L21-type (or DO3-type) Mn3Ga [53] and XA-type Cr2ZnZ (Z = Si, Ge, Sn) [54]. These materials represent three typical crystal structures of Heusler alloys for HMFCF. Although they have different numbers of magnetic ions, a common characteristic of these alloys is that the relationship between their NV (which is 18, 24 and 28 for CrMnSb, Mn3Ga, and Cr2ZnZ, respectively) and Mt (which is zero for all three alloys) follows the Slater–Pauling rule. The Mt = 0 and NV = 18 of Mn2Si, just like other HMFCFs that follow the Slater–Pauling rule in the form of Mt = NV − 18. It should be noted that Heusler alloys, which are not HMFCFs but have zero net magnetic moments, also exit, such as Fe2VAl and CoFeTiGa (Al) [55,56]. Since these alloys are not our focus, they will not be discussed in detail. In the following part, we studied the influence of the symmetry of crystal structure and chemical environments of magnetic ions on the electronic and magnetic properties of the Mn2Si alloy.
We firstly analyzed the electronic properties of three cases: C1b-type Mn2Si, C1-type Mn2Si, and ZnS-type Mn2 in a Heusler matrix. The Heusler matrix has four fcc sublattices, which are represented by four Wyckoff coordinates, A (0, 0, 0), B (0.25, 0.25, 0.25), C (0.5, 0.5, 0.5), and D (0.75, 0.75, 0.75). It should be noted that the C1b-type Mn2Si is energy stable according to the previous work, and the other two cases were employed for comparative discussion only. Figure 3 and Table 1 provide the information of crystal structures for the above three cases. The C1b-type configuration is the so-called half-Heusler, which has a non-centrosymmetric crystal structure (F-43m, space group no. 216). In the unit cell of this crystal structure, one Mn occupies site A (denoted as Mn (A)), the other Mn occupies site B (denoted as Mn (B)), no atom (void) occupies site C, and Si occupies site D. It can be seen that these two Mn atoms possess different chemical environments. There are four Mn (B) atoms and four Si atoms as the nearest neighbors of Mn (A). However, Mn (B) has only four Mn (A) atoms as the nearest neighbors. In the situation of C1-type Mn2Si, which has a centrosymmetric crystal structure (Fm-3m, space group no. 225), two Mn atoms occupy site A and site C (denoted as Mn (A) and Mn (C), respectively). In this structure, Mn (A) and Mn (C) have the same chemical environments. Each Mn has only four Si atoms as its nearest neighbors. Figure 3c shows that the Mn (A) and Mn (B) of Mn2Si form a ZnS (or diamond) structure as Si is removed from the C1b-type Mn2Si matrix. Due to the removing of Si atoms, the ZnS-type Mn2 matrix becomes a centrosymmetric structure (FD-3m, space group no. 227). In this case, these two Mn atoms have the same neighbor conditions: each Mn has four other Mn atoms as its nearest neighbors. The difference of ZnS-type Mn2 from C1-type Mn2Si is that the nearest neighbor for each Mn atom comprises four Mn atoms instead of four Si atoms. Next, we analyzed the electronic structures of these three types of configurations.
In a conventional AFM, the number of states (NOS) below the Fermi level in the spin-up channel is the same as that in the spin-down channel. The NOS can be obtained by integrating the DOS. As shown in Figure 4, both spin-up and spin-down channels have nine valance electrons per unit, which are from 1 × s of Si, 3 × p of Si, 2 × eg of Mn and 3 × t2g of Mn, below the Fermi level. It should be noted that the configurations of NOS for C1b-type Mn2Si are different in the two spin channels. There is a platform that exists in the spin-down channel. Within the range of this platform, the NOS is constant. For C1-type Mn2Si, the configurations of NOS for the two spin channels are completely identical, indicating a normal AFM state. In addition, the spin-up and spin-down channels of ZnS-type Mn2 also have the same number of NOS below the Fermi level, which was not shown in the figure.
Figure 5 shows the distribution of the partial density of states (PDOS) for the two Mn atoms and the net spin state of the unit cell as a function of energy. The PDOS of C1b-type Mn2Si (Figure 5a) exhibits two important characteristics. The first one is the spin-polarized PDOS of Mn (A) and Mn (B). The PDOS in the spin-down channel has an energy gap with a considerable size near the Fermi level, and that in the spin-up channel has a valley corresponding to a small energy gap. The second characteristic is that the distributions of PDOS in Mn (A) and Mn (B) are not identical, which results in a net spin distribution as a function of energy, as shown in the lower panel. This result corresponds to the different chemical environments between Mn (A) and Mn(B). For C1-type Mn2Si (Figure 5b) and ZnS-Mn2 (Figure 5c), the DOS of Mn (A) atoms are identical to that of Mn (B) atoms. The only difference is that the net spin of each Mn atom is opposite. The net spin of the total DOS is zero in all energy ranges, which indicates that no polarization can be obtained at any energy range.
As such, the distribution of the spin is strongly dependent on the symmetry of the crystal structure and the chemical environments of each Mn atom. As discussed above, C1b-type Mn2Si has a non-centrosymmetric crystal structure, and the first nearest neighbor environments of the Mn ions differ from each other. Mn (A) and Mn (C) in C1-Mn2Si have the same chemical environments. The situation is similar for ZnS-Mn2. The result is a similar situation where there is no spin polarization at all. In C1b-Mn2Si, the differing chemical environments of each Mn atom result in fully spin-polarized DOS, although the Si atoms are nonmagnetic. These results indicate that the asymmetrical chemical environments broke the rotational symmetry of the spins, which led to an asymmetrical distribution of spin-up and spin-down states as a function of energy. As such, broken spin rotational symmetry can be regarded as a key point to obtain HMFCF, which comprises a system without a spontaneous magnetization but with 100% spin polarization in the charge carriers at EF. Only C1b-type Mn2Si has the asymmetric configurations of DOS for the two spin channels.
In order to further reveal the influence of symmetry from the crystal structure on magnetic properties, we investigated the phase stability, the site preference, the tetragonal distortion, and the magnetic and electronic properties of full-Heusler Cr2YZ and Mn2YZ (Y = Ni, Cu, and Zn; Z = Ga, Ge, and As) alloys. There are two non-equivalent crystal lattices for full-Heusler alloys. When the Y atoms occupy site C of the Heusler matrix, as shown in Figure 3a, a non-centrosymmetric XA-type crystal structure, which has the same symmetry as C1b-type Mn2Si, will be formed. When Y atoms occupy site B of the Heusler matrix, as shown in Figure 3b, a centrosymmetric L21-type structure, which has the same symmetry as C1-type Mn2Si, will be formed.
In Heusler alloys, the atomic site preference is related to the NV of the 3d transition metallic elements. Generally, elements with lower NV tend to occupy site B of the Heusler matrix, while those with higher NV occupy site A and site C of the Heusler matrix. This site preference rule has been proved to be valid in many Heusler alloys [57,58,59,60,61]. According to such a rule, the selected Cr2-based and Mn2-based alloys in the present work should all be crystallized into the XA-type crystal structure.
