L2₁ and XA Ordering Competition in Hafnium-Based Full-Heusler Alloys Hf₂VZ (Z = Al, Ga, In, Tl, Si, Ge, Sn, Pb)

Abstract

For theoretical designing of full-Heusler based spintroinc materials, people have long believed in the so-called Site Preference Rule (SPR). Very recently, according to the SPR, there are several studies on XA-type Hafnium-based Heusler alloys X₂YZ, i.e., Hf₂VAl, Hf₂CoZ (Z = Ga, In) and Hf₂CrZ (Z = Al, Ga, In). In this work, a series of Hf₂-based Heusler alloys, Hf₂VZ (Z = Al, Ga, In, Tl, Si, Ge, Sn, Pb), were selected as targets to study the site preferences of their atoms by first-principle calculations. It has been found that all of them are likely to exhibit the L2₁-type structure instead of the XA one. Furthermore, we reveal that the high values of spin-polarization of XA-type Hf₂VZ (Z = Al, Ga, In, Tl, Si, Ge, Sn, Pb) alloys have dropped dramatically when they form the L2₁-type structure. Also, we prove that the electronic, magnetic, and physics nature of these alloys are quite different, depending on the L2₁-type or XA-type structures.

1. Introduction

Heusler alloys are a noticeable class of intermetallic materials that represent as usual by the formula X₂YZ (often called full-Heusler) [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15] or XYZ (usually named as half-Heusler) [16], where X, Y are transition-metal-element atoms and Z is a main group element. The structure of full-Heusler alloys consists of four interpenetrating fcc lattices with four equidistant sites as basis along the diagonal of the unit cell. According to the well-known Site Preference Rule (SPR) [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15], when the valence of the X is larger than that of Y, the atomic sequence is X^A-Y^B-X^C-Z^D and the structure is the well-known L2₁ one with prototype Cu₂MnAl. Otherwise, the alloys crystallize in the so-called XA structure, where the sequence of the atoms is then X^A-X^B-Y^C-Z^D and the prototype is Hg₂CuTi. The latter alloys are also named as inverse Heusler alloys.

To the best of our knowledge, SPR has been applied extensively in the theoretical design of full-Heusler alloys and in predictions of their electronic, magnetic, and transport behavior. Some XA-type full-Heusler alloys, Mn₂CoAl [16], Ti₂MnAl [17], and Ti₂CoSi [18], were predicted to be novel spin-gapless semiconductors (SGSs) [19,20]. Furthermore, lots of XA-type full-Heusler alloys, Sc₂-, V₂-, Cr₂-, Mn₂-, Ti₂-, Zr₂-, and even Hf₂-based alloys, were revealed to be excellent half-metallic materials (HMMs) [21]. Surprisingly, one counterexample after another, including X₂CuAl [22] and Ti₂FeZ (Z = Al, Ga) alloys [6], has been reported very recently, in which these alloys show the L2₁-type structure and disobey the SPR, so that Y with more electrons enters the B sites.

In this work, a systematic theoretical work has been carried out to examine whether the conventional SPR was suitable for the Hf₂-based highly-ordered full-Heusler alloys. To this end, the competition between the XA and L2₁ orderings of Hf₂VZ (Z = Al, Ga, In, Tl, Si, Ge, Sn, Pb) full-Heusler alloys has been studied through the first-principles calculations. Our current work shows that not all the full-Heusler alloys obey the SPR, and further, the SPR can not be considered as a concise judgement principle for the structure of highly-ordered full-Heusler alloys. Remarkably, we exhibit that atomic site occupation in these Hf₂VZ alloys is decisive in determining their electronic, magnetic, and Slater-Pauling properties. For the L2₁-type Hf₂VZ (Z = Al, Ga, In, Tl, Si, Ge, Sn, Pb) full-Heusler alloys, the phase stability from the aspect of the formation energy and mechanical behaviors has also been examined in term of theory. Details of the results are shown in the following discussion.

