Cite this Article
Wei Xiao-Ping, Cao Tie-Yi, Sun Xiao-Wei, Gao Qiang, Gao Peifeng, Gao Zhi-Lei, Tao Xiao-Ma. Structural, electronic, and magnetic properties of quaternary Heusler CrZrCoZ compounds: A first-principles study. Chinese Physics B, 2020, 29(7): 077105  
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Structural, electronic, and magnetic properties of quaternary Heusler CrZrCoZ compounds: A first-principles study
Wei Xiao-Ping1, †, Cao Tie-Yi1, Sun Xiao-Wei1, Gao Qiang2, Gao Peifeng3, Gao Zhi-Lei1, Tao Xiao-Ma4
The School of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou 730070, China
Institute of Materials Science, Technische Universitat Darmstadt, 64287 Darmstadt, Germany
Key Laboratory of Mechanics on Western Disaster and Environment, Ministry of Education, College of Civil Engineering and Mechanics, Key Laboratory of Special Function Materials and Structure Design of Ministry of Education, Lanzhou University, Lanzhou 730000, China
School of Physical Science and Technology, Guangxi University, Nanning 530004, China

 

† Corresponding author. E-mail: weixp2008@gmail.com

Project supported by the National Natural Science Foundation of China (Grant No. 11864021) and Foundation of A Hundred Youth Talents Training Program of Lanzhou Jiaotong University, China, the Key Talent Foundation of Gansu Province, China (Grant No. 2020RCXM100), and Excellent Research Team of Lanzhou Jiaotong University, China (Grant No. 201803).

Abstract

Using the first-principles calculations, we study the structural, electronic, and magnetic properties along with exchange interactions and Curie temperatures for CrZrCoZ (Z = Al, Ga, In, Tl, Si, Pb) quaternary Heusler alloys. The results show that the CrZrCoZ alloys are half-metallic ferrimagnets, and their total spin magnetic moments, which are mainly carried by the Cr atom, obey the Slater–Pauling rule. Analysis of local density of states confirms that the exchange splitting between eg and t2g states leads to the formation of half-metallic gap. According to the calculated Heisenberg exchange coupling parameters, it is found that the Cr(A)–Cr(A) and Cr(A)–Zr(B) exchanges dominate the appearance of ferrimagnetic states in CrZrCoZ (Z = Al, Ga, In, Tl, Pb) alloys, and it is the Cr(A)–Zr(B) and Zr(B)–Zr(B) exchanges for CrZrCoSi alloy. Finally, we estimate the Curie temperatures of CrZrCoZ by using mean-field approximation, it is found that the CrZrCoZ (Z = Al, Ga, In, Tl, Pb) alloys have noticeably higher Curie temperatures than room temperature. So, we expect that the CrZrCoZ alloys are promising candidates in spintronic applications in future.

PACS: ;71.20.Lp;;85.75.Dd;;75.50.-y;;77.80.B-;
Keyword:;quaternary Heusler alloys;;electronic structure;;magnetic properties;;Curie temperature;
1. Introduction

Heusler alloys have attracted wide attention over the last decades because of their multi-functional properties, such as half-metallic magnetism,[1–5] magnetic shape memory behavior,[6–11] spin gapless semiconductor,[12–15] giant magnetocaloric effect,[16,17] thermoelectric effects,[18,19] superconductivity,[20] and so on, these properties succeed in technological applications. For example, the half-metallicity has been used in spin injectors, magnetic tunnel junctions, spin valves, and spin torque transfer-random access memories.[21]

Usually, Heusler alloys can be classified into half-Heusler and full-Heusler alloys, the former has a general chemical formula XYZ, and crystallizes into C1b structure with space group, where X and Y represent transition metals, and Z is of sp element; the latter has stoichiometry X2YZ, and crystallizes into L21 structure with space group.[22–24] By substituting another transition metal X′ for X in full-Heusler alloys X2YZ, the quaternary Heusler alloys XX′YZ with LiMgPdSn prototype are formed.[25,26] The quaternary Heusler alloys have greater flexibilities on the choices of components and atomic arrangements, thus they may offer newer materials with target properties. So far, many quaternary Heusler alloys as spintronic candidates have been reported to be half-metallicity with high Curie temperature.[3,27–31]