To determine the theoretical lattice parameters and the site preference, we performed structural optimization calculations on the selected Cr2YZ and Mn2YZ (Y = void, Ni, Cu, and Zn; Z = Ga, Ge, and As) alloys, for both L21-type and XA-type configurations. The free energy difference (ΔE) between XA-type and L21-type structures, which is ΔE = E (XA) − E (L21), was selected as the y-axis, as shown in Figure 6, which shows the NV-dependent ΔE of the selected alloys. The fact that ΔE < 0 suggests that XA-type configuration is more energy stable than L21-type configuration, whereas L21 is more stable. The calculation results show that the selected Cr2YZ alloys, except for Cr2NiGa and Cr2NiGe, prefer to crystallize into the centrosymmetric L21-type structure, while the selected Mn2YZ alloys, except for Mn2CuAs, Mn2ZnGe, and Mn2ZnAs, tend to crystallize into a non-centrosymmetric XA-type structure. These results of the atomic occupations of the selected alloys apparently deviate from the conventional site preference rule of full-Heusler alloys. This result indicates the complexity of the phase stability of Heusler alloys. Similar situations also occurred in Ti2-based and V2-based full-Heusler alloys [62,63]. It should be noted that for each transition metal, such as Cr2CuZ, as Z progresses from Ga through Ge to As, the ΔE increases by about the same amount, except for the Mn2(Void)Z family, where there is a decrease. There might be some particular reason for this.
To confirm the thermal stability of the selected alloys, the formation energy (EF) is determined by the formula of EF = E(Cr or Mn) 2YZ − (2E(Cr or Mn) + EY + EZ) [64,65,66], where ECr2YZ and EMn2YZ are the total free energy of the selected alloys per unit cell at their equilibrium states, E(Cr or Mn), EY, and EZ are the free energy per atom of pure metals (Cr or Mn), Y and Z, respectively. Figure 7 gives the values of the EF of the selected alloys for the energy-stable state that was determined in Figure 6. The negative value of EF indicates most of the selected alloys can be obtained in experiments. The positive values of EF for Cr2Ga, Cr2Ge, Mn2Ga, Cr2CuGe, Cr2CuAs, Cr2ZnAs, and Mn2ZnAs indicate that the L21-type or XA-type cubic phase of these alloys are energetically unstable in experiments. For example, bulk Mn2Ga crystallizes into Al3Ti structure type, which belongs to the space group I4/mmm [24]. However, C1b-Mn2Ga films were recently successfully grown on V (001) epitaxial films [12], which indicates that the cubic phase of the above alloys also has a possibility to be realized in experiments using advanced preparation methods (e.g., molecular beam epitaxy technology).
As observed in Mn2Si, the symmetry of the crystal structure and the chemical environment of the magnetic ions significantly affect the magnetic properties. Table 2 shows the atomic and total magnetic moments of the selected alloys for both the XA-type and L21-type configurations. Note that XA-type Mn2NiGa has been experimentally confirmed [45]. The calculated lattice constant a = 5.89 Å of the XA-type structure is very close to the experimental one a = 5.91 Å. The calculated Mt = 1.01 μB is less than the experimental one of 1.41 μB (measured at 5K) because Mn2NiGa undergoes martensitic transformation at low temperatures. These results indicate the reliability of our calculations. The magnetic moments of Cr1 (MCr1) and Cr2 (MCr2) (or MMn1 and MMn2) are antiparallelly aligned in the centrosymmetric L21-type configuration, and the magnitude of the magnetic moments is exactly the same, forming a typical AFM. The Y and Z atoms contribute no magnetic moment at all, and the Mt of the alloys is zero. Magnetic structure becomes more complex in the non-centrosymmetric XA-type structure. The MCr1 and MCr2 (or MMn1 and MMn2) are still aligned in an antiparallel manner, but their magnitude is different. For example, in Mn2Ge, MMn1 and MMn2 are close, while in Mn2As, MMn1 and MMn2 differ greatly. The Mt of Mn2Ge is also different from that of Mn2As. The magnetism of the XA alloy is obviously closely related to the NV.
Figure 8 shows the NV dependence of Mt for the selected Mn2-based and Cr2-based alloys. Among them, there are 9 L21-type alloys (red) and 11 XA-type alloys (black). It can be seen that the Mt of the alloys with an L21-type structure is exactly zero, and totally independent of NV. This is because in a centrosymmetric L21-type structure, MCr1 and MCr2 (or MMn1 and MMn2) are exactly the same magnitude and aligned in an antiparallel manner, so MY and MZ have no contribution to Mt. However, the Mt and NV of XA-type alloys have obvious correlation, following the Slater–Pauling rule. The deviation of Mt in some alloys from the Slater–Pauling curve is because they are not ideal half metals. For Mn2Ge and Mn2As, which are half metals, the Mt and NV satisfy the rule of Mt = NV − 18. Therefore, Mn2YZ and Cr2YZ (Y = void, Ni, Cu, and Zn; Z = Ga, Ge, and As) alloys with L21-type structures are potential candidates for conventional AFM, while those with XA-type structures are promising candidates for compensated ferrimagnets.
Figure 9 shows the DOS of the alloys selected for study. Both the L21-type and XA-type structural configurations are provided for comparison. The DOS of alloys crystallized into L21-type configuration has no spin-polarized distribution, and the distributions of the DOS for the two spin directions are completely symmetric. The zero-spin polarization and zero net magnetic moment indicate all the nine alloys that crystallize into L21-type configurations are conventional AFMs. The DOS of alloys with XA-type configuration exhibit spin-polarized distribution. This is because in non-centrosymmetric XA-type Mn2YZ and Cr2YZalloys, Mn (or Cr) atoms at site A and B have different chemical environments, which results in an asymmetric distribution of DOS for the different Mn (or Cr) atoms and the DOS of these two Mn (or Cr) atoms cannot form a fully symmetric DOS without spin polarization. Therefore, the total DOS shows spin polarization, which is similar to C1b-type Mn2Si. Therefore, alloys with XA-type structural configurations have a chance to be HMFCF. However, not all selected alloys exhibit half-metallic and fully compensated ferrimagnetic properties. XA-type Cr2NiGa has an energy gap in the spin-down channel, but is far away from the Fermi level. Only XA-type Cr2NiGe, XA-type Mn2CuGa, and XA-type Cr2ZnGe have energy gaps near the Fermi level. Cr2NiGe has an NV of 26, so Mt is not zero but 1.82 μB, which is basically consistent with Mt = NV − 28. The NV of Cr2ZnGe and Mn2CuGa are both 28, so Mt is close to zero, and they can possibly form HMFCF. The HMFCF properties of XA-type Cr2ZnGe have also been confirmed by other calculations.
Figure 10 shows the total energy differences between the tetragonal-distorted phase and cubic phase (ΔE = Etot (c/a) − Etot (c/a = 1)) and total magnetic moment (Mt), as a function of c/a for alloys, including XA-type Cr2NiGa, XA-type Mn2CuGa, XA-type Mn2CuGe, and L21-type Cr2CuZ (Z = Ga, Ge and Sn). XA-type Cr2NiGa has two energy minimums at c/a = 1 and c/a = 1.32 during the tetragonal distortion. The fact of ΔE (c/a = 1.32) < 0 indicates that the tetragonal phase is more stable than the cubic phase. XA-type Mn2CuGa has no definite energy minimum in the range from 0.9 to 1.2 of c/a, indicating that the cubic phase and tetragonal phase may coexist in the same condition. XA-type Mn2CuGe has a stable tetragonal phase (c/a = 1.16) and a metastable cubic phase (c/a = 1). L21-type Cr2CuZ (Z = Ga, Ge and As) has an energy maximum at c/a = 1 and two energy minimums at c/a < 1 and c/a > 1. These results indicate that the cubic phase is an unstable state for L21-type Cr2CuZ (Z = Ga, Ge and As), which is consistent with the positive EF calculated for Cr2CuGa and Cr2CuAs (Figure 7). The tetragonal distortion lowers the energy in the cubic phase and leads to a stable tetragonal phase, which has a negative EF. The tetragonal phase that is distorted from the XA-type cubic phase is non-centrosymmetric, resulting in the Mt of the alloys being very sensitive to the lattice distortion. The tetragonal phase that is distorted from the L21-type cubic phase is still centrosymmetric; thus, these alloys have very robust antiferromagnetic properties, with respect to the tetragonal distortion.

