1. Introduction
Transition metals are discovered to possess high Curie temperature in a research study on an ordered double perovskite material,
A2BB'O
6 [
1]. The
A-site elements are alkaline-earth ions, such as Ca, Sr, Ba and
B,
B′ being [
2] the large low-field tunneling magneto resistance (TMR) [
3] at room temperature and half-metallicity [
4]. The earliest half-metallic (HM) compound, denoted as Sr
2FeMoO
6 [
4,
5,
6,
7,
8], was found by Kobayashi
et al. Half-metallic (HM) materials are metallic for one spin direction, while they act as an insulator for the other spin direction [
9,
10,
11,
12,
13,
14]. Thus, three characteristic properties can be found: (1) 100% spin polarization at the Fermi level; (2) quantization of the magnetic moment; and (3) a spin susceptibility of zero. HM materials with this interesting attribute can be used as computer memory, magnetic recorders, and so on.
Ordered double perovskites,
A2BB′O
6, are a good baseline structure for determining suitable HM material candidates, because there are a variety of combinations for substituting the
A-site or
B-site elements. In previous research, HM compounds were found, such as Sr
2FeWO
6 [
6], Sr
2FeReO6 [
15,
16], Sr
2MnMoO6 [
16], Sr
2CuOsO
6 [
17], Sr
2VOsO
6 [
18], Sr
2NiRuO
6 [
19], Sr
2FeTiO
6 [
20], Sr
2CrMoO
6 [
12,
21], Sr
2CoMoO
6 [
12], Sr
2CrReO
6 [
11,
15] and Sr
2CrWO
6 [
14,
22] for the same ion Sr on the
A-site with different combinations of
B-site elements. Sr
2MnMoO
6 [
8] and Ba
2MnMoO
6 [
23] represent
A-site substitution with fixed
B-site elements. From previous research, the electronic structure between Sr
2FeMoO
6 [
4,
5,
6,
7,
8] and Sr
2CrMoO
6 [
12,
21] is similar, near the Fermi level, which gives a clue about the combination required for double perovskite HM compounds.
In order to find more potential HM materials, this paper presents theoretical HM compound prediction on double perovskites,
A2BB′O
6, by systematically substituting the
A-site ion with fixed
BB′ combinations, such as Cr
M (
M = Mo, Re and W). For the
A-site ion, the similarity of the valence electrons between the IVA group elements and alkaline-earth elements (Ca, Sr and Ba), denoted as IIA(
s2) and IVA(
p2), motivated us to use IVA group elements instead of alkaline-earth elements [
24]. However, the elements in the IVA group are quite different from Sr, so an examination of the structural stability is still needed. All the compounds are stable, except the Si-based double perovskite structure. The double perovskite structure cannot be synthesized as the covalent bond of SiO
2, because the binding energy is too strong. Thus, calculations of density functional theory (DFT) were carried out, and new combinations of HM materials were found. The exempted HM materials in the preparation of this work are compounds of carbon, which represents the
A-site position (C
2Cr
MO
6). This exemption is due to the covalent bounding of carbon, which is too strong. The valence of carbon is +4 rather than +2. For
M = Mo and Re, all compounds can be HM materials, with Si
2CrMoO
6, Ge
2CrMo and Ge
2CrReO
6 needing strong correlation correction. For
M = W, only
A = Sn and Pb are possible candidates as HM materials.