2. Computational Details

First-principles electronic-structure calculations were performed using density function theory (DFT) implemented in the CASTEP code [23,24] according to the plane-wave pseudo-potential method. The generalized gradient approximation (GGA) [25] was adopted for the exchange-correlation functional. For the XA-type and L2₁-type Heusler alloys Hf₂VZ (Z = Al, Ga, In, Tl, Si, Ge, Sn, Pb), a Monkhorst-Pack special k-point mesh of 9 × 9 × 9 was used in the Brillouin zone integrations with a cutoff energy of 400 eV and a self-consistent field tolerance of 10⁻⁶ eV. The quality of the k-point separation for the band structure calculation is 0.01 Å⁻¹.

As shown in Figure 1, the crystal structures of XA and L2₁ types Hf₂VZ Heusler alloys have been given. For the former, the two Hf atoms sitting at the two inequivalent sites, and Hf 1, Hf 2, V and Z atoms occupy the 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), respectively; For the latter, the two Hf atoms are essentially equivalent, and they have the same atomic environment. Hf 1, Hf 2, V and Z atoms occupy the Wyckoff coordinates A (0, 0, 0), B (0.5, 0.5, 0.5), C (0.25, 0.25, 0.25) and D (0.75, 0.75, 0.75), respectively.

3. Results and Discussion

3.1. Competition of L2₁ and XA Structure Ordering in Full-Heusler Hf₂VZ Alloys

First, to confirm the ground state of Hf₂VZ (Z = Al, Ga, In, Tl, Si, Ge, Sn, Pb) full-Heusler alloys, the geometry optimization has been performed by calculating the total energies as functions of the lattice constant (cell volume) [26]. In this work, three possible magnetic states, i.e., paramagnetic (PM), ferrimangetic (FiM), and antiferromagnetic (AFM) states are taken into account. The PM (or nonmagnetic) state means that the constituent atoms of Hf₂VZ have no spin polarization. The FiM state implies that the spin magnetic moments of Hf atoms align anti-parallel to those of the V atoms, and the total magnetic moment is not equal to zero, while the AFM state means the spin magnetic moments of Hf atoms align antiparallelly to those of the V atoms, and the total magnetic moment is equal to zero. For our Hf₂VZ (Z = Al, Ga, In, Tl, Si, Ge, Sn, Pb) full-Heusler alloys, the FiM state is the most stable among three magnetic states.

As shown in Figure 2, the total energy as functions of lattice constants of full-Heusler Hf₂VZ (Z = Al, Ga, In, Tl, Si, Ge, Sn, Pb) alloys with two atomic occupation orderings, XA and L2₁, and with their most stable magnetic state, FiM, have been exhibited. Obviously, for all these alloys, the L2₁ state has lower energies than XA state. Therefore, Hf₂VZ alloys studied in current work prefer to form the L2₁-type structure as ground state with equilibrium lattice constant of 6.69 Å, 6.66 Å, 6.90 Å, 6.89 Å, 6.56 Å, 6.61 Å, 6.82 Å and 6.90 Å, respectively. The obtained equilibrium lattice constants of these alloys with XA-type and L2₁-type structures, respectively, and the energy differences between these two structures ($\Delta E=E_{XA}^{total}-E_{L{2}_1}^{total}$) for Hf₂VZ (Z = Al, Ga, In, Tl, Si, Ge, Sn, Pb) alloys are also listed in

Table 1. The calculated energy difference ΔE, lattice constant a, total and atomic spin magnetic moments for Hf₂VZ alloys with L2₁ and XA structures, respectively.