4. Conclusions

Due to the resulting zero net magnetic moment and high spin-polarization, HMFCFs are promising materials for future spintronic technology. In this paper, we discussed the phase stability, the site preference, the tetragonal distortion, and the influence of symmetry from the crystal structure and the chemical environments of magnetic ions on the magnetic properties of Mn2YZ and Cr2YZ (Y = void, Ni, Cu, and Zn; Z = Ga, Ge, and As) full-Heusler alloys. We found that the selected Cr2-based alloys, except for Cr2NiGa and Cr2NiGe, prefer to crystallize into the centrosymmetric L21-type structure, while the selected Mn2-based alloys, except for Mn2CuAs, Mn2ZnGe, and Mn2ZnAs, tend to crystallize into the non-centrosymmetric XA-type structure. The distributions of the DOS in the selected alloys under study are strongly dependent on the symmetry of the crystal structure and the chemical environments of each Mn or Cr atom. Alloys with magnetic ions in different chemical environments can achieve (fully) compensated ferrimagnetism, even when the number of magnetic ions is even and less than three. The AFM state of Mn2-based and Cr2-based alloys with centrosymmetric structure is very stable and cannot be destroyed by changing NV and tetragonal distortion, nor will spin-polarized DOS be generated. In contrast, the Mt of XA-type alloys with non-centrosymmetric structure depends heavily on NV, tetragonal, and generated spin-polarized DOS. These results indicate that the magnetism, distribution of DOS, and spin polarizability of the non-centrosymmetric XA-type alloys selected in this study are highly tunable. Therefore, the selected alloys with L21-type structure and their tetragonal-distorted structure are potential candidates for conventional AFMs, while those with XA-type structure and their tetragonal-distorted structure are promising candidates for (fully) compensated ferrimagnets.

Author Contributions

Conceptualization, Y.Z.; Data curation, Z.W.; Supervision, Z.L. and X.M.; Writing—original draft, Z.W.; Writing—review and editing, Y.Z. and X.M. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by the Science and Technology Plan Project of Guangzhou (No. 202102020934).

Data Availability Statement

Not applicable.

Conflicts of Interest

The authors declare no conflict of interest.

