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The Magnetic Band-Structures of Ordered Pt_{x}Fe_{1−x}, Pt_{x}Co_{1−x}, and Pt_{x}Ni_{1−x} (x = 0.25, 0.50, and 0.75)

## Abstract

**:**

_{2}and L1

_{0}phases of the Pt

_{x}M

_{1−x}(M = Fe, Co and Ni) alloys were investigated using spin-polarized density functional theory (DFT). The relative contributions of both itinerant (Stoner) and localized magnetism at the high-symmetry k-points were determined and discussed qualitatively. Significant directional effects were identified along the A and R directions of the L1

_{0}and L1

_{2}alloys, respectively, and are discussed in terms of charge channeling effects.

## 1. Introduction

_{x}M

_{1−x}(M = Fe, Co or Ni) alloys studied in the current work belong to a family of alloys that disorder at elevated temperatures but order at lower temperatures [11,12,13]. This structural complexity adds much to their significance because of the different properties the materials have depending on their short- and long-range order. For the alloys considered in the current work, these ordering effects can, for example, alter the optical [14], magnetic and structural properties [15,16,17,18], stress/strain relationships [19,20] and chemical reactivity [21] of the alloy. The stress–strain relationships in these alloys have recently shown [18,20] refs. therein] that these surfaces can be switched between mono-dentate and tri-dentate towards very common reaction species such as hydrogen and oxygen. The significance of this has only now started to be understood.

_{x}Pt

_{1−x}, (M = Co, Ni) [29] and of Ni

_{x}Pt

_{1−x}[30] and the more established experimental studies of Ni

_{3}Pt and NiPt

_{3}[31]. The importance of ordered phase studies as a contribution to the theoretical description of these magnetic bimetallic materials is because they remove the level of approximation required to describe a disordered, or mixed phase, material. This level of approximation is required in, for example, the CPA described in the previous paragraph. Further, it establishes a computational framework around which a more thorough theoretical description of these alloys may become possible, and ideally, these developments will be accompanied by appropriate experimental studies of the ordered crystalline systems.

_{x}Ni

_{1−x}. However, in terms of purely characterizing materials and mechanisms, the DFT is still very much the accepted level of theory.

_{x}M

_{1−x}(M = Fe, Co and Ni; x = 0.25, 0.50 and 0.75) alloys are investigated using DFT. The band structures will be elucidated and discussed in terms of the magnetic moment distribution through reciprocal space.

## 2. Results

_{0.75}M

_{0.25}(M = Fe, Co or Ni) and Pt

_{0.5}M

_{0.5}alloys considered in the current work are shown in Figure 1.

_{0.75}M

_{0.25}(M = Fe, Co or Ni) alloys considered in the current work are shown in Table 1. This table contains two sets of parameters: the first set are those determined using DFT in the current work, and the second set are experimental values. The values of the lattice constant determined for Pt

_{x}Ni

_{1−x}in the current work differ nominally from those in the recent studies [30] because of the pseudopotential. In the current work, the pseudopotentials are PAW, whereas, in recent studies, the pseudopotentials were norm-conserving. Because of this, lattice constants in the current work are more accurately estimated; the magnitude of the a parameters estimated in the current work differs from the experimental values by 0.011–0.055 Å, whereas the same parameters differed by 0.082–0.106 Å in [30].

#### 2.1. Spin-Resolved Electron Band Structures for the Ordered Pt_{x}m_{1−X} Alloys

_{x}M

_{1−x}(M = Fe, Co or Ni) alloys are shown in Figure 2. The band-resolved exchange splitting $\mathsf{\delta}{\epsilon}_{\mathrm{ex};\mathrm{j}}\left(k\right)$is defined in Equation (1):

_{Pt}and μ

_{M}of the ordered Pt

_{x}M

_{1−x}(M = Fe, Co or Ni) alloys are presented in Table 2. This analysis was performed in two ways: first, by integrating the projected densities of states (PDOS) and then second, by calculating the Lowdin charges [36]. Both methods show that the Fe, Co and Ni atoms carry the majority of the magnetic moment in each alloy.

#### 2.2. Exchange Splitting ΔE_{ex} for the Ordered Pt_{x}m_{1−X} Alloys

## 3. Discussion

_{x}M

_{1−x}(M = Fe, Co or Ni) alloys. Table 1 shows that the geometric model used in the current study is accurate within the levels of approximation used by density functional theory (DFT). This is evidenced by the low (<1%) percentage error in the DFT estimates of a and c compared to their experimental values. The significance of this low level of error between the experimental and theoretical values of a and c supports the choice of pseudopotential and exchange-correlation interaction used in the current simulations and described in full in the ‘Materials and Methods’ section of this work.

_{x}Ni

_{1−x}alloys. However, in this latter case, the Pt

_{0.75}Ni

_{0.25}alloy is significantly less magnetic than Pt

_{0.5}Ni

_{0.5}or Pt

_{0.25}Ni

_{0.75}. This reduces the moment carried by the d-bands.

