Next Article in Journal
Modeling and Predicting the Stress Relaxation of Composites with Short and Randomly Oriented Fibers
Previous Article in Journal
Time-Variant Reliability Analysis for Rubber O-Ring Seal Considering Both Material Degradation and Random Load
Article Menu
Issue 10 (October) cover image

Export Article

Materials 2017, 10(10), 1200; https://doi.org/10.3390/ma10101200

Article
L21 and XA Ordering Competition in Hafnium-Based Full-Heusler Alloys Hf2VZ (Z = Al, Ga, In, Tl, Si, Ge, Sn, Pb)
1
School of Physical Science and Technology, Southwest University, Chongqing 400715, China
2
Institute for Superconducting and Electronic Materials, University of Wollongong, Wollongong 2500, Australia
3
Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
*
Author to whom correspondence should be addressed.
Received: 16 September 2017 / Accepted: 15 October 2017 / Published: 20 October 2017

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.
Keywords:
site preference; Hf-based full-Heusler compounds; first-principles study; band structures; magnetic properties; mechanical properties

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 XA-YB-XC-ZD 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 XA-XB-YC-ZD 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 ( Δ E = E X A t o t a l E L 2 1 t o t a l ) for Hf2VZ (Z = Al, Ga, In, Tl, Si, Ge, Sn, Pb) alloys are also listed in Table 1. 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 V Z t o t a l ( E H f 1 b u l k + E H f 2 b u l k + E V b u l k + E Z b u l k )
where E H f 2 V Z t o t a l is the total energy of Hf2VZ per formula unit, and E H f 1 b u l k , E H f 2 b u l k , E V b u l k , and E Z b u l k 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, 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 Ef 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 Cij. For these alloys with cubic structure, only three independent elastic constants (C11, C12 and C44) are needed to be taken into consideration, and the Cij can be shown as below [31]:
( 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 )
For a small strain on a cubic system, the change of elastic energy ΔE can be shown as below [32]:
Δ E = V 2 i = 1 6 i = 1 6 C i j e i e j
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 ( 0 , 0 , 0 , δ , δ , δ ) , ( δ , δ , 0 , 0 , 0 , 0 ) and ( δ , δ , δ , 0 , 0 , 0 ) to obtain the elastic constants:
Δ E 1 V = 3 2 C 44 δ 2
Δ E 2 V = ( C 11 + C 12 ) δ 2
Δ E 3 V = 3 2 ( C 11 + C 12 ) δ 2
where Δ E 1 , Δ E 2 , Δ E 3 are the change of elastic energy for small strain tensors ( 0 , 0 , 0 , δ , δ , δ ) , ( δ , δ , 0 , 0 , 0 , 0 ) and ( δ , δ , δ , 0 , 0 , 0 ) , respectively.
Based on obtained Cij, the mechanical properties, including bulk modulus B, shear modulus G, Voigt’s shear modulus GV, Reuss’s shear modulus GR, Young’s modulus E, Pugh’s ratio B/G, anisotropy factor A, of Hf2VZ can be calculated by using the following equations [29]:
B = C 11 + 2 C 12 3
G = G R + G V 2
G V = C 11 C 12 + 3 C 44 5
G R = 5 ( C 11 - C 12 ) C 44 4 C 44 + 3 ( C 11 C 12 )
E = 9 G B 3 B + G
A = 2 C 44 C 11 C 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
( C 11 C 12 ) 2 > 0
B > 0
C 12 < B < C 11
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, 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 (3t2g and 2eg) and 5 non-bonding bands (2eu and 3t1u). 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 (2eu and 3t1u) still exist with no hybridizing. Finally, the distribution of the 15 d orbitals in the minority-spin direction can be determined, i.e., 3t2g, 2eg, 2eu, 3t1u, 3t2g, and 2eg, 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 Γ 15 and Γ 25 states, coming from the bonding t2g and antibonding t1u 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 = n ( E f ) n ( E f ) n ( E f ) + n ( E f ) × 100 %
where the n↑ (Ef) and n↓ (Ef) 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 μB/f.u. or 1 μB/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
where the Mt stands for the total magnetic moment of these materials and the Zt 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 μB/f.u. and 0.20 μB/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.

Acknowledgments

Funding for this research was proved by National Key R&D Program of China 2017YFA0206303, and National Natural Science Foundation of China, Grant No. 11574374.