References

  1. Šmejkal, L.; Mokrousov, Y.; Yan, B.; MacDonald, A.H. Topological antiferromagnetic spintronics. Nat. Phys. 2018, 14, 242. [Google Scholar] [CrossRef]
  2. Marrows, C. Addressing an antiferromagnetic memory. Science 2016, 351, 558. [Google Scholar] [CrossRef] [PubMed]
  3. Jungwirth, T.; Marti, X.; Wadley, P.; Wunderlich, J. Electrical switching of an antiferromagnet. Nat. Nanotech. 2016, 11, 231. [Google Scholar] [CrossRef] [PubMed]
  4. Lebrun, R.; Ross, A.; Bender, S.A.; Qaiumzadeh, A.; Baldrati, L.; Cramer, J.; Brataas, A.; Duine, R.A.; Klaui, M. Tunable long-distance spin transport in a crystalline antiferromagnetic iron oxide. Nature 2018, 561, 222. [Google Scholar] [CrossRef]
  5. Baltz, V.; Manchon, A.; Tsoi, M.; Moriyama, T.; Ono, T.; Tserkovnyak, Y. Antiferromagnetic spintronics. Rev. Mod. Phys. 2018, 90, 015005. [Google Scholar] [CrossRef] [Green Version]
  6. Van Leuken, H.; de Groot, R. Half-Metallic Antiferromagnets. Phys. Rev. Lett. 1995, 74, 1171. [Google Scholar] [CrossRef] [Green Version]
  7. Wurmehl, S.; Kandpal, H.C.; Fecher, G.H.; Felser, C. Valence electron rules for prediction of half-metallic compensated-ferrimagnetic behaviour of Heusler compounds with complete spin polarization. J. Phys. Condens. Matter 2006, 18, 6171. [Google Scholar] [CrossRef]
  8. Akai, H.; Ogura, M. Half-Metallic Diluted Antiferromagnetic Semiconductors. Phys. Rev. Lett. 2006, 97, 026401. [Google Scholar] [CrossRef] [Green Version]
  9. Hu, X. Half-Metallic Antiferromagnet as a Prospective Material for Spintronics. Adv. Mater. 2012, 24, 294. [Google Scholar] [CrossRef]
  10. Manna, K.; Sun, Y.; Muechler, L.; Kübler, J.; Felser, C. Heusler, Weyl and Berry. Nat. Rev. Mater. 2018, 3, 244. [Google Scholar] [CrossRef] [Green Version]
  11. Finley, J.; Lee, C.-H.; Huang, P.Y.; Liu, L. Cubic Mn2Ga Thin Films: Crossing the Spin Gap with Ruthenium. Adv. Mater. 2018, 31, 1805361. [Google Scholar] [CrossRef] [PubMed]
  12. Kurt, H.; Rode, K.; Stamenov, P.; Venkatesan, M.; Lau, Y.C.; Fonda, E.; Coey, J.M.D. Cubic Mn2Ga Thin Films: Crossing the Spin Gap with Ruthenium. Phys. Rev. Lett. 2014, 112, 027201. [Google Scholar] [CrossRef] [PubMed] [Green Version]
  13. Skaftouros, S.; Ozdogan, K.; Sasioglu, E.; Galanakis, I. Search for spin gapless semiconductors: The case of inverse Heusler compounds. Appl. Phys. Lett. 2013, 102, 022402. [Google Scholar] [CrossRef] [Green Version]
  14. Jia, H.Y.; Dai, X.F.; Wang, L.Y.; Liu, R.; Wang, X.T.; Li, P.P.; Cui, Y.T.; Liu, G.D. Doping effect on electronic structures and band gap of inverse Heusler compound: Ti2CrSn. J. Magn. Magn. Mater. 2014, 367, 33. [Google Scholar] [CrossRef]
  15. Jia, H.Y.; Dai, X.F.; Wang, L.Y.; Liu, R.; Wang, X.T.; Li, P.P.; Cui, Y.T.; Liu, G.D. Ti2MnZ (Z = Al, Ga, In) compounds: Nearly spin gapless semiconductors. Advances 2014, 4, 047113. [Google Scholar] [CrossRef] [Green Version]
  16. Zhang, Y.J.; Liu, Z.H.; Liu, E.K.; Liu, G.D.; Ma, X.Q.; Wu, G.H. Towards fully compensated ferrimagnetic spin gapless semiconductors for spintronic applications. Europhys. Lett. 2015, 111, 37009. [Google Scholar] [CrossRef]
  17. Fang, Q.-L.; Zhao, X.-M.; Zhang, J.-M.; Xu, K.-W. Magnetic properties and half-metallic in bulk and (001) surface of Ti2MnAl Heusler alloy with Hg2CuTi-type structure. Thin Solid Film. 2014, 558, 241. [Google Scholar] [CrossRef]
  18. Zhang, Y.J.; Liu, Z.H.; Liu, G.D.; Ma, X.Q.; Cheng, Z.X. Robust fully-compensated ferrimagnetism and semiconductivity in inverse Heusler compounds: Ti2VZ (Z = P, As, Sb, Bi). J. Magn. Magn. Mater. 2018, 449, 515. [Google Scholar] [CrossRef]
  19. Galanakis, I.; Özdoğan, K.; Şaşıoğlu, E.; Aktaş, B. Ab initio design of half-metallic fully compensated ferrimagnets: The case of Cr2MnZ (Z = P, As, Sb, and Bi). Phys. Rev. B 2007, 75, 172405. [Google Scholar] [CrossRef] [Green Version]
  20. Galanakis, I. First-principles electronic and magnetic properties of the half-metallic antiferromagnet Cr2MnSb. J. Magn. Magn. Mater. 2009, 321, L34. [Google Scholar]
  21. Galanakis, I.; Ozdogan, K.; Sasioglu, E.; Aktas, B. Doping of Mn2VAl and Mn2VSi Heusler alloys as a route to half-metallic antiferromagnetism. Phys. Rev. B 2007, 75, 092407. [Google Scholar] [CrossRef] [Green Version]
  22. Manuel, M.M.; Geisler, P. Phase stability of chromium based compensated ferrimagnets with inverse Heusler structure. J. Magn. Magn. Mater. 2013, 341, 72. [Google Scholar]