3. Results and Discussions
In an ordered double perovskite structure,
A2BB′B
6 (
Figure 1), four magnetic phases exist: ferromagnetic (FM), ferrimagnetic (FiM), antiferrimagnetic (AF) and nonmagnetic (NM). These phases are controlled by the spin state of the two
B and
B′ ions. In the ideal cubic structure (
Fm3m, no. 225) for FM and FiM states, the
B and
B′ ions can be described in the order of the NaCl configuration, i.e., a face-centered cubic (fcc) stacked by
B and
B′ ions with lattice constant 2
a. In the fcc cubic, each
B(
B′) is coordinated by
B′(
B), and each has an O ion between, so there are 6
B-O-
B′ bonds per unit cell with the length of
B-O and
B′-O being equal. To determine theoretical lattice constants and atomic positions by structural optimization calculations, a larger unit cell with 2 f.u. (formula unit) was considered to enable the structure to relax to a reduced symmetry. [
37] After full structural optimization, most of the ideal cubic structure (
Fm3m, no. 255) will reduce to a tetragonal (
I4/
mmm, no. 139) structure. In the tetragonal structure (
I4/
mmm), there are two nonequivalent types of O atoms, as shown in
Table 2. There are two O
1 atoms located on the
z-axis with
B and
B′ atoms sitting between, and the four O
2 atoms are located on the
xy-plane; the same as the
B and
B′ atoms (
Figure 1 and
Table 2). The angle of the
B-O-
B′ remained at 180° during structural optimization, whereas the lattice constant and bond length changed. The closeness of the
c/
a ratio to the ideal value,

, shows that the symmetry reduction is minor. Although some compounds remained in the ideal cubic structure (
Fm3m, no. 255) during full structural optimization, the O
1 and O
2 are equivalent to the
c/
a ratio,

. In the AF state, the tetragonal structure (
P4/
mmm, no. 123) remains the same in the full structural optimization.
Table 1.
Calculated physical properties of the A2CrMO6 in a double perovskite structure in the full structural optimization calculation of generalized gradient approximation (GGA) + U (Coulomb parameter).
Table 1.
Calculated physical properties of the A2CrMO6 in a double perovskite structure in the full structural optimization calculation of generalized gradient approximation (GGA) + U (Coulomb parameter).
Materials | U (Cr, M) | Spin magnetic moment (μB/f.u.) | d Orbital electrons ↑/↓ | N (EF) | Band gap | Spin-polarization | ΔE = FM − AF |
---|
A2Cr[M]O6 | MCr | MM | mtot | Cr | M | states/eV/f.u. | eV | N((↑ − ↓)/(↑ + ↓)) (%) | meV/f.u. |
---|
Si[Mo] | (0, 0) | 2.297 | −0.407 | 2.004 | 3.333/1.069 | 1.934/2.320 | ↑0.152/↓2.739 | | −89.5 | −27.9 |
(3, 2) | 2.618 | −0.701 | 2.000 | 3.479/0.898 | 1.764/2.442 | ↓3.620 | ↑0.925 | −100.0 | −197.9 |
Ge[Mo] | (0, 0) | 2.318 | −0.439 | 2.000 | 3.336/1.051 | 1.914/2.332 | ↓3.108 | ↑0.150 | −100.0 | −63.1 |