Alloy	Structure	ΔE (eV/cell)	a (Å)	Mt (μB/f.u.)	MHf 1 (μB)	MHf 2 (μB)	MV (μB)	MZ (μB)	Stable Structure
Hf2VAl	XA	0.64	6.71	2	−0.65	0.14	2.62	−0.12	L21
	L21		6.69	1.65	−0.21	−0.21	2.19	−0.11
Hf2VGa	XA	0.71	6.68	1.99	−0.59	0.15	2.55	−0.13	L21
	L21		6.66	1.68	−0.13	−0.13	2.06	−0.12
Hf2VIn	XA	0.39	6.88	2	−0.75	−0.01	2.88	−0.12	L21
	L21		6.90	1.70	−0.24	−0.24	2.31	−0.13
Hf2VTl	XA	0.32	6.92	1.98	−0.79	−0.1	3.0	−0.12	L21
	L21		6.89	1.61	−0.17	−0.17	2.06	−0.11
Hf2VSi	XA	0.63	6.57	1.02	−0.74	−0.25	2.08	−0.06	L21
	L21		6.56	0.24	−0.03	−0.03	0.31	−0.01
Hf2VGe	XA	0.58	6.64	1.03	−0.83	−0.34	2.26	−0.06	L21
	L21		6.61	0.2	−0.02	−0.02	0.26	−0.01
Hf2VSn	XA	0.18	6.85	1	−1.07	−0.51	2.64	−0.05	L21
	L21		6.82	0.49	−0.06	−0.06	0.63	−0.03
Hf2VPb	XA	0.11	6.94	1	−1.17	−0.63	2.87	−0.04	L21
	L21		6.90	0.32	−0.03	−0.03	0.41	−0.02

. A higher value of ΔE indicates the L2₁-type structure is more stable than XA-type. The highest positive value of 0.71 eV/cell appears in Hf₂VGa alloy, reflecting that the site preference of V for the B position is quite strong. Hence, compared to other Hf₂-based Heusler alloys, XA-type Hf₂VGa maybe more difficult to synthesize experimentally due to its largest ΔE.

We should point out here that, as we said in the part of Introduction, Hf₂VZ alloys or even all the Hf-based full Heusler alloys should exhibit XA-type Heusler structure on basis of the SPR. Surprisingly, in current work, our results break the traditional SPR. Also, the data in this work, together with the latest scientific findings in References [6,22,27,28], are sufficient to demonstrate that not all the full-Heusler alloys X₂YZ obey the well-known SPR, especially X are low-valent transition metals. Since the SPR can not be regarded as the only way to determine the competition of XA and L2₁ structural ordering in Heusler alloys, alloys with L2₁-type structure should also be taken into account in the previous works [1,2,3,7,8,9,10,11].

3.2. Structural and Mechanical Properties of Heusler-Based Hf₂VZ with L2₁-Type Ordering

Further, we aim to check the structural stability of the L2₁-type full-Heusler Hf₂VZ according to their calculated formation energies and mechanical properties. Similar methods [29,30] have been applied extensively to analyze the stability for Heusler alloys in term of theory. The formation energies of these alloys can be obtained from the following equation:

$$E_f=E_{H{f}_{2}VZ}^{total}-(E_{Hf1}^{bulk}+E_{Hf2}^{bulk}+E_V^{bulk}+E_Z^{bulk})$$

(1)

where $E_{H{f}_{2}VZ}^{total}$ is the total energy of Hf₂VZ per formula unit, and $E_{Hf1}^{bulk}$, $E_{Hf2}^{bulk}$, $E_V^{bulk}$, and $E_Z^{bulk}$ are the total energies per atom of each element in the bulk form for the Hf 1, Hf 2, V, and Z, respectively. The results have been given in

Table 2. Calculated elastic constants C_ij, bulk modulus B, shear modulus G, Young’s modulus E (GPa), Pugh’s ratio B/G, anisotropy factor A, and formation energy (eV) for Hf₂VZ alloys with L2₁ structure.