  23. Galanakis, I.; Şaşıoglu, E. High TC half-metallic fully-compensated ferrimagnetic Heusler compounds. Appl. Phys. Lett. 2011, 99, 052509. [Google Scholar] [CrossRef] [Green Version]
  24. Žic, M.; Rode, K.; Thiyagarajah, N.; Lau, Y.-C.; Betto, D.; Coey, J.M.D.; Sanvito, S.; O’Shea, K.J.; Ferguson, C.A.; MacLaren, D.A.; et al. Designing a fully compensated half-metallic ferrimagnet. Phys. Rev. B 2016, 93, 140202. [Google Scholar] [CrossRef] [Green Version]
  25. Zhang, Y.J.; Liu, Z.H.; Liu, G.D.; Ma, X.Q. Half-metallic fully compensated ferrimagnetism in C1b-type half Heusler compounds Mn2Si1-xGex. J. Magn. Magn. Mater. 2015, 387, 67. [Google Scholar] [CrossRef]
  26. Luo, H.Z.; Liu, G.D.; Meng, F.B.; Wang, W.H.; Wu, G.H.; Zhu, X.X.; Jiang, C.B. Half-metallicity and magnetic properties of half-Heusler type Mn2Sn: Ab initio predictions. Phys. B 2011, 406, 4245. [Google Scholar] [CrossRef]
  27. Wang, X.; Cheng, Z.; Wang, J.; Liu, G. A full spectrum of spintronic properties demonstrated by a C1b-type Heusler compound Mn 2 Sn subjected to strain engineering. J. Mater. Chem. C 2016, 4, 8535. [Google Scholar] [CrossRef]
  28. Galanakis, I.; Özdoğan, K.; Şaşıoglu, E. High-TC fully compensated ferrimagnetic semiconductors as spin-filter materials: The case of CrVXAl (X= Ti, Zr, Hf) Heusler compounds. J. Phys. Condens. Matter 2014, 26, 086003. [Google Scholar] [CrossRef]
  29. Venkateswara, Y.; Gupta, S.; Samatham, S.S.; Varma, M.R.; Suresh, K.G.; Alam, A. Competing magnetic and spin-gapless semiconducting behavior in fully compensated ferrimagnetic CrVTiAl: Theory and experiment. Phys. Rev. B 2018, 97, 054407. [Google Scholar] [CrossRef] [Green Version]
  30. Zhang, Y.J.; Liu, Z.H.; Wu, Z.G.; Ma, X.Q. Prediction of fully compensated ferromagnetic spin-gapless semiconducting FeMnGa/Al/In half Heusler alloys. IUCrJ 2019, 6, 610. [Google Scholar] [CrossRef] [Green Version]
  31. Lamrani, A.F.; Ouchri, M.; Benyoussef, A.; Belaiche, M.; Loulidi, M. Half-metallic antiferromagnetic behavior of double perovskite Sr2OsMoO6: First principle calculations. J. Magn. Magn. Mater. 2013, 345, 195. [Google Scholar] [CrossRef]
  32. Heusler, F.; Dtsch, V. Über magnetische manganlegierungen. Phys. Ges. 1903, 5, 219. [Google Scholar]
  33. Graf, T.; Felser, C.; Parkin, S.S.P. Simple rules for the understanding of Heusler compounds. Prog. Solid State Chem. 2011, 39, 1. [Google Scholar] [CrossRef]
  34. Galanakis, I.; Dederichs, P.; Papanikolaou, N. Origin and properties of the gap in the half-ferromagnetic Heusler alloys. Phys. Rev. B 2002, 66, 134428. [Google Scholar] [CrossRef] [Green Version]
  35. Galanakis, I.; Dederichs, P.; Papanikolaou, N. Slater-Pauling behavior and origin of the half-metallicity of the full-Heusler alloys. Phys. Rev. B 2002, 66, 174429. [Google Scholar] [CrossRef] [Green Version]
  36. Skaftouros, S.; Özdoğan, K.; Şaşıoğlu, E.; Galanakis, I. Generalized Slater-Pauling rule for the inverse Heusler compounds. Phys. Rev. B 2013, 87, 024420. [Google Scholar] [CrossRef]
  37. Ozdogan, K.; Galanakis, I.; Sasioglu, E.; Atkas, B. Search for half-metallic ferrimagnetism in V-based Heusler alloys Mn2VZ (Z = Al, Ga, In, Si, Ge, Sn). J. Phys. Condens. Matter 2006, 18, 2905. [Google Scholar] [CrossRef] [Green Version]
  38. Yousuf, S.; Gupta, D.C. Robustness in spin polarization and thermoelectricity in newly tailored Mn2-based Heusler alloys. Indian J. Phys. 2018, 92, 855. [Google Scholar] [CrossRef]
  39. Nayak, A.K.; Kumar, V.; Ma, T.; Werner, P.; Pippel, E.; Sahoo, R.; Damay, F.; Rößler, U.K.; Felser, C.; Parkin, S.S.P. Magnetic antiskyrmions above room temperature in tetragonal Heusler materials. Nature 2017, 548, 561. [Google Scholar] [CrossRef]
  40. Liu, Z.H.; Zhang, Y.J.; Zhang, H.G.; Zhang, X.J.; Ma, X.Q. Giant exchange bias in Mn2FeGa with hexagonal structure. Appl. Phys. Lett. 2016, 109, 032408. [Google Scholar] [CrossRef]
  41. Nayak, A.K.; Nicklas, M.; Chadov, S.; Khuntia, P.; Shekhar, C.; Kalache, A.; Baenitz, M.; Skourski, Y.; Guduru, V.K.; Puri, A.; et al. Design of compensated ferrimagnetic Heusler alloys for giant tunable exchange bias. Nat. Mater. 2015, 14, 679. [Google Scholar] [CrossRef] [PubMed] [Green Version]
  42. Ouardi, S.; Fecher, G.H.; Felser, C. Realization of Spin Gapless Semiconductors: The Heusler Compound Mn2CoAl. Phys. Rev. Lett. 2013, 110, 100401. [Google Scholar] [CrossRef] [PubMed] [Green Version]
  43. Kreiner, G.; Kalache, A.; Hausdorf, S.; Alijani, V.; Qian, J.-F.; Shan, G.; Burkhardt, U.; Ouardi, S.; Felser, C. New Mn2-based Heusler compounds. Z. Anorg. Allg. Chem. 2014, 640, 738. [Google Scholar] [CrossRef]