(3, 2) | 2.267 | −0.729 | 2.000 | 3.474/0.885 | 1.746/2.451 | ↓3.592 | ↑1.100 | −100.0 | −109.3 |
Sn[Mo] | (0, 0) | 2.370 | −0.503 | 2.000 | 3.328/0.992 | 1.864/2.344 | ↓2.759 | ↑0.675 | −100.0 | −117.3 |
(3, 2) | 2.656 | −0.806 | 2.000 | 3.447/0.829 | 1.690/2.471 | ↓3.143 | ↑1.675 | −100.0 | −118.1 |
Pb[Mo] | (0, 0) | 2.381 | −0.513 | 2.000 | 3.324/0.977 | 1.850/2.340 | ↓2.767 | ↑0.700 | −100.0 | −123.7 |
(3, 2) | 2.659 | −0.806 | 2.000 | 3.439/0.818 | 1.680/2.461 | ↓3.160 | ↑1.825 | −100.0 | −120.8 |
Si[Re] | (0, 0) | 2.294 | −0.958 | 1.095 | 3.319/1.059 | 1.659/2.593 | ↑0.541/↓2.357 | | −62.7 | −158.1 |
(3, 2) | 2.624 | −1.364 | 1.000 | 3.463/0.877 | 1.446/2.783 | ↓2.243 | ↑1.125 | −100.0 | −176.3 |
Ge[Re] | (0, 0) | 2.314 | −1.030 | 1.042 | 3.323/1.043 | 1.623/2.628 | ↑0.338/↓4.108 | | −84.8 | −253.9 |
(3, 2) | 2.631 | −1.399 | 1.000 | 3.458/0.865 | 1.427/2.798 | ↓2.303 | ↑1.300 | −100.0 | −292.9 |
Sn[Re] | (0, 0) | 2.360 | −1.125 | 1.000 | 3.317/0.992 | 1.568/2.666 | ↓2.772 | ↑0.800 | −100.0 | −301.9 |
(3, 2) | 2.652 | −1.469 | 1.000 | 3.433/0.819 | 1.384/2.824 | ↓2.842 | ↑1.850 | −100.0 | −276.3 |
Pb[Re] | (0, 0) | 2.372 | −1.143 | 1.000 | 3.314/0.977 | 1.553/2.669 | ↓2.865 | ↑0.875 | −100.0 | −308.7 |
(3, 2) | 2.656 | −1.473 | 1.000 | 3.426/0.808 | 1.376/2.282 | ↓2.875 | ↑1.825 | −100.0 | −273.9 |
Si[W] | (0, 0) | 2.377 | −0.268 | 2.617 | 3.359/1.015 | 1.951/2.199 | ↑0.545/↓2.226 | | −60.7 | −28.9 |
(3, 2) | 2.640 | −0.429 | 2.165 | 3.464/0.861 | 1.826/2.235 | ↑0.550/↓2.300 | | −61.4 | −53.6 |
Ge[W] | (0, 0) | 2.374 | −0.311 | 2.131 | 3.355/1.014 | 1.925/2.215 | ↑0.473/↓2.273 | | −65.5 | −70.6 |
(3, 2) | 2.635 | −0.496 | 2.119 | 3.458/0.859 | 1.790/2.625 | ↑0.438/↓2.449 | | −69.7 | −85.6 |
Sn[W] | (0, 0) | 2.349 | −0.454 | 2.000 | 3.320/1.004 | 1.842/2.273 | ↓3.149 | ↑0.825 | −100.0 | −166.2 |
(3, 2) | 2.623 | −0.703 | 2.000 | 3.422/0.834 | 1.683/2.360 | ↓3.177 | ↑1.900 | −100.0 | −143.2 |
Pb[W] | (0, 0) | 2.352 | −0.464 | 2.000 | 3.313/0.993 | 1.831/2.271 | ↓2.890 | ↑1.025 | −100.0 | −180.2 |
(3, 2) | 2.622 | −0.716 | 2.000 | 3.411/0.825 | 1.671/2.360 | ↓3.099 | ↑2.725 | −100.0 | −151.9 |
Figure 1.
An ideally ordered double perovskite structure, A2CrMO6.
Figure 1.
An ideally ordered double perovskite structure, A2CrMO6.
Table 2.
Structural parameters in the fully optimized structure (I4/mmm, no. 139 and Fm3m, no. 255), where A (x, y, z) = (0, 0.5, 0.75), Fe (x, y, z) = (0, 0, 0), M (x, y, z) = (0, 0, 0.5), O1 (x, y,z) = (0, 0, O1z) and O2 (x, y, z) = (O2x, O2y, 0.5).
Table 2.
Structural parameters in the fully optimized structure (I4/mmm, no. 139 and Fm3m, no. 255), where A (x, y, z) = (0, 0.5, 0.75), Fe (x, y, z) = (0, 0, 0), M (x, y, z) = (0, 0, 0.5), O1 (x, y,z) = (0, 0, O1z) and O2 (x, y, z) = (O2x, O2y, 0.5).
