1. bookVolume 34 (2016): Issue 2 (June 2016)
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Site preference and electronic structure of Mn2RhZ (Z = Al, Ga, In, Si, Ge, Sn, Sb): a theoretical study

Published Online: 09 May 2016
Volume & Issue: Volume 34 (2016) - Issue 2 (June 2016)
Page range: 251 - 259
Received: 08 Apr 2015
Accepted: 05 Feb 2015
Journal Details
License
Format
Journal
eISSN
2083-134X
First Published
16 Apr 2011
Publication timeframe
4 times per year
Languages
English
Abstract

The electronic structure and magnetism of Mn2RhZ (Z = Al, Ga, In, Si, Ge, Sn, Sb) Heusler alloys have been studied by using first-principles calculations. Three half-metallic ferromagnets, namely, Mn2RhAl, Mn2RhGe and Mn2RhSb have been considered. The calculated equilibrium lattice constant increases with increasing atomic number of Z atoms lying in same column of periodic table. The calculated total magnetic moments Mtot are 2 µB/f.u. for Mn2RhAl and Mn2RhGa, 3 µB/f.u. for Mn2RhSi, Mn2RhGe and Mn2RhSn, and 4 µB/f.u. for Mn2RhSb, which agrees with the Slater-Pauling curve quite well. In all these compounds, except for Mn2RhSb, the moments of Mn (A) and Mn (B) are antiparallel to each other. The total magnetic moments of the three considered half-metals assume integral values in a wide range of equilibrium lattice parameters.

Keywords

Introduction

In recent years, Mn-based Heusler alloys have attracted much attention due to their physical properties as half-metallic materials and potential applications [13] in the fields of magnetic actuators and spintronics devices. The so-called half-metallic materials have an energy gap in one spin direction at the Fermi level EF whereas the other spin band is strongly metallic. As a result, a 100 % spin polarization can be obtained [4] and half-metals can be used as spin injectors for magnetic random access memories and other spin-dependent devices [5]. Many Mn-based Heusler alloys have been studied experimentally for their half-metallic band structure and high Curie temperatures. At the same time, many Mn-based Heusler alloys have been theoretically predicted to be half-metals, and experiments have been carried out to establish their half-metallic band structure and magnetic properties, which make them suitable for technical applications [612].

Since the time when de Groot et al. predicted NiMnSb to be a half-metal as well as half-Heusler alloy, much attention has been paid to the Heusler alloy family for their unique character [4]. Among these materials, Mn2-based Heusler alloy with half-metallicity attracted much attention. Among Mn2YZ alloys, Mn2VAl was the first half-metal predicted by the band theory calculation in Mnbased Heusler alloys [13]. Later Mn2CoZ (Z = Al, Si, Ge, Sn, Sb) [1416], Mn2FeZ (Z = Al, Sb) [17] and Mn2CrZ (Z = Al, Sb) [18] also attracted much attention due to their half-metallicity. Recently, Mn2CuZ (Z = Si, Sb, Ge) [1921] alloys have been reported to exhibit half-metallic character in their electronic structures. These Mn2YZ alloys (where Y denotes 3d elements) have been thoroughly investigated. However, the electronic structure as well as half-metallicity of Heusler alloys containing 4d or 5d elements is not very clear. Recently, Abada et al. [22] have found Mn2ZrZ (Z = Si, Ge) alloy exhibiting half-metallicity in theory, and Endo et al. [23] have synthesized a new Heusler alloy Mn2RuSn, which besides Mn contained a 4d-element Ru, indicating the possibility to find new functional materials in Heusler alloys consisting of 4d elements in theory and experiment.

In this paper, we studied the site preference in Mn2RhZ (Z = Al, Ga, In, Si, Ge, Sn, Sb) by first-principles calculations. The electronic structure and the magnetic properties of the compounds were investigated in details. Three half-metals, namely, Mn2RhAl, Mn2RhGe and Mn2RhSb, were predicted. For all these alloys, except Mn2RhSb, the antiparallel alignment between the large local moments of the Mn atoms is observed. The total magnetic moments of the three half-metals with Z atoms lying in the same columns assume integral values for a wide range of equilibrium lattice parameters. Furthermore, we also explored the influence of tetragonal distortion on the half-metallicity property of Mn2RhAl as an example for the subsequent research.