  44. Faleev, S.V.; Ferrante, Y.; Jeong, J.; Samant, M.G.; Jones, B.; Parkin, S.S.P. Origin of the tetragonal ground state of Heusler compounds. Phys. Rev. Appl. 2017, 7, 034022. [Google Scholar] [CrossRef] [Green Version]
  45. Liu, G.D.; Chen, J.L.; Liu, Z.H.; Dai, X.F.; Wu, G.H.; Zhang, B.; Zhang, X.X. Martensitic transformation and shape memory effect in a ferromagnetic shape memory alloy: Mn2NiGa. Appl. Phys. Lett. 2005, 87, 262504. [Google Scholar] [CrossRef]
  46. Hafner, J. Ab-initio simulations of materials using VASP: Density-functional theory and beyond. J. Comput. Chem. 2008, 29, 2044. [Google Scholar] [CrossRef]
  47. Perdew, J.P. Density-functional approximation for the correlation energy of the inhomogeneous electron gas. Phys. Rev. B 1986, 33, 8822. [Google Scholar] [CrossRef]
  48. Perdew, J.P.; Yue, W. Accurate and simple density functional for the electronic exchange energy: Generalized gradient approximation. Phys. Rev. B 1986, 33, 8800. [Google Scholar] [CrossRef]
  49. Mavropoulos, P.; Galanakis, I.; Popescu, V.; Dederichs, P.H. The influence of spin–orbit coupling on the band gap of Heusler alloys. J. Phys. Condens. Matter 2004, 16, S5759. [Google Scholar] [CrossRef]
  50. Blaha, P.; Schwarz, K.; Madsen, G.K.; Kvasnicka, D.; Luitz, J. WIEN2K: An Augmented Plane Wave Plus Local Orbitals Program for Calculating Crystal Properties; Vienna University of Technology: Vienna, Austria, 2001. [Google Scholar]
  51. Wang, X.L. Proposal for a new class of materials: Spin gapless semiconductors. Phys. Rev. Lett. 2008, 100, 156404. [Google Scholar] [CrossRef] [Green Version]
  52. Fujii, S.; Ishida, S.; Asano, S. High Spin Polarization of Ferrimagnetic Half-Heusler Compounds FeCrZ and MnYZ (Y = Mn, Cr; Z = IIIb, IVb, Vb Elements). J. Phys. Soc. Jpn. 2010, 79, 124702. [Google Scholar] [CrossRef]
  53. Balke, B.; Fecher, G.H.; Winterlik, J.; Felser, C. Mn3Ga, a compensated ferrimagnet with high Curie temperature and low magnetic moment for spin torque transfer applications. Appl. Phys. Lett. 2007, 90, 152504. [Google Scholar] [CrossRef]
  54. Ma, J.; He, J.; Mazumdar, D.; Munira, K.; Keshavarz, S.; Lovorn, T.; Ghosh, A.W.; William, H.B. Computational investigation of inverse Heusler compounds for spintronics applications. Phys. Rev. B 2018, 98, 094410. [Google Scholar] [CrossRef] [Green Version]
  55. Guo, G.Y.; Botton, G.A.; Nishino, Y. Electronic structure of possible 3d ‘heavy-fermion’ compound. J. Phys. Condens. Matter 1998, 10, L119. [Google Scholar] [CrossRef]
  56. Zhang, Y.J.; Liu, Z.H.; Liu, G.D.; Ma, X.Q. Structural, electronic and magnetic properties of CoFeTiGa1− xSbx compounds. J. Magn. Magn. Mater. 2017, 422, 32. [Google Scholar] [CrossRef]
  57. Tan, J.G.; Liu, Z.H.; Zhang, Y.J.; Li, G.T.; Zhang, H.G.; Liu, G.D.; Ma, X.Q. Site preference and tetragonal distortion of Heusler alloy Mn-Ni-V. Results Phys. 2009, 321, L34. [Google Scholar] [CrossRef]
  58. Liu, G.; Dai, X.; Liu, H.; Chen, J.; Li, Y.; Xiao, G.; Wu, G. Mn2CoZ (Z = Al, Ga, In, Si, Ge, Sn, Sb) compounds: Structural, electronic, and magnetic properties. Phys. Rev. B 2008, 77, 014424. [Google Scholar] [CrossRef] [Green Version]
  59. Zhang, Y.J.; Li, G.J.; Liu, E.K.; Chen, J.L.; Wang, W.H.; Wu, G.H. Ferromagnetic structures in Mn2CoGa and Mn2CoAl doped by Co, Cu, V, and Ti. J. Appl. Phys. 2013, 113, 123901. [Google Scholar]
  60. Geiersbach, U.; Bergmann, A.; Westerholt, K. Structural, magnetic and magnetotransport properties of thin films of the Heusler alloys Cu2MnAl, Co2MnSi, Co2MnGe and Co2MnSn. J. Magn. Magn. Mater. 2002, 240, 546. [Google Scholar] [CrossRef]
  61. Amari, S.; Dahmane, F.; Omran, S.B.; Doumi, B. Theoretical investigation of the structural, magnetic and band structure characteristics of Co2FeGe1−xSix (x = 0, 0.5, 1) full-Heusler alloys. J. Korean Phys. Soc. 2016, 69, 1462. [Google Scholar] [CrossRef]
  62. Wang, X.; Cheng, Z.; Yuan, H.; Khenata, R. L21 and XA ordering competition in titanium-based full-Heusler alloys. J. Mater. Chem. C 2017, 5, 11559. [Google Scholar] [CrossRef]
  63. Du, X.; Zhang, Y.; Liu, Z.; Wu, Z.; Xu, S.; Ma, X. Site preference, magnetic and electronic properties of half-metallic Vanadium-based full Heusler alloys. J. Magn. Magn. Mater. 2021, 517, 167379. [Google Scholar] [CrossRef]
  64. Berri, S. Electronic structure and half-metallicity of the new Heusler alloys PtZrTiAl, PdZrTiAl and Pt0. 5Pd0. 5ZrTiAl. Chin. J. Phys. 2017, 55, 195. [Google Scholar] [CrossRef]