Author Contributions

Zhenxiang Cheng conceived and designed the project; Xiaotian Wang performed the calculations and prepared this manuscript; Xiaotian Wang analyzed the data; Wenhong Wang and Zhenxiang Cheng discussed the results.

Conflicts of Interest

The authors declare no conflict of interest.

References

  1. Hu, Y.; Zhang, J.M. The structural, electronic, magnetic, elastic properties of new Heusler alloys Hf2CrZ (Z = Al, Ga, In): A first-principles study. Solid State Commun. 2017, 259, 1–6. [Google Scholar] [CrossRef]
  2. Zhang, L.; Gao, Y.C. Electronic structures, magnetic properties and half-metallicity in the heusler alloy Hf2VAl. Chin. J. Phys. 2017, 55, 1466–1472. [Google Scholar] [CrossRef]
  3. Hu, Y.; Zhang, J.M. First-principles study of the Hf-based Heusler alloys: Hf2CoGa and Hf2CoIn. J. Magn. Magn. Mater. 2017, 421, 1–6. [Google Scholar] [CrossRef]
  4. Dahmane, F.; Mogulkoc, Y.; Doumi, B.; Tadjer, A.; Khenata, R.; Omran, S.B.; Rai, D.P.; Murtaza, G.; Varshney, D. Structural, electronic and magnetic properties of Fe2-based full Heusler alloys: A first principle study. J. Magn. Magn. Mater. 2016, 407, 167–174. [Google Scholar] [CrossRef]
  5. Bayar, E.; Kervan, N.; Kervan, S. Half-metallic ferrimagnetism in the Ti2CoAl Heusler compound. J. Magn. Magn. Mater. 2011, 323, 2945–2948. [Google Scholar] [CrossRef]
  6. Zhang, X.J.; Liu, Z.H.; Zhang, Y.J.; Liu, H.Y.; Liu, G.D.; Cui, Y.T.; Ma, X.Q. Theoretical and experimental study of the phase formation for Ti2YAl and Ti2Y’Ga (Y = Co, Fe; Y’ = Cr, Fe). Intermetallics 2016, 73, 26–30. [Google Scholar] [CrossRef]
  7. Wang, X.T.; Lin, T.T.; Rozale, H.; Dai, X.F.; Liu, G.D. Robust half-metallic properties in inverse heusler alloys composed of 4d transition metal elements: Zr2RhZ (Z = Al, Ga, In). J. Magn. Magn. Mater. 2016, 402, 190–195. [Google Scholar] [CrossRef]
  8. Deng, Z.Y.; Zhang, J.M. Half-metallic and magnetic properties of full-heusler alloys Zr2CrZ (Z = Ga, In) with Hg2CuTi-type structure: A first-principles study. J. Magn. Magn. Mater. 2016, 397, 120–124. [Google Scholar] [CrossRef]
  9. Fang, Q.L.; Zhang, J.M.; Xu, K.W. Magnetic properties and origin of the half-metallicity of Ti2MnZ (Z = Al, Ga, In, Si, Ge, Sn) Heusler alloys with the Hg2CuTi-type structure. J. Magn. Magn. Mater. 2014, 349, 104–108. [Google Scholar] [CrossRef]
  10. Li, J.; Zhang, Z.; Lu, Z.; Xie, H.; Fang, W.; Li, S.; Liang, C.; Yin, F. The strain induced band gap modulation from narrow gap semiconductor to half-metal on Ti2CrGe: A first principles study. AIP Adv. 2015, 5, 156404. [Google Scholar] [CrossRef]
  11. Skaftouros, S.; Ozdogan, K.; Sasioglu, E.; Galanakis, I. Generalized Slater-Pauling rule for the inverse Heusler compounds. Phys. Rev. B 2013, 87, 4469–4487. [Google Scholar] [CrossRef]
  12. Skaftouros, S.; Ozdogan, K.; Sasıoglu, 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]
  13. Zhang, X.M.; Xu, G.Z.; Du, Y.; Liu, E.K.; Liu, Z.Y.; Liu, G.D.; Wang, W.H.; Wu, G.H. Phase stability, magnetism and generalized electron-filling rule of vanadium-based inverse Heusler compounds. EPL 2013, 104, 27012. [Google Scholar] [CrossRef]