A2Cr[M]O6 | a | c/a | V0 (Å3/f.u.) | O1z | O2x | O2y |
---|
Si[Mo] | 5.4616 | 1.4138 | 115.17 | 0.24755 | 0.25238 | 0.25238 |
Ge[Mo] | 5.4816 | 1.4135 | 116.41 | 0.24779 | 0.25210 | 0.25210 |
Sn[Mo] * | 5.5653 |  | 121.89 | 0.24896 | 0.24896 | 0.24896 |
Pb[Mo] | 5.5879 | 1.4146 | 123.41 | 0.24923 | 0.25077 | 0.25077 |
Si[Re] | 5.4561 | 1.4134 | 114.78 | 0.24926 | 0.25075 | 0.25075 |
Ge[Re] | 5.4768 | 1.4136 | 116.11 | 0.24931 | 0.25071 | 0.25071 |
Sn[Re] | 5.5614 | 1.3920 | 119.72 | 0.24974 | 0.25025 | 0.25025 |
Pb[Re] * | 5.5837 |  | 123.10 | 0.25000 | 0.25000 | 0.25000 |
Si[W] | 5.4719 | 1.4137 | 115.80 | 0.24951 | 0.25047 | 0.25047 |
Ge[W] | 5.4892 | 1.4147 | 116.99 | 0.24966 | 0.25038 | 0.25038 |
Sn[W] * | 5.5721 |  | 122.33 | 0.25059 | 0.25059 | 0.25059 |
Pb[W] | 5.5952 | 1.4147 | 123.90 | 0.25109 | 0.24884 | 0.24884 |
The investigation of structural stability is necessary, because Si, Ge, Sn and Pb are quite different from Sr. The structural stability can be examined by the energy difference between the double perovskite structure and existing materials, such as
AO
(2) (
A = Si, Ge, Sn and Pb),
MO
2 (
M = Mo, Re and W) and CrO
(2). Thus, the energy difference can be written as Equations (1) and (2), where E
tot(f.u.) represents the energy of each compound.
SiO
2, GeO
2, SnO
2, PbO
2, SnO, PbO, CrO, CrO
2 and
MO
2 are existing materials that can be easily calculated. The case of
MO is a bit tricky, because
MO
2 actually exists, but not
MO. Thus, the energy of
MO can be calculated by the average of
MO
2 and
M bulk, which can be expressed as:
The result shows that all compounds are stable, except the Si-based double perovskite structure. The energy differences (ΔE) for Si, Ge, Sn and Pb of A2CrMO6 are 8 eV/f.u. to 10 eV/f.u., −0.5 eV/f.u. to −2 eV/f.u., −10 eV/f.u. to −12 eV/f.u., and −9 eV/f.u. to −13 eV/f.u., respectively. These results show that after oxidation, the crystal ionic radii for Si4+ are too small, so the bond of the electrons remains close to the Si4+ ion, thereby producing a covalent bond. The binding energy of SiO2 (covalent bond) is too strong, such that the double perovskite structure cannot be synthesized. Sr2FeMoO6 is also stable. We present the Si-based double perovskite structure to ensure the integrity of our work.
In the FM and FiM states, the B and B′ ions each have the same spin state, that is FM (B, B, B′, B′) = (m, m, m′, m′) and FiM (B, B, B′, B′) = (m, m, −m′, −m′), which can lead us to the assumption of the HM state. For the AF state, the spin state can be shown as (B, B, B′, B′) = (m, −m, m′, −m′). The induced equivalence in the charges is Q↑ [B (B′)] = Q↓ [B (B′)]. In the total density of state (DOS), symmetrical spin-up and spin-down electron distribution can be observed, but no HM features are evident. In the NM phase, there is no spin polarization effect, which results in the absence of magnetic properties. To ascertain which magnetic phase is the most stable, calculations for all four magnetic phases are performed. The result shows that spin-polarized calculated total energies are always lower than those without spin-polarization and the initial FM and FiM state all converge to the FiM state. To guarantee the accuracy of the calculation result, a self-consistent process with higher convergence criteria is also performed.