  65. Paudel, R.; Zhu, J.C. Investigation of half-metallicity and magnetism of bulk and (111)-surfaces of Fe2MnP full Heusler alloy. Vacuum 2019, 164, 336. [Google Scholar] [CrossRef]
  66. Paudel, R.; Kaphle, G.C.; Batouche, M.; Zhu, J.C. Half-metallicity, magnetism, mechanical, thermal, and phonon properties of FeCrTe and FeCrSe half-Heusler alloys under pressure. Int. J. Quantum Chem. 2020, 120, e26417. [Google Scholar] [CrossRef]
Symmetry 14 00988 g001 550
Figure 1. Schematic diagram of density of states for conventional AFM (a) and (b) HMFCF. EF is the Fermi Energy.
Figure 1. Schematic diagram of density of states for conventional AFM (a) and (b) HMFCF. EF is the Fermi Energy.
Symmetry 14 00988 g001
Symmetry 14 00988 g002 550
Figure 2. Band structure of C1b-type Mn2Si calculated by pseudopotential method (a) and by full-potential method (b) at the equilibrium state, (c) spin-polarized DOS, (d) magnetic moments as functions of the lattice parameter for Mn (A), Mn (B) and Mn2Si.
Figure 2. Band structure of C1b-type Mn2Si calculated by pseudopotential method (a) and by full-potential method (b) at the equilibrium state, (c) spin-polarized DOS, (d) magnetic moments as functions of the lattice parameter for Mn (A), Mn (B) and Mn2Si.
Symmetry 14 00988 g002
Symmetry 14 00988 g003 550
Figure 3. Schematic representation of C1b-Mn2Si (a), ZnS-Mn2 (b), and C1-Mn2Si (c) in a Heusler matrix. Here, C site is vacant for C1b-Mn2Si, C and D sites are vacant for ZnS-Mn2, and B site is vacant for C1-Mn2Si. Mn1 and Mn2 in C1-Mn2Si are denoted at Mn (A) and Mn (C), while those in C1b-Mn2Si and ZnS-Mn2 are denoted as Mn (A) and Mn (B) since they occupy the different Wyckoff coordinates.
Figure 3. Schematic representation of C1b-Mn2Si (a), ZnS-Mn2 (b), and C1-Mn2Si (c) in a Heusler matrix. Here, C site is vacant for C1b-Mn2Si, C and D sites are vacant for ZnS-Mn2, and B site is vacant for C1-Mn2Si. Mn1 and Mn2 in C1-Mn2Si are denoted at Mn (A) and Mn (C), while those in C1b-Mn2Si and ZnS-Mn2 are denoted as Mn (A) and Mn (B) since they occupy the different Wyckoff coordinates.
Symmetry 14 00988 g003
Symmetry 14 00988 g004 550
Figure 4. Integrated DOS of C1b-Mn2Si (a) and C1-Mn2Si (b).
Figure 4. Integrated DOS of C1b-Mn2Si (a) and C1-Mn2Si (b).
Symmetry 14 00988 g004
Symmetry 14 00988 g005 550
Figure 5. Atomic-resolved PDOS and the net DOS of total as a function of energy for C1b-Mn2Si (a), C1-Mn2Si (b), and ZnS-Mn2 (c).
Figure 5. Atomic-resolved PDOS and the net DOS of total as a function of energy for C1b-Mn2Si (a), C1-Mn2Si (b), and ZnS-Mn2 (c).
Symmetry 14 00988 g005
Symmetry 14 00988 g006 550
Figure 6. Free energy difference (ΔE) between XA-type and L21-type structures, ΔE = E (XA) − E (L21), for Cr2YZ and Mn2YZ (Y = void, Ni, Cu, and Zn; Z = Ga, Ge, and As). The case of ΔE < 0 suggests the XA-type structure of the alloy is more energy stable, while ΔE > 0 indicates a L21-type structure of the alloy.
Figure 6. Free energy difference (ΔE) between XA-type and L21-type structures, ΔE = E (XA) − E (L21), for Cr2YZ and Mn2YZ (Y = void, Ni, Cu, and Zn; Z = Ga, Ge, and As). The case of ΔE < 0 suggests the XA-type structure of the alloy is more energy stable, while ΔE > 0 indicates a L21-type structure of the alloy.
Symmetry 14 00988 g006
Symmetry 14 00988 g007 550
Figure 7. Formation energy vs NV for Cr2YZ and Mn2YZ (Y = void, Ni, Cu, and Zn; Z = Ga, Ge, and As) alloys.
Figure 7. Formation energy vs NV for Cr2YZ and Mn2YZ (Y = void, Ni, Cu, and Zn; Z = Ga, Ge, and As) alloys.
Symmetry 14 00988 g007
Symmetry 14 00988 g008 550
Figure 8. Mt vs NV for Cr2YZ and Mn2YZ (Y = void, Ni, Cu, and Zn; Z = Ga, Ge, and As) full-Heusler alloys. The dotted lines represent the three typical Slater–Pauling curves.
Figure 8. Mt vs NV for Cr2YZ and Mn2YZ (Y = void, Ni, Cu, and Zn; Z = Ga, Ge, and As) full-Heusler alloys. The dotted lines represent the three typical Slater–Pauling curves.
Symmetry 14 00988 g008
Symmetry 14 00988 g009 550
Figure 9. DOS per formula unit for both XA-type (yellow fill) and L21-type (red dotted line) structure configurations of (a) Cr2NiGa, (b) Cr2NiGe, (c) Mn2CuGe, (d) Cr2CuGe, (e) Mn2CuGa, (f) Cr2ZnGe, (g) Cr2CuGa, (h) Cr2NiAs, (i) Cr2ZnAs, (j) Mn2ZnGe, (k) Mn2ZnGa, (l) Mn2ZnAs, (m) Mn2NiGa, (n) Mn2NiAs, (o) Cr2ZnGa, (p) Mn2NiGe, (q) Mn2CuAs and (r) Cr2CuAs.