Computational methods

The electronic structure calculations were carried out by using the CASTEP code based on pseudopotential method with a plane-wave basis set [24, 25]. The electronic exchange-correlation energy was treated under the local-density approximation (LDA) [26, 27]. The interactions between the atomic core and the valence electrons were described by the ultrasoft pseudopotential [28]. We considered two cases of initial spin moments of Mn (A) and Mn (B) atoms: antiparallel and parallel, and set the spins as –1, +2 and +2, +2, respectively. The plane wave basis set cut-off was used as 500 eV for all the cases. A mesh of 15 × 15 × 15 k-points was employed for Brillouin zone integrations for all the cases. The convergence tolerance for the calculations was selected as a difference of total energy within 1 × 10–6 eV/atom. These parameters ensured good convergence for total energy and the calculations were based on the theoretical equilibrium lattice parameters.

Results and discussion
Lattice parameter

Heusler alloy has a structure of four interpenetrating face-center-cubic (fcc) lattices and a stoichiometric composition of X2YZ, where X and Y atoms are transition metal elements, and Z atom is an element of group III, IV or V. Generally, Heusler alloy has two possible structures: Hg2CuTi-type and Cu2MnAl-type. In Hg2CuTi-type structure, the X atoms occupy the A (0, 0, 0) and B (1/4, 1/4, 1/4) sites, and the Y atom enters the C (1/2, 1/2, 1/2) site, with Z atom located at the D (3/4, 3/4, 3/4) site in the Wyckoff coordinates. While in Cu2MnAl-type structure, the X atoms occupy the A and C sites, Y atom enters the B site, leaving the D site to Z atom.

For 3d transition metal elements, the preference of transition metal elements is determined by the number of their valence electrons. And the elements with less electrons prefer to occupy the B sites, while others prefer to enter the (A, C) sites [29]. According to the previous studies on Mn2RhZ (Z = Al, Ga, Sn), Felser et al. studied the tetragonal distortion based on the cubic Heusler alloys in Hg2CuTi-type structure [30]. Thus, we considered Mn2RhZ (Z = Al, Ga, In, Si, Ge, Sn and Sb) with Hg2CuTi-type structure in this paper, and performed calculations for two cases of different initial spin moments for Mn atoms: antiparallel and parallel.

All these calculations concerning total energy as a function of lattice parameters, except for Mn2RhIn alloy, are completely identical in both cases, and the calculated equilibrium lattice constants are listed in Table 1. In Table 1, the equilibrium lattice constants increase with increasing atomic number of Z atoms from the same column of the periodic table. The theoretical lattice constant of Mn2RhAl is smaller than that of Mn2RhGa because the Al atoms have a smaller atomic radius than Ga atoms. The radius of In atom is much larger than those of Al and Ga atoms. Thus, the lattice constant of Mn2RhIn is the largest in the series of Mn2RhZ (Z = Al, Ga and In). Similarly, Mn2RhSn has the largest lattice constant among Mn2RhZ (Z = Si, Ge and Sn) compounds and the lattice constants increase with the rising of the atomic number of Si → Ge → Sn.

The calculated equilibrium lattice constants, gap width, the calculated total and partial magnetic moments for the Mn2CrZ (Z = Al, Ga, In, Si, Ge, Sn and Sb) alloys.

Compounda [Å]Gap width [eV]mtotalB]mMn(A)B]mMn(B)B]mRhB]mZB]
Mn2RhAl5.800.202.00–1.563.180.380.00
Mn2RhGa5.822.02–1.643.300.360.00
Mn2RhIn6.101.61–2.623.840.380.04
Mn2RhSi5.730.173.00–0.503.160.320.02
Mn2RhGe5.790.163.00–0.583.240.340.00
Mn2RhSn6.010.113.00–0.803.460.320.00
Mn2RhSb6.020.414.000.143.520.360.00
Electronic structure and magnetic properties

The electronic structure of the Mn2RhZ (Z = Al, Ga, In, Si, Ge, Sn, Sb) alloys has been studied by first-principles calculations. The calculated total and partial densities of states (DOSs) of Mn2RhZ are shown in Fig. 1. In Fig. 1, we chose the majority-spin states as positive values and the minority-spin states as negative values. The states between –5 eV and –6 eV are mainly from the p electrons of Z atoms in the occupied valence states, which hybridize with the d electrons of Mn and Rh atoms and determine the degree of occupation of the p-d orbitals. The total DOSs of –4 eV to +2 eV arise from d states of transition metal atoms. The widely spread d states are mainly from covalent hybridization between Mn and Rh atoms. The covalent hybridization between the lower-energy d states of the higher-valence transition metal atom, like Rh, and the higher-energy d states of the lowervalence transition metal atom, like Mn, can be strong and lead to the formation of bonding and antibonding bands [31].