Based on the energy difference in
Table 1, the FiM state is the most stable magnetic phase in all compounds. In
A2Cr
MO
6, the half-metallic characteristics can be obtained by the energy gap at the spin-up channel with a total magnetic moment (
mtot) of 2.0, 1.0 and 2.0 μ
B for
M = Mo, Re and W in the GGA calculation, respectively. Except (1) for
M = Mo and Re, Si
2CrMoO
6, Ge
2CrMoO
6 and Ge
2CrReO
6 need a strong correlation correction (GGA +
U), and (2) for
M = W. Only
A = Sn and Pb are possible candidates as HM materials. The lattice constant and volume of the unit cell will rise with the
A site atom from silicon to lead (
Table 2), and it will narrow down the electron band structure. The energy gap will appear and become larger (
Table 1). The maintenance of the conductivity at the spin-down channel and the behavior between the size of the
A-site ion and energy gap at the spin-up channel cause the success of
A = Sn and Pb as candidate HM materials. The IVA group elements near the top of the Periodic Table show their strong covalent characteristics. The ionic characteristics become stronger as we go deeper. The positive charge of the atomic nucleus will gain attraction to the inner shell (
s-orbital) of valence electrons. Thus, the valence from C (2
s22
p2) to Pb (6
s26
p2~6
p2) can be denoted as +4 to +2, whereas the others are in between. Thus, IVA group elements, like Pb (6
p2), are similar to Sr (5
s2) in the outer valence electron. This is also the reason for the absence of half metallicity in the carbon stand for the
A site in each compound. The value of the effective parameter,
U (
UCr,
UM), is tuned up from (3, 2) to (5, 3). All results were the same: (1) in all compounds, the FiM magnetic phase is still the most stable state compared with the AF state by about 10
1 meV to 10
2 meV; and (2) the HM compound still carries its original characteristics, and the non-HM compound (Si
2CrWO
6, Ge
2CrWO
6) has no spin-polarized gap.
Figure 2 presents the density of states (DOS) of
A2CrMoO
6,
A2CrReO
6 and
A2CrWO
6 in GGA calculations. The electronic structures are very similar to each other in that they share the same mechanism of HM characteristics. Below the Fermi level (E
F), the O
2p orbital extends from about −8 eV to −2 eV, and hybrids with the
M t2g orbital are in the same energy region. The Cr
t2g orbital in the spin-up channel extends from about −6 eV to −3 eV, which hybrids with the O
2p orbital, and from −2 eV to −0.5 eV below E
F. With the
M t2g and Cr
eg orbitals above the E
F, the band gap appears at the spin-up channel. In the spin-down channel, the
M t2g orbital extends from about −1 eV to 1 eV, which dominates the conductivity of the whole compound. In the ionic picture, the formal valence of Cr
M is +8, and the electron configuration is Cr
3+ (
t2g3eg0), S = 3/2; Mo
5+ (
t2g1eg0), S = −1/2 for
A2CrMoO
6, and Cr
3+ (
t2g3eg0), S = 3/2; Re
5+ (
t2g2eg°), S = −1 for
A2CrReO
6, and Cr
3+ (
t2g3eg0), S = 3/2; W
5+ (
t2g1eg0), S = −1/2 for
A2CrWO
6; according to the calculated electron numbers. The number of valence electrons for Re
5+ (
t2g2eg0) is greater than that for Mo (W)
5+(
t2g1eg0) by one, whereas the local magnetic moment of Re is about two times larger than that of Mo (W). The value of spin polarization is important for applications in spintronics, which can be defined as (N↑ − N↓)/(N↑ + N↓), where N denotes the spin-up (↑) and spin-down (↓) components in the DOS at the E
F. In all HM compounds of this work, spin polarizations are all −100%. The negative sign indicates that the metallic behavior can only be obtained in the spin-down channel. For other materials, they still show high spin polarization more than −65%, such as Si
2CrMoO
6 (−89.5%), Ge
2CrReO
6 (−84.8%) and Ge
2CrWO
6 (−65.5%) in the GGA scheme, indicating that they may still be applied in spintronics.