Alloy	C11	C12	C44	B	G	E	B/G	Formation Energy	Anisotropy Factor
Hf2VAl	164.4	117.6	73.0	133.2	46.8	124.6	2.8	−0.54	3.10
Hf2VGa	143	104.3	72.9	117.9	43.0	115.10	2.7	−1.19	3.76
Hf2VIn	179.8	154.3	49.7	162.8	28.9	82.0	5.6	−0.05	3.89
Hf2VTl	110.4	100.1	46.8	103.5	20.5	57.9	5.0	−0.85	9.08
Hf2VSi	191	146.9	63.7	161.6	41.6	115.12	3.8	−0.66	2.88
Hf2VGe	158.2	123.1	54.1	134.8	34.5	95.4	3.9	−0.41	3.08
Hf2VSn	187.5	155.6	42.2	166.2	28.5	81.1	5.8	−0.07	2.64
Hf2VPb	109.1	97.9	18.2	101.6	11.4	32.9	8.9	−0.91	3.25

, we can see that all these alloys have negative formation energies. Furthermore, the total energy of L2₁-type Hf₂VZ is lower than XA-type Hf₂VZ (see Figure 1), and thus, the E_f of L2₁-type Hf₂VZ should be lower than XA-type Hf₂VZ. This implies that L2₁-type Hf₂VZ should be more stable than the XA-type Hf₂VZ alloys.

Next, we come to the mechanical properties of these alloys and examine their stability based on achieved elastic constants C_ij. For these alloys with cubic structure, only three independent elastic constants (C₁₁, C₁₂ and C₄₄) are needed to be taken into consideration, and the C_ij can be shown as below [31]:

$$\left(\begin{array}{cccccc}{C}_{11}& C_{12}& C_{12}& 0& 0& 0\\ C_{12}& C_{11}& C_{12}& 0& 0& 0\\ C_{12}& C_{12}& C_{11}& 0& 0& 0\\ 0& 0& 0& C_{44}& 0& 0\\ 0& 0& 0& 0& C_{44}& 0\\ 0& 0& 0& 0& 0& C_{44}\end{array}\right)$$

(2)

For a small strain on a cubic system, the change of elastic energy ΔE can be shown as below [32]:

$$\Delta E=\frac{V}{2}{\displaystyle \sum _{i=1}^6}{\displaystyle \sum _{i=1}^6{C}_{ij}{e}_i{e}_j}$$

(3)

where V stands for the volume of the unit cell. The strain tensors are always symmetric, and they can therefore be expressed more compactly as 6-component vectors, using the so-called Voigt notation. We select three strain tensors $\left(0,0,0,\delta ,\delta ,\delta \right)$, $\left(\delta ,\delta ,0,0,0,0\right)$ and $\left(\delta ,\delta ,\delta ,0,0,0\right)$ to obtain the elastic constants:

$$\frac{\Delta E_1}{V}=\frac{3}{2}{C}_{44}{\delta }^2$$

(4)

$$\frac{\Delta E_2}{V}=(C_{11}+C_{12}){\delta }^2$$

(5)

$$\frac{\Delta E_3}{V}=\frac{3}{2}(C_{11}+C_{12}){\delta }^2$$

(6)

where $\Delta E_1,\Delta E_2,\Delta E_3$ are the change of elastic energy for small strain tensors $\left(0,0,0,\delta ,\delta ,\delta \right)$, $\left(\delta ,\delta ,0,0,0,0\right)$ and $\left(\delta ,\delta ,\delta ,0,0,0\right)$, respectively.

Based on obtained C_ij, the mechanical properties, including bulk modulus B, shear modulus G, Voigt’s shear modulus G_V, Reuss’s shear modulus G_R, Young’s modulus E, Pugh’s ratio B/G, anisotropy factor A, of Hf₂VZ can be calculated by using the following equations [29]:

$$B=\frac{C_{11}+2C_{12}}{3}$$

(7)

$$G=\frac{G_R+G_V}{2}$$

(8)

$$G_V=\frac{C_{11}-C_{12}+3C_{44}}{5}$$

(9)

$$G_R=\frac{5(C_{11}-C_{12})C_{44}}{4C_{44}+3(C_{11}-C_{12})}$$

(10)

$$E=\frac{9GB}{3B+G}$$

(11)

$$A=\frac{2C_{44}}{C_{11}-C_{12}}$$

(12)