Figure 9. DOS per formula unit for both XA-type (yellow fill) and L21-type (red dotted line) structure configurations of (a) Cr2NiGa, (b) Cr2NiGe, (c) Mn2CuGe, (d) Cr2CuGe, (e) Mn2CuGa, (f) Cr2ZnGe, (g) Cr2CuGa, (h) Cr2NiAs, (i) Cr2ZnAs, (j) Mn2ZnGe, (k) Mn2ZnGa, (l) Mn2ZnAs, (m) Mn2NiGa, (n) Mn2NiAs, (o) Cr2ZnGa, (p) Mn2NiGe, (q) Mn2CuAs and (r) Cr2CuAs.
Symmetry 14 00988 g009
Symmetry 14 00988 g010 550
Figure 10. The total energy differences (black solid circle) between tetragonal-distorted phase and cubic phase (ΔE = Etot (c/a) − Etot (c/a = 1)) and the corresponding magnetic moment (red hollow circle) as a function of c/a for (a) XA-type Cr2NiGa, (b) XA-type Mn2CuGa, (c) XA-type Mn2CuGe, (d) L21-type Cr2CuGa, (e) L21-type Cr2CuGe, and (f) L21-type Cr2CuAs alloys. The zero point corresponds to the energy in the cubic phase.
Figure 10. The total energy differences (black solid circle) between tetragonal-distorted phase and cubic phase (ΔE = Etot (c/a) − Etot (c/a = 1)) and the corresponding magnetic moment (red hollow circle) as a function of c/a for (a) XA-type Cr2NiGa, (b) XA-type Mn2CuGa, (c) XA-type Mn2CuGe, (d) L21-type Cr2CuGa, (e) L21-type Cr2CuGe, and (f) L21-type Cr2CuAs alloys. The zero point corresponds to the energy in the cubic phase.
Symmetry 14 00988 g010
Table
Table 1. The crystal structure and corresponding chemical environments of first nearest neighbors (Fnn) of each Mn atom of studied compounds.
Table 1. The crystal structure and corresponding chemical environments of first nearest neighbors (Fnn) of each Mn atom of studied compounds.
Compd.StructureSpace GroupGroup No.Fnn of Mn1Fnn of Mn2
Mn2SiC1bF-43m2164Mn2, 4Si4Mn1
Mn2SiC1Fm-3m2254Si4Si
Mn2ZnSFD-3m2274Mn24Mn1
Table
Table 2. NV, equilibrium lattice constants, atomic-resolved and total magnetic moments per unit cell in μB for Cr2YZ and Mn2YZ (Y = void, Ni, Cu, and Zn; Z = Ga, Ge, and As) full-Heusler alloys. Both the XA-type and L21 type structures were considered.
Table 2. NV, equilibrium lattice constants, atomic-resolved and total magnetic moments per unit cell in μB for Cr2YZ and Mn2YZ (Y = void, Ni, Cu, and Zn; Z = Ga, Ge, and As) full-Heusler alloys. Both the XA-type and L21 type structures were considered.
AlloysNVStructureaMX1MX2MYMZMt
Cr2Ga15XA5.924.11−4.47-−0.09−0.45
L216.01−4.764.76-00
Cr2Ge16XA5.823.67−4.07-0.07−0.46
L215.90−4.374.37-00
Cr2As17XA5.77−3.773.57-0.03−0.17
L215.87−4.314.31-00
Mn2Ga17XA5.76−3.873.73-−0.05−0.19
L215.86−4.384.38-00
Mn2Ge18XA5.63−3.123.13-−0.010
L215.84−4.294.29-00
Mn2As19XA5.64−2.703.64-0.061.00
L215.79−3.983.98-00
Cr2NiGa25XA5.93−3.634.050.350.050.83
L216.03−4.234.23000
Cr2NiGe26XA5.85−2.423.750.460.041.82
L215.96−3.793.79000
Cr2CuGa26XA6.05−4.084.3500.030.30
L216.10−4.444.44000
Cr2NiAs27XA5.89−2.954.010.260.081.40
L215.94−3.513.51000
Cr2CuGe27XA6.02−3.934.20−0.010.030.30
L216.05−4.204.20000
Cr2ZnGa27XA6.13−4.264.2900.010.04
L216.15−4.484.48000
Mn2NiGa27XA5.89−3.324.050.210.071.01
L215.93−3.823.82000
Cr2CuAs28XA6.08−4.134.31−0.010.040.21
L216.08−4.234.23000
Cr2ZnGe28XA6.11−4.064.170.030.010.15
L216.16−4.454.45000
Mn2CuGa28XA5.93−3.64−3.96−0.030.030.32
L215.99−3.993.99000
Mn2NiGe28XA5.87−3.203.930.060.080.88
L215.87−3.383.38000
Mn2NiAs29XA5.90−3.484.140.010.090.93
L215.88−3.253.25000
Mn2CuGe29XA5.96−3.474.02−0.060.10.59
L215.98−3.763.76000
Mn2ZnGa29XA6.05−3.754.01−0.060.090.29
L216.08−3.963.96000
Cr2ZnAs29XA6.23−4.274.480.050.010.28
L216.25−4.504.50000
Mn2CuAs30XA5.98−2.954.07−0.040.131.21
L216.00−3.643.64000
Mn2ZnGe30XA6.09−3.574.02−0.080.100.46
L216.07−3.703.70000
Mn2ZnAs31XA6.12−3.134.10−0.110.141.01
L216.08−3.513.51000
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Share and Cite

MDPI and ACS Style

Wu, Z.; Zhang, Y.; Liu, Z.; Ma, X. Influence of Symmetry from Crystal Structure and Chemical Environments of Magnetic Ions on the Fully Compensated Ferrimagnetism of Full Heusler Cr2YZ and Mn2YZ Alloys. Symmetry 2022, 14, 988. https://doi.org/10.3390/sym14050988

AMA Style

Wu Z, Zhang Y, Liu Z, Ma X. Influence of Symmetry from Crystal Structure and Chemical Environments of Magnetic Ions on the Fully Compensated Ferrimagnetism of Full Heusler Cr2YZ and Mn2YZ Alloys. Symmetry. 2022; 14(5):988. https://doi.org/10.3390/sym14050988

Chicago/Turabian Style

Wu, Zhigang, Yajiu Zhang, Zhuhong Liu, and Xingqiao Ma. 2022. "Influence of Symmetry from Crystal Structure and Chemical Environments of Magnetic Ions on the Fully Compensated Ferrimagnetism of Full Heusler Cr2YZ and Mn2YZ Alloys" Symmetry 14, no. 5: 988. https://doi.org/10.3390/sym14050988

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Metrics

Back to TopTop