  14. Jakobsson, A.; Mavropoulos, P.; Sasioglu, E.; Blugel, S.; Lezaic, M.; Sanyal, B.; Galanakis, I. First-principles calculations of exchange interactions, spin waves, and temperature dependence of magnetization in inverse-Heusler-based spin gapless semiconductors. Phys. Rev. B 2015, 91, 2445–2447. [Google Scholar] [CrossRef]
  15. Li, J.; Jin, Y. Half-metallicity of the inverse heusler alloy Mn2CoAl (001) surface: A first-principles study. Appl. Surf. Sci. 2013, 283, 876–880. [Google Scholar] [CrossRef]
  16. Wang, X.T.; Cheng, Z.X.; Liu, G.D. Largest magnetic moments in the half-Heusler alloys XCrZ (X = Li, K, Rb, Cs; Z = S, Se, Te): A first-principles study. Materials 2017, 10, 1078. [Google Scholar] [CrossRef] [PubMed]
  17. Galanakis, I.; Ozdogan, K.; Sasioglu, E.; Blugel, S. Conditions for spin-gapless semiconducting behavior in Mn2CoAl inverse heusler compound. J. Appl. Phys. 2014, 115, 668. [Google Scholar] [CrossRef]
  18. Feng, W.; Fu, X.; Wan, C.; Yuan, Z.; Han, X.; Quang, N.V.; Cho, S. Spin gapless semiconductor like Ti2MnAl film as a new candidate for spintronics application. Phys. Status Solidi RRL Rapid Res. Lett. 2016, 9, 641–645. [Google Scholar] [CrossRef]
  19. Birsan, A.; Palade, P.; Kuncser, V. Prediction of half metallic properties in Ti2CoSi Heusler alloy based on density functional theory. J. Magn. Magn. Mater. 2013, 331, 109–112. [Google Scholar] [CrossRef]
  20. Bainsla, L.; Mallick, A.I.; Raja, M.M.; Coelho, A.A.; Nigam, A.K.; Johnson, D.D.; Alam, A.; Suresh, K.G. Origin of spin gapless semiconductor behavior in CoFeCrGa: Theory and experiment. Phys. Rev. B Condens. Matter. 2015, 92, 45201. [Google Scholar] [CrossRef]
  21. Groot, R.A.D.; Mueller, F.M.; Engen, P.G.V.; Buschow, K.H.J. New class of materials: Half-metallic ferromagnets. Phys. Rev. Lett. 1983, 50, 2024–2027. [Google Scholar] [CrossRef]
  22. Luo, H.; Xin, Y.; Liu, B.; Meng, F.; Liu, H.; Liu, E.; Wu, G. Competition of L21, and XA structural ordering in heusler alloys X2CuAl (X = Sc, Ti, V, Cr, Mn, Fe, Co, Ni). J. Alloys Compd. 2016, 665, 180–185. [Google Scholar] [CrossRef]
  23. Payne, M.C.; Teter, M.P.; Allan, D.C.; Arias, T.A.; Joannopoulos, J.D. Iterative minimization techniques for ab initio total-energy calculations: Molecular dynamics and conjugate gradients. Rev. Mod. Phys. 1992, 64, 1045. [Google Scholar] [CrossRef]
  24. Segall, M.D.; Lindan, P.J.; Probert, M.A.; Pickard, C.J.; Hasnip, P.J.; Clark, S.J.; Payne, M.C. First-principles simulation: Ideas, illustrations and the CASTEP code. J. Phys. Condens. Matter 2002, 14, 2717–2744. [Google Scholar] [CrossRef]
  25. Perdew, J.P.; Burke, K.; Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 1996, 77, 3865–3868. [Google Scholar] [CrossRef] [PubMed]
  26. Qin, G.; Wu, W.; Hu, S.; Tao, Y.; Yan, X.; Jing, C.; Li, X.; Gu, H.; Cao, S.; Ren, W. Effect of swap disorder on the physical properties of the quaternary heusler alloy PdMnTiAl: A first-principles study. IUCrJ 2017, 4, 506–511. [Google Scholar] [CrossRef] [PubMed]
  27. Galehgirian, S.; Ahmadian, F. First principles study on half-metallic properties of heusler compounds Ti2VZ (Z = Al, Ga, and In). Solid State Commun. 2015, 202, 52–57. [Google Scholar] [CrossRef]