_{0.50}Co

_{0.50}and the more magnetic Pt

_{0.25}Co

_{0.75}panels; the linearity is complex. In both cases at m

_{j}< 0.1, a linear segment appears to exist but will not intersect with points at larger m

_{j}. This complexity discourages a deduction of the Stoner parameter from the data presented in the current work. It does, however, show that the Stoner model is complex for these alloys and that this complexity is dependent on both structure and total magnetism. It is of note that the largest deviation from conventional Stoner magnetism is apparent for the Co-bearing alloys, whose total magnetization lies between that of the Fe- and Ni-bearing alloys. The precise reason for this is, however, not clear.

_{2}and L1

_{0}phases, the R and A points, respectively, lie significantly far from these predictions. This behavior is seen less clearly for the Pt

_{0.75}Ni

_{0.25}alloy though this latter observation is expected to be a consequence of the low magnetic moment seen for this particular phase, which was discussed in the previous paragraphs. The directionality of these deviations is significant. For both the L1

_{2}and L1

_{0}cases, these deviations occur in a direction, which is parallel to the [111] crystalline direction. By considering the real-space alloy structures in Figure 1, this direction has the greatest interatomic direction between nearest-neighbors as the closest pairs of atoms are identical and lie at the fractional (0,0,0) and (1,1,1) positions of the unit cell. Because of the large inter-atomic distance along the [111] direction, it is unlikely that a significant delocalized bond exists between these identical atoms. This reduction in bonding is then plausibly responsible for the significant breakdown of the itinerant model and offers a very convenient insight into the Stoner mechanism. Rather than considering the mechanism as entirely anisotropic, the DFT predictions in the current work have shown that it is more appropriate to consider the delocalization along the binding directions of the crystal. In addition to this, the earlier qualitative analyses have shown that the Stoner parameter is nonconstant throughout k-space, though a model for the parameter is not proposed.

## 4. Materials and Methods

_{2}structures, the equation of state for each alloy with that structure was calculated, and the minimum of that equation of state determined numerically. The equation of state was obtained by plotting the density functional theory (DFT) Kohn–Sham energy as a function of a. For the L1

_{0}structures, a similar approach was used; however, for each value of a, the lattice was allowed to relax along the c direction. The equilibrium lattice parameters obtained in this way for both the L1

_{2}and L1

_{0}structures were used throughout the remainder of this work. The band structures were subsequently calculated using the equilibrium lattice parameters and the orthogonal unit cells shown in Figure 1.

## 5. Conclusions

_{2}and L1

_{0}phases of the binary Pt

_{x}M

_{1−x}(M = Fe, Co and Ni; x = 0.25, 0.50 and 0.75) alloys were investigated using spin-polarized density functional theory (DFT). The investigations have quantified the magnetic moment of each of these ordered alloys and have shown that the magnetization is strongest in the Fe-carrying alloys and weakest in those containing Ni. The spin-resolved magnetic band-structures for these alloys reflect these quantitative observations and support a model of magnetism that is primarily localized and generated by charge transfer between the d-orbitals.

_{2}(L1

_{0}) structures. It is suggested that charge channeling may be the reason for this failure.

## Funding

## Conflicts of Interest

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**Figure 1.**Structures of the ordered (

**a**) L1

_{2}Pt

_{0.75}M

_{0.25}(M = Fe, Co or Ni) and (

**b**) L1

_{0}Pt

_{0.5}M

_{0.5}alloys. The ordered Pt

_{0.25}M

_{0.75}alloys are formed by replacing each Pt atom with an M atom and vice versa, in the structure shown in panel (

**a**). The lattice parameters a for L1

_{2}and L1

_{0}and c (for L1

_{0}only) are shown.

**Figure 2.**Spin-resolved electronic band structures of the (

**a**–

**c**) Pt

_{x}Fe

_{1−x}, (

**d**–

**f**) Pt

_{x}Co

_{1−x}and (

**g**–

**i**) Pt

_{x}Ni

_{1−x}ordered alloys. The Pt

_{0.50}Fe

_{0.50}, Pt

_{0.50}Co

_{0.50}and Pt

_{0.50}Ni

_{0.50}alloys are ordered L1

_{2}structures, whereas the remaining alloys are L1

_{0}. E

_{F}is the Fermi level, and the legends show the exchange splitting $\mathsf{\delta}{\epsilon}_{\mathrm{ex};\mathrm{j}}\left(k\right)$ in eV.

**Figure 3.**Exchange splitting ΔE

_{ex}for the ordered (

**a**–

**c**) Pt

_{x}Fe

_{1−x}, (

**d**–

**f**) Pt

_{x}Co

_{1−x}and (

**g**–

**i**) Pt

_{x}Ni

_{1−x}ordered alloys. The labels correspond to high symmetry positions in the electron band structure, and m

_{j}is the normalized total magnetic moment. The line included in each graph is a guide for the eye.