Fig. 1

The calculated spin-projected total and partial DOS plots for (a) Mn2RhAl, (b) Mn2RhGa, (c) Mn2RhIn, (d) Mn2RhSi, (e) Mn2RhGe, (f) Mn2RhSn, and (g) Mn2RhSb.

The total DOSs of Mn2RhZ (Z = Al, Ga, In, Si, Ge, Sn, Sb) shown in Fig. 2 can be divided into three groups according to valence electrons of Z atoms: Al, Ga and In alloys belonging to the first group, Si, Ge and Sn alloys attributed to the second group, and Mn2RhSb as the third group. All total DOSs of the three-group compounds are similar in shape. The low energy states are mainly contributed by s electrons of Z atoms, and the s states are separated from the d states by a gap in the DOSs in both spin directions. Thus, we have ignored the effect of s electrons on the d-d interaction and have not presented s states in Fig. 1. Similar behavior has also been observed in Mn2FeZ (Z = Al, Ga, Si, Ge, Sb) [32] alloys.

Fig. 2

The calculated total DOS of Mn2RhZ (Z = Al, Ga, In, Si, Ge, Sn, Sb).

In order to clarify the variation trend of the DOSs of the three groups, we analyzed the DOSs of Mn2RhIn, Mn2RhSn and Mn2RhSb alloys, which are shown in the Fig. 2 (the 3rd, 6th and 7th panel). The DOSs obviously move to the lower energy as the number of valence electrons increases. Sn atom has one more p state valence electron in comparison to In atom, and Mn2RhSb also has an extra valence electron compared with Mn2RhSn. The p electrons of the main group element hybridize with the d states of Mn and Rh, determining the degree of occupation of p-d orbitals and influencing the position of the majority spin states relative to the Fermi level due to the extra electrons from Z atoms filling in only majority spin states [33]. The majority states move to a lower energy as Z element is varying from In to Sn. The Fermi level locates at a deep DOS valley in the Mn2RhSi, and moves to a large DOS peak in Mn2RhSn. However, the change in minority states is not so obvious. The minority states are slightly shifted to a lower energy with respect to the Fermi level. Similar results have also been found in Mn2FeAl and Mn2FeSi alloys [32]. In the same way, the majority states of Mn2RhSb shift to a lower energy with respect to the Fermi level. Meanwhile, the extra charge leads to a change in their spin moments. Mn (A) has four Z atoms as nearest neighbors, so the hybridization between Mn (A) and Z is strong. Thus, Mn (A) spin moment increases intensely.

It is worth noting that Sb atom has the largest number of valence electrons among the main group atoms Z (Z = Al, Ga, In, Si, Ge, Sn, Sb), and the majority DOS of Mn2RhSb moves to a lower energy relative to the other Mn2RhZ (Z = Al, Ga, In, Si, Ge, Sn) due to the extra electrons filling in only majority spin states. Mn (A) is the nearest neighbor atom of Sb atom, and the magnetic moment of Mn (A) reveals the most obvious change in Table 1. The moments of Mn (A) and Mn (B) for Mn2RhSb are parallel instead of antiparallel ones for the other Mn2RhZ (Z = Al, Ga, In, Si, Ge, Sn). Thus, the magnetic structure of Mn2RhSb is ferromagnetic.