  28. Lukashev, P.; Kharel, P.; Gilbert, S.; Staten, B.; Hurley, N.; Fuglsby, R.; Huh, Y.; Valloppilly, S.; Zhang, W.; Yang, K.; et al. Investigation of spin-gapless semiconductivity and half-metallicity in Ti2MnAl-based compounds. Appl. Phys. Lett. 2016, 108, 156404. [Google Scholar] [CrossRef]
  29. Wang, X.; Cheng, Z.; Wang, J.; Liu, G. A full spectrum of spintronic properties demonstrated by a C1b-type Heusler compound Mn2Sn subjected to strain engineering. J. Mater. Chem. C 2016, 4, 8535–8544. [Google Scholar] [CrossRef]
  30. Benkaddour, K.; Chahed, A.; Amar, A.; Rozale, H.; Lakdja, A.; Benhelal, O.; Sayede, A. First-principles study of structural, elastic, thermodynamic, electronic and magnetic properties for the quaternary Heusler alloys CoRuFeZ (Z = Si, Ge, Sn). J. Alloys Compd. 2016, 687, 211–220. [Google Scholar] [CrossRef]
  31. Gooch, J.W. Elastic Constant; Springer: New York, NY, USA, 2011. [Google Scholar]
  32. Zhao, J.S.; Gao, Q.; Li, L.; Xie, H.H.; Hu, X.R.; Xu, C.L.; Deng, J.-B. First-principles study of the structure, electronic, magnetic and elastic properties of half-Heusler compounds LiXGe (X = Ca, Sr and Ba). Intermetallics 2017, 89, 65–73. [Google Scholar] [CrossRef]
  33. Born, M.; Huang, K. Dynamical Theory of Crystal Lattices; Clarendon Press: Oxford, UK, 1954. [Google Scholar]
  34. Wang, X.T.; Cheng, Z.X.; Wang, J.L.; Rozale, H.; Wang, L.Y.; Yu, Z.Y.; Liu, G.D. Strain-induced diverse transitions in physical nature in the newly designed inverse Heusler alloy Zr2MnAl. J. Alloys Compd. 2016, 686, 549–555. [Google Scholar] [CrossRef]
  35. Wang, X.; Cheng, Z.; Khenata, R.; Wu, Y.; Wang, L.; Liu, G. Lattice constant changes leading to significant changes of the spin-gapless features and physical nature in a inverse heusler compound Zr2MnGa. J. Magn. Magn. Mater. 2017, 444, 313–318. [Google Scholar] [CrossRef]
  36. Ahmadian, F. Half-metallic ferromagnetism in the Ti2FeGe Heusler compound: A first-principles study. J. Supercond. Nov. Magn. 2013, 26, 381–388. [Google Scholar] [CrossRef]
  37. Galanakis, I.; Dederichs, P.H.; Papanikolaou, N. Slater-Pauling behavior and origin of the half-metallicity of the full-Heusler alloys. Phys. Rev. B 2002, 66, 553–562. [Google Scholar] [CrossRef]
Figure 1. Crystal structures of XA and L21 types Hf2VZ (Z = Al, Ga, In, Tl, Si, Ge, Sn, Pb) alloys. This 1 × 1 × 1 super-cell system contains 16 atoms, i.e., 8 × Hf, 4 × V and 4 × Z.
Figure 1. Crystal structures of XA and L21 types Hf2VZ (Z = Al, Ga, In, Tl, Si, Ge, Sn, Pb) alloys. This 1 × 1 × 1 super-cell system contains 16 atoms, i.e., 8 × Hf, 4 × V and 4 × Z.
Materials 10 01200 g001
Figure 2. Calculated total energies of XA and L21 types Hf2VAl (a); Hf2VGa (b); Hf2VIn (c); Hf2VTl (d); Hf2VSi (e); Hf2VGe (f); Hf2VSn (g); Hf2VPb (h) alloys with respect to the lattice constant.
Figure 2. Calculated total energies of XA and L21 types Hf2VAl (a); Hf2VGa (b); Hf2VIn (c); Hf2VTl (d); Hf2VSi (e); Hf2VGe (f); Hf2VSn (g); Hf2VPb (h) alloys with respect to the lattice constant.
Materials 10 01200 g002
Figure 3. Calculated band structures of XA and L21 types Hf2VZ (Z = Al, Ga, In) alloys.
Figure 3. Calculated band structures of XA and L21 types Hf2VZ (Z = Al, Ga, In) alloys.