As shown in Table 2, we can see that all these alloys with L2₁-type structure are mechanical stable due to their calculated elastic constants follow the generalized elastic stability criteria [33]:

$$C_{44}>0$$

(13)

$$\frac{(C_{11}-C_{12})}{2}>0$$

(14)

$$B>0$$

(15)

$$C_{12}<B<C_{11}$$

(16)

Moreover, some special mechanical properties of these alloys can also be observed from Table 2. We addressed some as follows: (1) the values of B/G of these alloys are all larger than 1.75, reflecting they are ductile according to Pugh’s criteria; (2) the values of A for all these alloys are not equal to 1, meaning the fact that they are anisotropic; (3) as is known, the higher the value of E, the stiffer is the materials, and therefore, the relative stiffer order of these Hf₂VZ materials is Hf₂VAl > Hf₂VSi > Hf₂VGa > Hf₂VGe > Hf₂VIn > Hf₂VSn > Hf₂VTl > Hf₂VPb.

3.3. Calculated Electronic Behaviors of L2₁ and XA Types Hf₂VZ

Electronic-structure calculation theory has clearly indicated [20,34,35] that the spintonic properties of materials among Heusler family has an unusual sensibility to the atomic occupation in crystal cell. Hence, in this section, a discussion about the spintronic property differences, including the magnetic, electronic, half-metallic, and the spin polarization ratio (p), between the XA and L2₁ Hf₂VZ (Z = Al, Ga, In, Tl, Si, Ge, Sn, Pb) full-Heusler alloys should be performed. Figure 3, Figure 4 and Figure 5 show the calculated band structures for XA and L2₁ types Hf₂VZ (Z = Al, Ga, In, Tl, Si, Ge, Sn, Pb) at their equilibrium lattice constants.

For XA atomic ordering, Hf₂VZ (Z = Al, Ga, In, Tl, Si, Sn) alloys are excellent half-metallic materials since there is a semiconducting-type band gap in the spin-up direction and the Fermi level locates between the opened gap. However, in the spin-down direction, the semiconducting-type band gap disappeared: although an opened gap can be observed near the Fermi level, the Fermi level has overlapped with the spin-down bands in varying degrees. In the spin-up channel, the significant factors, including the calculated valence band maximum (VBM), conduction band minimum (CBM), semiconducting band gap, and spin-flip/half-metallic gap, have been given in

Table 3. Calculated valence band maximum (VBM), conduction band minimum (CBM), semiconducting band gaps, spin-flip gap/half-metallic gaps and spin polarization ratio (p) of Hf₂VZ alloys with L2₁ and XA structures, respectively.

Alloy	Structure	CBM	VBM	Band Gap	Half-Metallic Gap	p (%)
Hf2VAl	XA	0.23	−0.23	0.46	0.23	100
	L21	-	-	-	-	48
Hf2VGa	XA	0.36	−0.23	0.59	0.23	100
	L21	-	-	-	-	52
Hf2VIn	XA	0.37	−0.22	0.59	0.22	100
	L21	-	-	-	-	36
Hf2VTl	XA	0.47	−0.17	0.64	0.17	100
	L21	-	-	-	-	40
Hf2VSi	XA	0.07	−0.28	0.35	0.07	100
	L21	-	-	-	-	39
Hf2VGe	XA	-	-	-	-	88
	L21	-	-	-	-	33
Hf2VSn	XA	0.04	−0.26	0.30	0.04	100
	L21	-	-	-	-	56
Hf2VPb	XA	-	-	-	-	94
	L21	-	-	-	-	36