Among these quaternary Heusler alloys, the Cr-based quaternary Heusler alloys, as a subset of these materials, have been studied extensively from the perspectives of theory and experiment.[13,32–40] For instance, it has been confirmed experimentally that the CoFeCrGe alloy crystalizes with LiMgPdSn structure, and has an integer magnetic moment of 3.0 μB/f.u. together with a Curie temperature of 866 K.[41] The CoFeCrZ (Z = P, As, Sb) and CoFeCrZ (Z = Al, Si, Ga, Ge) alloys were predicted to be half-metals.[27,42,43] From these reports, we notice that there are two magnetic atoms in these quaternary Heusler alloys, only a few studies focus on the quaternary Heusler alloys including one magnetic atom.[32,38,44,45] Therefore, it is necessary to design new quaternary Heusler alloys with half-metallicity to understand the effects of the presence of one magnetic atom. Apart from the quest, another motivation is to explore whether new half-metallic magnets with high Curie temperature can be derived from the family of Heusler alloys by changing the compositions and thus expand the database of materials for spintronic applications.

Up to now, the Cr2CoZ (Z = Al, Ga, In, Si, Ge, Sn, Pb) alloys have been predicted to be half-metals with low magnetic moments.[46–48] Accordingly, the half-metallicity is also observed in Zr2CoZ (Z = Al, Ga, In, Tl, Si, Ge, Sn, Pb) alloys.[49–51] Therefore, it is worth to explore whether the quaternary Heusler alloys with Cr, Zr, and Co elements still have the half-metallicity and high Curie temperature or other fancy properties. In this paper, we systematically study the structural, electronic, and magnetic properties together with exchange interactions and Curie temperatures of CrZrCoZ (Z = Al, Ga, In, Tl, Si, Pb) quaternary Heusler alloys.

2. Calculation details

In the present calculations, the full-potential local-orbital minimum-basis band-structure method (FPLO) is used to calculate the structural, electronic, and magnetic properties.[52,53] In this scheme, the scalar relativistic Dirac equation is solved, and the number of k-points in the Brillouin zone is set to 20 × 20 × 20 in order to ensure the convergence of the calculated parameters. The convergence of the total energy is set to be better than 10−8 Hartree together with that of the density better than 10−6 in internal unit of the code, more details can be found in Ref. [50]. Heisenberg exchange coupling parameters can be obtained by the spin-polarized relativistic Korringa–Kohn–Rostoker package Munich SPRKKR.[54] An angular momentum cutoff of ℓmax = 3 and 22 × 22 × 22 k-points in each Brillouin zone direction are used in the calculations. In order to further improve the charge convergence with respect to lmax, the Lloyd’s formula is used to determine the Fermi level,[55,56] the details of calculations can be found from our previous works.[61,62] The elastic constants are calculated by using the Vienna ab initio simulation package (VASP).[57,58] The projected augmented wave (PAW) potentials and the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm are applied to fully relax all atomic positions, cell shapes, and cell volumes to achieve the minimum total energies of the systems. The cutoff energy for the plane wave expansion is 600 eV for all models, and the k-mesh is chosen as 20 × 20 × 20 in each Brillouin zone. The total energy and force convergences are 10−5 eV per atom and 0.01 eV⋅Å−1, respectively. In addition, the Perdew–Burke–Ernzerhof in the frame of generalized gradient approximation is used to describe the exchange–correlation potential in all calculations.[59]

3. Results and discussion
3.1. Structural properties

Quaternary Heusler alloys CrZrCoZ (Z = Al, Ga, In, Tl, Si, Pb) usually crystallize with three possible structures as shown in Table 1.

Table 1.

Three possible structures for quaternary Heusler alloys XX′YZ (CrZrCoZ).

.

Therefore, we compare the total energies of the three possible structures in Fig. 1 to obtain the stable structure. It is found that the type I structure is the most stable. For the type I structure, we make a comparison of the total energies between nonmagnetic (NM) and ferrimagnetic (FIM) phases in Fig. 2, and find that the ferrimagnetic ground states have lower total energies. Accordingly, we tabulate the calculated lattice parameters in Table 2. Furthermore, we check the magnetic stability of the CrZrCoZ alloys by using Stoner model. In the simplified version of this theory, the Stoner criterion for spin polarization is IN0 > 1, where I is the exchange integral (Stoner parameter) and N0 is the density of states (DOS) of the ground state at the Fermi level. The exchange integral is determined by using the following formula:

where the susceptibility is calculated by

The E(m) curves in Eq. (2) are calculated by the fixed spin moment method. The DOS N0 at the Fermi level and the exchange integral I are estimated from Eq. (1), and their products (IN0) are listed in Table 2. As is shown, the products IN0 of the CrZrCoZ alloys are larger than 1, indicating that these alloys have magnetic ground states, it is also well consistent with our obtained total energy curves in Fig. 2. In addition, because the exchange splitting is proportional to Im2, thus the larger exchange constants can generally lead to higher Curie temperature according to the mean field consideration.[60] Based on the calculated exchange integral and total magnetic moments, it is believed that the CrZrCoPb alloy has the highest Curie temperature among the CrZrCoZ alloys. In the following, we discuss the structural stability of CrZrCoZ alloys from the point views of thermodynamics and mechanical properties.

Fig. 1. Changes of total energies as a function of lattice constants for three possible structures of CrZrCoZ alloys.
Fig. 2. Total energies of nonmagnetic and ferrimagnetic phases as a function of lattice constants for CrZrCoZ alloys.

Firstly, the thermodynamic stability is measured by the formation energy Ef,

where Etot is the calculated total energy per formula unit of the ground state, μi is the chemical potential of element i, and xi is the quantity of element i in the alloys. From the Ef summarized in Table 2, we can see that the CrZrCoZ (Z = Al, Ga, In, Si) alloys have negative formation energies, indicating that they can be easily fabricated in experiment. However, the positive formation energies for CrZrCoTl and CrZrCoPb alloys imply that they are thermodynamically unstable, but they may can be synthesized in the Heusler phase in a non-equilibrium process.[63]

Table 2.

Lattice constant at equilibrium (acal), density of states (N0) at the Fermi level in nonmagnetic states, and the exchange integral (I) of CrZrCoZ alloys and its products with N0 (IN0). The spin-flip gap (Esfg), band gap (Eg), and formation energy (Ef) per atom as well as Curie temperatures (TC) are also shown.

.

The mechanical properties of material are crucial in engineering technological applications. Therefore, we calculate the elastic constants of the CrZrCoZ alloys, which not only offer related mechanical parameters, but also can be used to check their mechanical stability by using the Born–Huang criteria as follows:

We can find from Table 3 that the CrZrCoZ (Z = Al, Ga, In, Tl, Si) alloys satisfy the stability conditions. Furthermore, these elastic constants are applied to evaluate the elastic moduli, including bulk modulus B, shear modulus G, etc. in the following relations:

where GR and GV are the Reuss’s and Voigt’s shear moduli, and their arithmetic average is estimated by using the Voigt–Reuss–Hill approximation in the following equation:

In addition, Young’s modulus E, Poisson’s ratio υ, and anisotropy factor A are obtained as follows:

In Table 3, we list these moduli parameters. Among them, the bulk modulus B is used to measure the material’s resistance against compression, whereas Young’s modulus E provides the information about stiffness where the higher value corresponds to a stiffer material. The results show that the B, E, and G values of CrZrCoSi are highest among the alloys. The ratio of the transverse contraction to the longitudinal extension in the direction of elastic loading is defined as Poisson’s ratio υ, where the lower limit of υ for most metals is about 0.25. Apart from the CrZrCoPb alloy, the calculated υ values range from 0.329 to 0.366. In addition, it should be mentioned that the B, E, G, and υ values of CrZrCoSi are higher than those of Zr2CoSi, and its A is lower than that of Zr2CoSi.[64] The calculated anisotropy factor A deviates from unity, indicating an anisotropic elastic behavior. According to the Pugh’s criteria, the CrZrCoZ alloys show a ductile behavior. Furthermore, the melting temperature is obtained in the following empirical relation:[65]

It is found that the melting temperature of CrZrCoSi is highest due to its highest bulk modulus (see Table 3).

Table 3.

The calculated elastic constants Cij (in GPa), bulk modulus B (in GPa), Young’s modulus E (in GPa), isotropic shear modulus G (in GPa), Poisson’s ratios υ, anisotropy factor A, Pugh’s ratio B/G, and the melting temperature Tmelt (in K) of CrZrCoZ (Z = Al, Ga, In, Tl, Si, Pb) alloys.