The total DOS of Mn2RhAl compound is shown in Fig. 1a. A three-peak structure is in the majority spin states. The peak above EF is mainly composed of the antibonding peak of Mn (A) and the occupied bonding peak results from the d states of Rh atom. It was reported that the bonding hybrids are localized mainly at the higher-valence transition metal atom site, such as Rh, and the unoccupied antibonding states are mainly from the lower-valence transition metal atom site, such as Mn [31]. A two-peak structure (bonding and antibonding peak), which is separated by an energy gap at the EF, can be observed in the minority spin states. The PDOS shapes of Mn (A) and Mn (B) are opposite to each other. In the majority states of Mn (B), the two peaks are occupied and lie below the Fermi level. In the minority of PDOS, the exchange splitting moves the antibonding peak high above EF, resulting in a large local moment of Mn (B). Large Mn moment has also been observed in other Heusler alloys, such as Mn2FeZ (Z = Al, Ga, Si, Ge, Sb) [32]. In contrast, the partial DOS of Mn (A) lies below EF in the minority spin states, and above EF in the majority spin states. Therefore, the contributions to the total DOS from Mn (A) and Mn (B) are opposite to each other, indicating an antiparallel configuration of their spin moments [34]. With the help of the calculated total and partial magnetic moments in Table 1, it can be inferred that the spin splitting of Mn (B) is large, resulting in a large local moment of Mn (B). The atomic energy level E(A;C) < E(B) has been reported by Bansil et al. [34]. They stated that the spin down bonding states preferentially reside at Rh (C) site and produce a negative spin density in the occupied states, resulting in strong bonding states in the minority spin states of Rh and causing a small exchange splitting. As a result, Rh shows only a small magnetic moment. The total moment of Mn2RhAl is 2.00 μB, which is mainly determined by the antiparallel aligned Mn (A) and Mn (B) spin moments.

The DOSs of Mn2RhGa and Mn2RhIn are shown in Fig. 1b and Fig. 1c, and are similar in shape to the one of Mn2RhAl. The two-peak structures are in the minority spin states and are separated by the energy gaps around EF. In the majority spin states, the Fermi level falls in a pseudogap, dividing the majority states into bonding and antibonding parts. The PDOSs of Mn (A) and Mn (B) contribute to both the bonding and antibonding parts in the total DOS. Mn atoms show spin splitting in both sites, and a different trend in their PDOS is observed. In Fig. 2, we gathered the calculated total DOS of Mn2RhZ (Z = Al, Ga, In, Si, Ge, Sn, Sb). Mn(A) is the nearest neighbor atom of the main group atoms (Al, Ga, In), thus, the influence of Al ! In on the Mn(A) atom is the largest. The DOSs of Mn2RhZ (Z = Al, Ga, In) move to a lower energy as Z is varying from Al to In because of the p-d hybridization between Z and Mn (A) atoms. From the total DOSs of Mn2RhZ (Z = Si, Ge, Sn) in Fig. 1d, Fig. 1e and Fig. 1f, an energy gap is in the minority states at the EF. And in the majority spin states, the antibonding peak mainly from Mn (A), is just located at the Fermi level. The energy gap variation trend, which is shown in Fig. 2 (the 4th to 6th panel), is similar to the trend in Mn2RhAl, Mn2RhGa and Mn2RhIn. The DOS moves to a lower energy as Z varies from Si to Sn, and the EF falls throughout the energy gap. In contrast, the bonding peak in the majority states moves to a higher energy with the increase in atomic number of Si → Ge → Sn. The Z atom plays an important role in Heusler alloys, which causes s-p states to hybridize with d electrons and determines the degree of the p-d orbital occupation [31]. The splitting of Z atom p states electrons occurs at higher energy when Z varies from Si to Sn (–2.9 eV for Si, –2.8 eV for Ge and –2.5 eV for Sn). Thus, the bonding hybridization between p and d electrons can be observed at the higher energy. The total magnetic moment of the Mn2RhZ (Z = Si, Ge, Sn) alloys is 3.00 μB, which is in line with the s-p rule of Mt = Zt – 24. Mn (B) spin moment is large and increases with increasing atomic number of Z atoms. Mn (A) and Rh atom moments are small and contribute little to the total moment.

The calculated total and partial DOS for Mn2RhSb is shown in Fig. 1g. There are three-peak structures in both spin directions. The bonding peak around –3.2 eV in the minority spin states is mainly determined by the Rh atom. And in the majority spin states, the bonding peak around –3 eV mainly arises from the hybridization of Mn and Rh atoms. The majority states are similar to the one of Mn2RhSi, the Fermi level is also located on the antibonding part. In this paper, Mn2RhSb is a ferromagnet with a small positive magnetic moment at the Mn (A) site, different from the ferrimagnet of the others. The magnet moment strongly depends on the atomic order of the nearest neighbor shell. The moment of Mn (B) is nearly constant because the Mn (A) and Rh atoms are the nearest atoms to Mn (B). In contrast, the moment of Mn (A) atom changes considerably as the nearest atom of Sb provides p electrons to hybridize with d states and determines the degree of p-d orbital occupation [30]. The PDOS of Mn (A) moves to a lower energy, resulting in the spin moment parallel to that of Mn (B).