Materials 10 01200 g003
Figure 4. Calculated band structures of XA and L21 types Hf2VZ (Z = Tl, Si, Ge) alloys.
Figure 4. Calculated band structures of XA and L21 types Hf2VZ (Z = Tl, Si, Ge) alloys.
Materials 10 01200 g004aMaterials 10 01200 g004b
Figure 5. Calculated band structures of XA and L21 types Hf2VZ (Z = Sn and Pb) alloys.
Figure 5. Calculated band structures of XA and L21 types Hf2VZ (Z = Sn and Pb) alloys.
Materials 10 01200 g005
Figure 6. Calculated band structures of XA type Hf2VAl (a); L21 type Hf2VAl (b); XA type Hf2VSi (c); L21 type Hf2VSi (d); respectively.
Figure 6. Calculated band structures of XA type Hf2VAl (a); L21 type Hf2VAl (b); XA type Hf2VSi (c); L21 type Hf2VSi (d); respectively.
Materials 10 01200 g006aMaterials 10 01200 g006b
Figure 7. (a,b) Calculated total and atomic spin magnetic moments of Hf2VZ (Z = Al, Ga) alloys as functions of the lattice constant. The red boxes show the high-spin polarization regions.
Figure 7. (a,b) Calculated total and atomic spin magnetic moments of Hf2VZ (Z = Al, Ga) alloys as functions of the lattice constant. The red boxes show the high-spin polarization regions.
Materials 10 01200 g007
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.
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.
AlloyStructureΔE (eV/cell)a (Å)MtB/f.u.)MHf 1B)MHf 2B)MVB)MZB)Stable Structure
Hf2VAlXA0.646.712−0.650.142.62−0.12L21
L216.691.65−0.21−0.212.19−0.11
Hf2VGaXA0.716.681.99−0.590.152.55−0.13L21
L216.661.68−0.13−0.132.06−0.12
Hf2VInXA0.396.882−0.75−0.012.88−0.12L21
L216.901.70−0.24−0.242.31−0.13
Hf2VTlXA0.326.921.98−0.79−0.13.0−0.12L21
L216.891.61−0.17−0.172.06−0.11
Hf2VSiXA0.636.571.02−0.74−0.252.08−0.06L21
L216.560.24−0.03−0.030.31−0.01
Hf2VGeXA0.586.641.03−0.83−0.342.26−0.06L21
L216.610.2−0.02−0.020.26−0.01
Hf2VSnXA0.186.851−1.07−0.512.64−0.05L21
L216.820.49−0.06−0.060.63−0.03
Hf2VPbXA0.116.941−1.17−0.632.87−0.04L21
L216.900.32−0.03−0.030.41−0.02
Table 2. Calculated elastic constants Cij, 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.
Table 2. Calculated elastic constants Cij, 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.
AlloyC11C12C44BGEB/GFormation EnergyAnisotropy Factor
Hf2VAl164.4117.673.0133.246.8124.62.8−0.543.10
Hf2VGa143104.372.9117.943.0115.102.7−1.193.76
Hf2VIn179.8154.349.7162.828.982.05.6−0.053.89
Hf2VTl110.4100.146.8103.520.557.95.0−0.859.08
Hf2VSi191146.963.7161.641.6115.123.8−0.662.88
Hf2VGe158.2123.154.1134.834.595.43.9−0.413.08
Hf2VSn187.5155.642.2166.228.581.15.8−0.072.64
Hf2VPb109.197.918.2101.611.432.98.9−0.913.25
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.
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.
AlloyStructureCBMVBMBand GapHalf-Metallic Gapp (%)
Hf2VAlXA0.23−0.230.460.23100
L21----48
Hf2VGaXA0.36−0.230.590.23100
L21----52
Hf2VInXA0.37−0.220.590.22100
L21----36
Hf2VTlXA0.47−0.170.640.17100
L21----40
Hf2VSiXA0.07−0.280.350.07100
L21----39
Hf2VGeXA----88
L21----33
Hf2VSnXA0.04−0.260.300.04100
L21----56
Hf2VPbXA----94
L21----36

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
Materials EISSN 1996-1944 Published by MDPI AG, Basel, Switzerland RSS E-Mail Table of Contents Alert
Back to Top