Figure 2.
Calculated total, spin and site partial density of states in GGA of (a) A2CrMoO6; (b) A2CrReO6; and (c) A2CrWO6. DOS, density of states.
Figure 2.
Calculated total, spin and site partial density of states in GGA of (a) A2CrMoO6; (b) A2CrReO6; and (c) A2CrWO6. DOS, density of states.
The exchange correlation correction effects (GGA +
U) are similar in all compounds. Thus, we present the GGA +
U calculations for Pb
2Cr
MO
6 (
M = Mo, Re and W) in
Figure 3. When the exchange correlation effect is induced, it will give rise to the electronic structure. For Pb
2Cr
MO
6, the Cr
t2g orbital is pushed down deeper, and all the orbitals become more localized, which enhances the local magnetic moment (LMM). Therefore, the band gap at the spin-up channel and the LMM becomes larger. For example, the band gap of Pb
2CrMO
6 grows from 0.70 (0.875, 1.03) eV to 1.83 (1.83, 2.73) eV for
M = Mo (Re, W) in the GGA and GGA +
U processes, respectively. The LMM improves from 2.381 (2.372, 2.352) to 2.659 (2.656, 2.622) for Fe and −0.513 (−1.143, −0.464) to −0.806 (−1.473, −0.716) for Mo (Re, W) in GGA and GGA +
U, respectively. With the consideration of the exchange correlation effect, Si
2CrMoO
6, Ge
2CrMo and Ge
2CrReO
6 appear to be HM materials, where
mtot turned into an integer from 2.004, 1.095 and 1.042 μ
B to 2.0, 1.0 and 1.0 μ
B, respectively. Each opened a gap at the spin-up channel of 3.62, 2.24 and 2.303 eV, respectively (
Table 1). However, for Si
2CrWO
6 and Ge
2CrWO
6, even the effective parameter,
U (
UCr,
UM), tuned up from (3, 2) to (5, 3). The HM characteristic never appears.
Figure 3.
Calculated total, spin and site partial density of states of Pb2CrMO6 (M = Mo, Re and W) in GGA + U (3, 2).
Figure 3.
Calculated total, spin and site partial density of states of Pb2CrMO6 (M = Mo, Re and W) in GGA + U (3, 2).
For the mechanism of the half metallicity and ferrimagnetic phase, Terakura
et al. [
38] proposed an F(i)M stabilization mechanism, while a nonmagnetic element is in between magnetic elements with the
p-d hybridization and double exchange interaction. For example, magnetic elements with a full spin splitting orbital with the E
F located in the middle are denoted as
d-states. The non-magnetic elements located at E
F and between the spin-polarized
d-states are denoted as
p-states. In the spin-up (down) channel, the
d-state will push the
p-state upward (downward). Thus, the E
F will be unequal in both spin states. To keep the E
F common, some electrons will switch spin states that move nonmagnetic elements to contribute negative moments and stabilize the F(i)M state. This phenomenon is called
p-d hybridization. If such hybridization is strong enough, the
p-state can be pushed above the E
F at one spin channel. With the double exchange effect, the band extends at the opposite spin channel with the E
F. Half metallicity and ferrimagnetic appear spontaneously. These behaviors do not exist in the AF configurations; thus, the FiM states are more stable than AF states. In our work, the Cr
t2g orbital represents the spin-split
d-states. The
M t2g orbital stands for the
p-state nonmagnetic elements, whereas weak-magnetic elements are also suitable for
p-state elements.