Half-metallicity

In the total DOSs of Mn2RhZ, the energy gap can be obtained in the minority spin around the Fermi level. However, in the majority states, the DOSs are quite large, indicating a high spin polarization at the EF. The spin polarization ratios of Mn2RhZ (Z = Al, Ge, Sb) are 100%due to the fact that the alloys do not have DOSs at EF in the minority gap. Here, the spin polarization ratio is calculated as the value of (N – N)/(N+ N), where N and N are the majority-spin and minorityspin DOSs at the EF, respectively. So, Mn2RhAl, Mn2RhGe and Mn2RhSb are “true” half-metals.

With the help of the band structure, the majority-spin and minority-spin band structures under the theoretical equilibrium lattice parameters, 5.80 Å for Mn2RhAl, 5.79 Å for Mn2RhGe and 6.02 Å for Mn2RhSb, are shown in Fig. 3. In the majority-spin states, the top of valence band overlaps with the EF, exhibiting metallic character at the EF. However, the minority-spin states exhibit an indirect G-X gap, and the EF is exactly located in the band gap, indicating semiconducting nature. The values of the band gap are shown in Table 1. In summary, Mn2RhAl, Mn2RhGe and Mn2RhSb alloys have an ideal half-metallic band structure.

Fig. 3

The calculated band structure plots for (a) Mn2RhAl, (b) Mn2RhSi, and (c) Mn2RhSb.

In the half-metallic Heusler alloys, the total moments Mt are integral values and follow the Slater-Pauling rule of Mt = Zt– 24, where Zt is the number of valence electrons [33]. Mn2RhAl, Mn2RhGe and Mn2RhSb have 26, 27 and 28 valence electrons, respectively. The total magnetic moment should be 2.00 μB, 3.00 μB and 4.00 μB, respectively, which agrees with our results in Table 1 quite well.

The calculations for all the cases have been performed under their theoretical equilibrium lattice constants and Mn2RhZ (Z = Al, Ge, Sb) alloys exhibit half-metallic characters. However, a small change in the lattice parameter may shift the EF with respect to the minority gap, which obviously affects the half-metallic character [35]. In experimental studies, such as melt spinning or ball milling, the strain may be quite large and will make the lattice constant deviate from the ideal one. Furthermore, the growth of thin film materials becomes possible by modern techniques. The lattice constant of the thin films is strongly influenced by the relationship between the half-metallic character and the lattice constant for a given material from both theoretical and technical point of view. Therefore, it is meaningful to study the dependence of the magnetic properties and spin polarization on the lattice parameters for Mn2RhAl, Mn2RhGe and Mn2RhSb.

Here, we have investigated the relationship between the dependence of the magnetic moments and lattice parameters for the three half-metals. The total magnetic moments as functions of the lattice parameter for Mn2RhZ (Z = Al, Ge, Sb) are shown in Fig. 4. A moderate variation in the lattice parameter does not significantly change the total magnetic moments in these compounds as the movement of the Fermi level usually takes place within the minority gap. The half-metallic properties of Mn2RhZ (Z = Al, Ge, Sb) can be obtained within the range of 5.56 Å to 5.96 Å, 5.69 Å to 6.06 Å, 5.81 Å to 6.49 Å, respectively. Thus, Mn2RhAl, Mn2RhGe and Mn2RhSb can be considered as new candidates for spintronics applications.

Fig. 4

The calculated total magnetic moments as functions of the Mn2RhZ (Z = Al, Ge, Sb) Heusler alloys.

Furthermore, we have shown the influence of tetragonal distortion on the half-metallicity of Mn2RhAl as an example of the later research of tetragonal distortion in the Mn2RhZ. We have investigated the relative stability of the unit cell geometry by calculating the total energy as a function of the c/a ratio at a constant volume and the respective graph is shown in Fig. 5. The lowest total energy can be observed at the point of c/a = 1, indicating the stability of cubic phase in the structure.