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3.2. Electronic structures

In Fig. 3, we give the total density of states (DOS) and partial density of states (PDOS) of CrZrCoZ (Z = Al, Ga, In, Tl, Si, Pb) alloys at their respective equilibrium states. It is evident that the CrZrCoZ alloys preserve a half-metallic behavior. An energy gap appears at the Fermi level in the minority-spin channel for the CrZrCoZ (Z = Al, Ga, In, Tl, Pb) alloys, while there is nonzero DOS in the majority-spin channel. By contrast, for the CrZrCoSi alloy, the gap is pinned in the majority-spin channel, and the states exist in the minority-spin channel. The striking difference is ascribed to its smallest exchange integral (see Table 2), which leads to the incomplete exchange splitting between eg and t2g states (see Fig. 5), and then the gap of CrZrCoSi is pinned in the majority-spin channel. In fact, a bigger exchange integral can make the exchange splitting stronger, and the gap larger. Particularly, for the CrZrCoSi alloy, the Cr-3d states offer a major contribution to the total DOS in the energy range between –1 eV and 0.5 eV in both spin channels. While for the CrZrCoZ (Z = Al, Ga, In, Tl, Pb) alloys, the strong hybridizations between Zr-4d and Co-3d states lead to an energy distribution spreading from 0 eV to 1 eV in the spin-down states, and they almost have the same contribution to the total DOS in the minority-spin states above the Fermi level, it is mainly governed by the Co-3d states from –3 eV to 0 eV below the Fermi level, and the spin-up states are briefly occupied by Co-3d and Cr-3d states. More evidently, there is a saddle point from Γ to X direction in the majority-spin of energy dispersion pictures for CrZrCoZ (Z = Al, Ga, In, Tl) alloys (see Fig. 4), indicating high DOS values. Overall, in the CrZrCoZ alloys, the obvious hybridizations between d–d states lead to the formation of the gap.

Fig. 3. Total and paritial density of states for CrZrCoZ alloys.
Fig. 4. Band structures for CrZrCoZ alloys. Note that the spin-flip gap is labeled.
Fig. 5. Local density of states of CrZrCoZ alloys.

In order to understand the electronic properties in depth, we analyze the local density of states (LDOS) in Fig. 5. It is clear that these alloys can be in quality classified into two types according to their gaps in the spin channels. For the CrZrCoZ (Z = Al, Ga, In, Tl, Pb) alloys, taking the CrZrCoAl as an example, we can notice that the d states are divided into t2g and eg states for Cr, Zr, and Co atoms in the octahedral crystal field, the difference between eg and t2g states for Cr and Co atoms is bigger than that of Zr atom. Thus, we conclude that the exchange splitting between eg and t2g states from Cr and Co atoms leads mainly to the formation of the spin-flip gap. In detail, both t2g and eg are unoccupied in the spin down channel for the Cr atom. In the spin up channel, the t2g states are fully occupied, leading to 3 μB contribution to the magnetic moment. As to the eg state, it is partially occupied, making 0.5 μB contribution. Thus, the magnetic moment of the Cr atom is about 3.5 μB. On the other hand, the Co magnetic moment, which arises from the complex crystal splitting, is 0.88 μB, the hybridization between Co and Cr atoms is so strong that we can not use an atomic model to analysis it. While for the CrZrCoSi alloy, the same situation is observed. Also, it is found that the t2g and eg states almost have equivalent splitting strengths for the Co atom. In addition, we can see from the band pictures in Fig. 4 that the indirect band gap from K to L direction is observed in CrZrCoZ (Z = Ga, In, Tl, Pb) alloys, and it is a direct band gap for CrZrCoAl and CrZrCoSi alloys from K to Γ direction and Γ point, respectively. Accordingly, the spin-flip gap, which is defined as the minimum energy required to flip the minority-spin electrons of conduction band minimum (CBM) to valence band for CrZrCoZ (Z = Al, Ga, In, Tl, Pb) alloys, and it is the minimum energy required to flip the majority-spin electrons of valence band maximum (VBM) to conduction band for CrZrCoSi, is labeled in the bands. In Table 2, we tabulate the band gap and the spin-flip gap, and observe that the CrZrCoSi alloy has the lowest band gap up to 0.269 eV because of its smallest exchange integral, and the spin-flip gap is the biggest for the CrZrCoPb alloy.