**Table 1.**Lattice parameters determined in the current work (‘Theory’) for each of the ordered L1

_{2}Pt

_{0.25}M

_{0.75}(M = Fe, Co or Ni) and Pt

_{0.75}M

_{0.25}alloys and for the ordered L1

_{0}Pt

_{0.5}M

_{0.5}alloy together with corresponding experimental values (‘Exptl’). The lattice parameters a and c are defined in Figure 1. All dimensions are in Å.

Alloy | Phase | a | c | ||
---|---|---|---|---|---|

Theory | Exptl | Theory | Exptl | ||

Pt_{0.75}Fe_{0.25} | L1_{2} | 3.911 | 3.866 [33] | - | - |

Pt_{0.50}Fe_{0.50} | L1_{0} | 3.894 | 3.852 [33] | 3.705 | 3.713 [33] |

Pt_{0.25}Fe_{0.75} | L1_{2} | 3.740 | 3.750 [33] | - | - |

Pt_{0.75}Co_{0.25} | L1_{2} | 3.890 | 3.831 [34] | - | - |

Pt_{0.50}Co_{0.50} | L1_{0} | 3.817 | 3.810 [35] 3.812 [34] | 3.727 | 3.710 [35] 3.708 [34] |

Pt_{0.25}Co_{0.75} | L1_{2} | 3.666 | 3.66 [34] | - | - |

Pt_{0.75}Ni_{0.25} | L1_{2} | 3.878 | 3.837 [33] | - | - |

Pt_{0.50}Ni_{0.50} | L1_{0} | 3.895 | 3.840 [35] | 3.540 | 3.610 [35] |

Pt_{0.25}Ni_{0.75} | L1_{2} | 3.657 | 3.646 [33] | - | - |

**Table 2.**Magnetic moments μ

_{Pt}and μ

_{M}for the Pt and metal (Fe, Co or Ni) atoms, respectively determined in the current work for each of the ordered L1

_{2}Pt

_{0.25}M

_{0.75}and Pt

_{0.75}M

_{0.25}alloys, and for the ordered L1

_{0}Pt

_{0.5}M

_{0.5}alloy. The unbracketed and bracketed moments were determined using the Lowdin and projected densities of states (PDOS) methods, respectively (see text). All magnetic moments are in μ

_{B}.

Alloy | Phase | μ_{Pt} | μ_{M} |
---|---|---|---|

Pt_{0.75}Fe_{0.25} | L1_{2} | 0.378 (0.388) | 3.309 (3.386) |

Pt_{0.50}Fe_{0.50} | L1_{0} | 0.408 (0.400) | 3.007 (3.051) |

Pt_{0.25}Fe_{0.75} | L1_{2} | 0.401 (0.381) | 2.762 (2.677) |

Pt_{0.75}Co_{0.25} | L1_{2} | 0.337 (0.324) | 2.041 (2.135) |

Pt_{0.50}Co_{0.50} | L1_{0} | 0.423 (0.411) | 1.965 (1.908) |

Pt_{0.25}Co_{0.75} | L1_{2} | 0.406 (0.386) | 1.848 (1.838) |

Pt_{0.75}Ni_{0.25} | L1_{2} | 0.168 (0.161) | 0.749 (0.712) |

Pt_{0.50}Ni_{0.50} | L1_{0} | 0.376 (0.387) | 0.792 (0.762) |

Pt_{0.25}Ni_{0.75} | L1_{2} | 0.360 (0.364) | 0.752 (0.764) |

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**MDPI and ACS Style**

Shuttleworth, I.
The Magnetic Band-Structures of Ordered Pt_{x}Fe_{1−x}, Pt_{x}Co_{1−x}, and Pt_{x}Ni_{1−x} (x = 0.25, 0.50, and 0.75). *Magnetochemistry* **2020**, *6*, 61.
https://doi.org/10.3390/magnetochemistry6040061

**AMA Style**

Shuttleworth I.
The Magnetic Band-Structures of Ordered Pt_{x}Fe_{1−x}, Pt_{x}Co_{1−x}, and Pt_{x}Ni_{1−x} (x = 0.25, 0.50, and 0.75). *Magnetochemistry*. 2020; 6(4):61.
https://doi.org/10.3390/magnetochemistry6040061

**Chicago/Turabian Style**

Shuttleworth, Ian.
2020. "The Magnetic Band-Structures of Ordered Pt_{x}Fe_{1−x}, Pt_{x}Co_{1−x}, and Pt_{x}Ni_{1−x} (x = 0.25, 0.50, and 0.75)" *Magnetochemistry* 6, no. 4: 61.
https://doi.org/10.3390/magnetochemistry6040061