, the semiconducting band gap [36] is the sum of the absolute values of CBM and VBM, and the spin-flip/half-metallic gap [36] is defined as the minimum value of the two absolute values of CBM and VBM. The semiconducting band gap values of these alloys are 0.46 eV for Hf₂VAl, 0.59 eV for Hf₂VGa, 0.59 eV for Hf₂VIn, 0.64 eV for Hf₂VTl, 0.35 eV for Hf₂VSi, 0.30 eV for Hf₂VSn, respectively. The spin-flip/half-metallic gap values of these alloys are 0.23 eV for Hf₂VAl, 0.23 eV for Hf₂VGa, 0.22 eV for Hf₂VIn, 0.17 eV for Hf₂VTl, 0.07 eV for Hf₂VSi, 0.04 eV for Hf₂VSn, respectively. For XA atom ordering Hf₂VGe and Hf₂VPb alloys, semiconducting band gaps in the both directions disappeared and both alloys exhibit common metallic properties.

For Hf₂VZ (Z = Al, Ga, In, Tl, Si, Sn) alloys with the XA structure, the V and Hf atoms occupy sites with same symmetry in XA-type Heusler structure, and the hybridization of their d orbitals creates 5 bonding bands (3t_2g and 2e_g) and 5 non-bonding bands (2e_u and 3t_1u). Then, the 5 V-Hf bonding d hybrids hybridize in turn with d orbitals of Hf, again forming bonding and anti-bonding bands, while the 5 non-bonding bands (2e_u and 3t_1u) still exist with no hybridizing. Finally, the distribution of the 15 d orbitals in the minority-spin direction can be determined, i.e., 3t_2g, 2e_g, 2e_u, 3t_1u, 3t_2g, and 2e_g, from the high-energy level to the low-energy level. Also, we cannot ignore that Z creates 1s and 3p bands which are totally occupied in Hf₂VZ and are also below the above-mentioned 15 d orbitals. Therefore, their semiconducting-type band gaps are created by the separated ${\Gamma }_{15}$ and ${\Gamma }_{25}$ states, coming from the bonding t_2g and antibonding t_1u states.

The calculated total and partial density of states (TDOS and PDOS) for the six Hf₂VZ (Z = Al, Ga, In, Tl, Si, Sn) alloys with XA and L2₁ atomic orderings and for equilibrium lattice constants have also been calculated in this work to deepen the understanding of their electronic properties. As an example, the results of Hf₂VAl and Hf₂VSi alloys are given in Figure 6. Clearly, from the figure, a semiconducting band gap can be found in spin-up direction for XA-type Hf₂VZ (Z = Al, Ga, In, Tl, Si, Sn) alloys, which is in a good agreement with above-mentioned band structures.

The DOS can be widely used to analyze the bonding/anti-bonding states and the gap formation and similar analytical approach can be observed in References [19,29]. In the spin-up channel, the main peaks of Hf 2 and V atoms occurred at around −0.5 eV. In the spin-down channel, in the similar energy region (around −0.5 eV), for V and Hf 2 atoms, such hybridized peaks appeared at the same time. Therefore, the hybridization between the V and Hf 2 atoms that formed strong bonding states at around −0.5 eV. Above the Fermi level, in the spin-up channel, the anti-bonding peak can be found at around 1 eV mainly arise from the Hf 1-d electrons, and in the spin-down channel, no opposite energy states are observed. Moreover, in the spin-up channel, the corresponding bonding-antibonding states led to the formation of an opened band gap, and the Fermi level, exactly, locates between the gap.

For L2₁ type atomic ordering, the band structures and the DOS have a big difference with XA type atomic ordering. Namely, all the Hf₂VZ alloys investigated in this work show conventional metallic behaviors without semiconducting-type band gaps at Fermi level in both spin channels. As shown in Figure 3, Figure 4, Figure 5 and Figure 6, both spin-up and spin-down bands are crossed by the Fermi level.