The half-metallic properties usually can be affected by lattice constants. Moreover, the half-metal is often used in the form of thin films in spintronic devices, where the lattice parameters are strongly determined by the growth substrate. Therefore, it is necessary to study the robustness of the half-metallicity on the lattice constants. In Fig. 6, we offer the CBM and VBM in the spin down channel as a function of the lattice constants for CrZrCoZ (Z = Al, Ga, In, Tl, Pb) alloys. By contrast, they are from the spin-up channel for the CrZrCoSi alloy. It can be seen from Fig. 6 that the half-metallicity can be preserved within the range of 6.212–7.053 Å, 6.169–6.997 Å, 6.189–7.055 Å, 6.163–7.009 Å, and 6.651–6.883 Å for CrZrCoAl, CrZrCoGa, CrZrCoIn, CrZrCoTl, and CrZrCoPb alloys, respectively. While for the CrZrCoSi alloy, the half-metallicity is kept in the lattice constants of 5.864–6.149 Å and 6.577–6.878 Å, where the disappearance of half-metallicity between 6.149 Å and 6.577 Å may ascribe to the occurrence of phase transition. For the CrZrCoZ (Z = Al, Ga, In, Tl) alloys, when the lattice constants are larger than 7.053 Å, 6.997 Å, 7.055 Å, and 7.009 Å, the Fermi level has an intersection with the valence bands, which leads to the loss of the half-metallicity although the band gap is still kept in the spin down channel. It is the intersection of the Fermi level and the conduction bands when the lattice constants are compressed to 6.212 Å, 6.169 Å, 6.189 Å, and 6.163 Å, also leading to the loss of the half-metallicity, which is different from the cases of the lattice expansion. Thus, we can say that the CrZrCoZ (Z = Al, Ga, In, Tl) alloys have a good robustness of half-metallic properties against the lattice change and are possible ideal candidates for applying in a quite wider temperature range.

Fig. 6. Changes of VBM and CBM for CrZrCoZ alloys depend on the lattice constants.
3.3. Magnetic properties

In this section, we focus on the magnetic moments of CrZrCoZ alloys, as expected, the calculated total magnetic moments are an integer because of their itinerant valence electrons, which is also a unique property for half-metal, and the calculated total magnetic moments obey the Slater–Pauling rule.[67,68] Besides, we also notice that the total magnetic moments for all alloys are mainly carried by the Cr atoms, and the other atoms offer rather small contribution. In addition, the magnetic moments of the Zr and Z atoms are anti-parallel with the Cr and Co magnetic moments, thus the CrZrCoZ alloys are ferrimagnetic half-metals. For the CrZrCoZ (Z = Al, Ga, In, Tl, Pb) alloys, their integral magnetic moments can be illustrated by the occupied electronic states in Fig. 7(a), i.e, the Cr and Co atoms located at A and C sites have the octahedral symmetry (see Fig. 7(b)), and their d-orbitals hybridize with each other, then creating five bonding triple-degenerated t2g and double-degenerated eg states, and five anti-bonding triple-degenerated t1u and double-degenerated eu states, and the energies of the t2g and eu states are higher than those of eg and t1u. In turn, the five bonding d-hybrids hybridize with the Zr-d orbitals, creating again bonding and anti-bonding states. In addition, the Z element provides a single s state and triple-degenerated p states, and partially accommodate d-charge from the Cr, Zr, and Co atoms, therefore, there are nine occupied electronic states in total, thus their total magnetic moments follow the Slater–Pauling rule Mt = Zt – 18, where Mt is the total magnetic moment, and Zt is the total valence number per formula unit. While for the CrZrCoSi alloy, the occupation of electronic states is the same as that of the CrZrCoZ (Z = Al, Ga, In, Tl, Pb) alloys, the triple-degenerated t1u states are only localized in the occupied states due to lower exchange splitting energy, then leading to twelve occupied electronic states, thus its total magnetic moment obeys the Slater–Pauling rule Mt = 24 – Zt (see Fig. 7(c)).

Fig. 7. Orbital hybridizations under the crystal field for (a) CrZrCoZ (Z = Al, Ga, In, Tl, Ge, Pb) alloys, (c) CrZrCoSi alloy, and (b) crystal structure and field symmetry in CrZrCoZ alloys.
Table 4.

Total and atomic magnetic moments (in μB) for CrZrCoZ (Z = Al, Ga, In, Tl, Si, Pb) alloys.