Fig. 5

Total energy difference D Etotal = Etotal(c/a) – Etotal(c/a = 1) between cubic (c/a = 1) and tetragonally-distorted phases, calculated as a function of c/a ratio along the constant volume for Mn2RhAl Heusler alloy.

Conclusions

First-principles calculations were performed for the Mn2RhZ (Z = Al, Ga, In, Si, Ge, Sn, Sb) Heusler alloys. We predicted three half-metallic ferromagnets, namely: Mn2RhAl, Mn2RhGe and Mn2RhSb. The calculated equilibrium lattice constant increases with increasing atomic number of Z atoms lying in the same column of the periodic table. The calculated total magnetic moments Mtot are 2 μB/f.u. for Mn2RhAl and Mn2RhGa, 3 μB/f.u. for Mn2RhSi, Mn2RhGe and Mn2RhSn, and 4 μB/f.u. for Mn2RhSb, which agrees with the Slater-Pauling curve quite well. In all these compounds, except for Mn2RhSb, the moments of Mn (A) and Mn (B) are antiparallel to each other. Mn2RhSb is a ferrimagnet due to the influence of the extra valence electron filling in the majority spin states. It has been found that the total magnetic moments of Mn2RhAl, Mn2RhGe and Mn2RhSb can assume an integral value within the range of 5.56 Å to 5.96 Å, 5.69 Å to 6.06 Å, 5.81 Å to 6.49 Å, respectively. As an example, we also investigated the effect of tetragonal distortion on the half-metallicity of Mn2RhAl and found that the structure of cubic phase is stable.

Fig. 1

The calculated spin-projected total and partial DOS plots for (a) Mn2RhAl, (b) Mn2RhGa, (c) Mn2RhIn, (d) Mn2RhSi, (e) Mn2RhGe, (f) Mn2RhSn, and (g) Mn2RhSb.
The calculated spin-projected total and partial DOS plots for (a) Mn2RhAl, (b) Mn2RhGa, (c) Mn2RhIn, (d) Mn2RhSi, (e) Mn2RhGe, (f) Mn2RhSn, and (g) Mn2RhSb.

Fig. 2

The calculated total DOS of Mn2RhZ (Z = Al, Ga, In, Si, Ge, Sn, Sb).
The calculated total DOS of Mn2RhZ (Z = Al, Ga, In, Si, Ge, Sn, Sb).

Fig. 3

The calculated band structure plots for (a) Mn2RhAl, (b) Mn2RhSi, and (c) Mn2RhSb.
The calculated band structure plots for (a) Mn2RhAl, (b) Mn2RhSi, and (c) Mn2RhSb.

Fig. 4

The calculated total magnetic moments as functions of the Mn2RhZ (Z = Al, Ge, Sb) Heusler alloys.
The calculated total magnetic moments as functions of the Mn2RhZ (Z = Al, Ge, Sb) Heusler alloys.

Fig. 5

Total energy difference D Etotal = Etotal(c/a) – Etotal(c/a = 1) between cubic (c/a = 1) and tetragonally-distorted phases, calculated as a function of c/a ratio along the constant volume for Mn2RhAl Heusler alloy.
Total energy difference D Etotal = Etotal(c/a) – Etotal(c/a = 1) between cubic (c/a = 1) and tetragonally-distorted phases, calculated as a function of c/a ratio along the constant volume for Mn2RhAl Heusler alloy.

The calculated equilibrium lattice constants, gap width, the calculated total and partial magnetic moments for the Mn2CrZ (Z = Al, Ga, In, Si, Ge, Sn and Sb) alloys.

Compounda [Å]Gap width [eV]mtotalB]mMn(A)B]mMn(B)B]mRhB]mZB]
Mn2RhAl5.800.202.00–1.563.180.380.00
Mn2RhGa5.822.02–1.643.300.360.00
Mn2RhIn6.101.61–2.623.840.380.04
Mn2RhSi5.730.173.00–0.503.160.320.02
Mn2RhGe5.790.163.00–0.583.240.340.00
Mn2RhSn6.010.113.00–0.803.460.320.00
Mn2RhSb6.020.414.000.143.520.360.00

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