Moreover, from the obtained total DOS, we calculated the spin polarization (p) at Fermi energy of XA and L2₁ types Hf₂VZ. The spin polarization p (%) that can be defined as the ratio of the difference to sum of the DOS values of spin up and spin down version at the Fermi-level [13,16], represented by mathematical formulation as a percent;

$$p=\frac{n\uparrow \left(E_f\right)-n\downarrow \left(E_f\right)}{n\uparrow \left(E_f\right)+n\downarrow \left(E_f\right)}\times 100\%$$

(17)

where the n↑ (E_f) and n↓ (E_f) stand for the spin-dependent DOS around the Fermi level. The results have been given in Table 3, we can see that the p of XA-type Hf₂VZ are quite high (>88%), even some alloys (such as Hf₂VAl) have completely spin polarization (100%). High-spin-polarization materials are very useful in spintronic application. However, the L2₁-type Hf₂VZ exhibit pretty low spin polarization (<56%). This also implies that the novel spintronic properties (such as half-metallic properties) in XA type structure are disappeared as these alloys form L2₁ type structure.

3.4. Magnetic and Slater-Pauling Properties of L2₁ and XA Types Hf₂VZ

Finally, in this section, we come to discuss the magnetism and Slater-Pauling rule of L2₁ and XA types Hf₂VZ (Z = Al, Ga, In, Tl, Si, Ge, Sn, Pb) alloys. The total and atomic magnetic moments of these alloy with XA and L2₁ atomic ordering are shown in Table 1. Obviously, all these alloys, with either XA or L2₁ type structures, exhibit FiM properties with the atomic magnetic moments of Hf antiparalled aligned with that of V atoms. Based on Table 1, V carries the largest moment and therefore, V atom makes the most contribution to the total magnetic moment.

For XA-type Hf₂VZ full-Heusler materials, their total magnetic moments are almost integer values (2 μ_B/f.u. or 1 μ_B/f.u.), reflecting their high spin polarization properties. Furthermore, from Table 1, we can see that their total magnetic moments follow the well-known Slater-Pauling rule [37]:

$$M_t=Z_t-18$$

(18)

where the M_t stands for the total magnetic moment of these materials and the Z_t means the number of total of valence electrons in Hf₂VZ alloys. For L2₁ type Hf₂VZ full-Heusler materials, their total magnetic moments are all largely deviated from integer values, indicating the half-metallic properties have been totally destroyed when these alloys are in L2₁ type structure. For Hf₂VSi and Hf₂VGe, both alloys have a very weak moment of 0.24 μ_B/f.u. and 0.20 μ_B/f.u., respectively. Therefore, the L2₁ type Hf₂VZ full-Heusler alloys do not obey the above-mentioned Slater-Pauling rule.

The magnetism for XA and L2₁ types full-Heusler alloys Hf₂VZ (Z = Al, Ga, In, Tl, Si, Sn) alloys at their strained lattice constants has also been examined. As an example, the results of Hf₂VAl and Hf₂VGa have been given in Figure 7. According to the Slater-Pauling and generalized electron-filling rules [11,13], the integer values of total magnetic moments which follow the Slater-Pauling rule indicate the half-metallic properties of materials and thus stand for their high spin polarization. From Figure 7, we can see that the high spin polarization properties of XA type Hf₂VZ (Z = Al, Ga, In, Tl, Si, Sn) alloys are very robust, however, the region with high spin polarization (integer values of total magnetic moments) of L2₁-type Hf₂VZ alloys is extraordinarily narrow. For the atomic magnetic moments of XA and L2₁ types Hf₂VZ, the values of Hf decreases, whereas the V atom, it increases.

4. Conclusions

To sum up, the atomic occupation of a series of Hf₂-based full-Heusler alloys was investigated by theoretical first-principle calculations. We observed that all Hf₂V-based alloys studied in current work are likely to form the L2₁ structure instead of XA structure. Our results also indicate that the spintronic properties (half-metallic or high spin-polarization properties), and Slater-Pauling behaviors that exist in XA for Hf₂VZ are fully destroyed in L2₁. With our study, it is clear that not all full-Heusler alloys X₂YZ obey the SPR, especially when X are low valence metal elements.