.
3.4. Exchange interactions and Curie temperatures

In Fig. 8, we show the calculated Heisenberg exchange coupling parameters Jij.[66] It is notable that Jij of the CrZrCoZ alloys is tightly confined to clusters of radius r ≤ 2.0a, and the Co(C)–Cr(A) exchange values decrease slightly as the atomic radius and electronegativity increase when Z is of III group. In particular, for the CrZrCoAl alloy, the first, second, and third nearest neighbors of Co(C)–Cr(A) interactions show positive coupling values, implying ferromagnetic exchanges. While for the Cr(A)–Cr(A) exchange, the first and third nearest neighbors are positive values, it is negative for the second nearest neighbor, indicating their anti-parallel coupling. For the CrZrCoGa, CrZrCoIn, and CrZrCoTl alloys, these exchanges are the same as those in the CrZrCoAl alloy, thus leaving them no discussion. For the CrZrCoSi alloy, the Co(C)–Cr(A) and Co(C)–Co(C) exchanges indicate the obvious oscillation behavior, and the three nearest neighbors for Cr(A)–Cr(A) exchange show negative values, implying anti-ferromagnetic interactions. In the CrZrCoPb alloy, the Co(A)–Cr(A) exchange plays a leading role in interactions, its first, second, and third interactions are positive, and notably higher than those in CrZrCoSi because of the larger local magnetic moments. The other exchanges are not shown in Fig. 8 due to their smaller interactions. Accordingly, the oscillatory behavior in all interactions indicates a RKKY exchange.[69] In fact, a positive Jij can improve the Curie temperature, and a negative Jij can reduce the Curie temperature. Furthermore, we offer the sum J0 of exchange coupling parameters Jij, these values are given in Table 5. It can be seen that the Cr(A)–Cr(A) and Cr(A)–Zr(B) exchanges lead to the appearance of ferrimagnetic states in CrZrCoZ (Z = Al, Ga, In, Tl, Pb) alloys. By comparison, the Cr(A)–Zr(B) and Zr(B)–Zr(B) exchanges play a leading role in CrZrCoSi alloy. According to the calculated J0, we construct a 4× 4 J0 matrix to obtain the Curie temperature (see Table 2), the details of calculations can be found from Ref. [19]. It finds that the CrZrCoZ (Z = Al, Ga, In, Tl, Pb) alloys have noticeably higher Curie temperatures than room temperature, and the tendency of the Curie temperatures is well consistent with the Im2.

Fig. 8. Exchange interactions Jij for CrZrCoZ alloys with the interatomic distance r.
Table 5.

Exchange coupling parameters J0 (in meV) between the constituents are tabulated. Note that the subscripts 1, 2, 3, and 4 correspond to the Cr, Co, Zr, and Z atoms, respectively.

.
4. Summary and conclusions

The first-principle calculations are used to study the structural, electronic, magnetic properties along with exchange interactions and Curie temperatures of quaternary Heusler alloys CrZrCoZ (Z = Al, Ga, In, Tl, Si, Pb). The results show that the CrZrCoZ alloys are ferrimagnetic half-metals, and their integral magnetic moments obey the Salter–Pauling rule. Stability analysis indicates that the CrZrCoZ alloys have magnetic ground states, and the CrZrCoAl, CrZrCoGa, CrZrCoIn, and CrZrCoSi alloys are likely to be prepared in experiment. Occupation of electronic states shows that there are nine occupied states for CrZrCoZ (Z = Al, Ga, In, Tl, Pb) alloys, thus their total magnetic moments obey the Salter–Pauling rule Mt = Zt – 18. For the CrZrCoSi alloy, the total magnetic moment follows the Sater–Pauling rule Mt = 24 – Zt due to its twelve occupied states, the exchange splitting between eg and t2g states from the Cr and Co atoms leads mainly to the formation of spin-flip gap. In addition, a good half-metallic robustness against the lattice changes is also observed in CrZrCoZ (Z = Al, Ga, In, Tl) alloys. According to the calculated Heisenberg exchange coupling parameters, we find that the Cr(A)–Cr(A) and Cr(A)–Zr(B) exchanges lead to the formation of ferrimagnetic states of CrZrCoZ (Z = Al, Ga, In, Tl, Pb) alloys, and it is the Cr(A)–Zr(B) and Zr(B)–Zr(B) exchanges for CrZrCoSi alloy. Finally, we estimate the Curie temperatures of CrZrCoZ by using mean-field approximation, the Curie temperatures of CrZrCoZ (Z = Al, Ga, In, Tl, Pb) alloys are noticeably higher than room temperature. Thus, our results may offer a valuable hint for CrZrCoZ alloys in future spintronic applications.

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