^{1}

^{*}

^{2}

^{1}

^{3}

^{1}

In introduction we presented a short historical survey of the discovery of superconductivity (SC) up to the Fe-based materials that are not superconducting in a pure state. For this type of material, the transition to SC state occurs in presence of different dopants. Recently in the Fe-based materials at high pressures, the SC was obtained at room critical temperature. In this paper, we present the results of calculations of the isolated cluster representing infinitum crystal with Rh and Pd as dopants. All calculations are performed with the suite of programs Gaussian 16. The obtained results are compared with our previous results obtained for embedded cluster using Gaussian 09. In the case of embedded cluster our methodology of the Embedded Cluster Method at the MP2 electron correlation level was applied. In the NBO population analysis two main features are revealed: the independence of charge density transfer from the spin density transfer and, the presence of orbitals with electron density but without spin density. This is similar to the Anderson’s spinless holon and confirms our conclusions in previous publications that the possible mechanism for superconductivity can be the RVB mechanism proposed by Anderson for high T
_{c} superconductivity in cuprates.

In 1911 Kamerlingh Onnes [_{c} = 4.19 K), while he was doing his experiments on the resistivity of gold and mercury wires at low temperature. At that time, Kamerlingh was the only one who could reach very low temperatures because he was the first who obtained the liquid helium. At that moment, a new state of the matter was discovered, the SC state. For explaining the mechanism of SC, it was required the creation of quantum mechanics (1925), see [

The BCS theory was based on creation of Cooper pairs (pair of electrons that attract each other, instead of repelling, through the interaction with the lattice vibrations). Later on, Gor’kov [

For many years the critical temperature T_{c} was low, the maximum critical temperature was obtained for Nb3Sn, T_{c} = 18.5 K. In 1986 Bednorz and Müller [_{c} (~30 K) SC in the cuprates family. For YBa_{2}Cu_{3}O_{7} the lowest T_{c} = 65 K [_{c}.

One of the long-standing challenges was the observation of room-temperature SC. For many years numerous laboratories failed to increase T_{c}. The progress arises after Drozdov et al. [_{c} = 203 K. At last, in 2020 Snider et al. [_{c} = 287.7 K (15˚C) for a photochemically transformed carbonaceous sulfur hydride. They used the diamond anvil cell with a palladium thin film that assisted the synthesis by protecting the sputtered yttrium from oxidation and promoting subsequent hydrogenation. These types of materials are characterized by high frequencies vibration that increases the electron-phonon coupling, which is needed for high T_{c} phonon mediated SC, that is, for conventional SC.

The discovery in 2008 by Hosono and coworkers [_{(1-x)} F_{x}]FeAs with (x = 0.05 − 0.12) represented the rise of a new era with the family of high-T_{c} Fe-based superconductors (Fe-SC), which are also named as iron-based superconductors (IBSC). This family is composed by six groups of IBSC compounds [_{2}As_{2}, are widely used [

IBSC materials where intensively studied by theorists, see [_{c} SC, although the latter has rather different mechanism of SC. We recommend the readers the popular and comprehensive reviews by Norman [

From the first year of the discovery of the IBSC, it has been accepted that the superconductivity in these materials is non-conventional, presenting an anti-ferromagnetic (AFM) order. As was proposed by Mazin et al. [_{±}. At the same time, Kuroki et al. [_{±}-wave symmetry [

The parent compound in IBSC can be considered as some kind of Mott insulator [_{±} pairing [

The RVB theory [_{c} superconductors was proposed after the discovery of the cuprates. In this theory the antiferromagnetic lattice is melted into a spin-liquid phase composed by singlet pairs. When doping is applied, the singlets become charged giving rise to the superconducting state. This theory takes into account the separation between spin and charge, then the electronic excitation spectra can be presented as two different branches: charged spinless holons and chargeless spinons [

In our previous publications devoted to IBSC [_{4}Fe_{5}As_{8} cluster and doped with substitutions of Fe atom by two pairs of dopants Co, Ni and Rh, Pd. The Embedded Cluster Method at the Möller-Plesset second order electron correlation level (ECM-MP2) [

In this article we study the electronic structure of the isolated Ba4Fe5As8 cluster doped by Rh and Pd using unrestricted Möller-Plesset second order (MP2) method. The presented results obtained by the GAUSSIAN 2016 A.03 [

The embedded cluster method at the Möller-Plesset second order electron correlation level (ECM-MP2) was used. The ECM-MP2 methodology includes two stages. At the first stage, the cluster representing the crystal is selected and the quantum-mechanical MP2 calculations are performed with the unrestricted Hartree-Fock (UHF) method, as the zero-order approximation. A detailed description of MP2 is given in Appendix 3 of book [

The complete structural information is taken from [

At the second stage, the cluster is embedded in a background charges that reproduce the Madelung potential for the infinite crystal. Two conditions must be fulfilled: 1) the symmetry of the crystal must be preserved; 2) the cluster with the background charges must be neutral. The background charges are taken from our previous studies [

The calculations are performed with the Gaussian 2016 A.03 suite of programs [

In _{4}Fe_{5}As_{8} calculated by GAUSSIAN 2016 A.03 [^{2} operator, S^{2} = S(S + 1), which is used for checking the spin contamination of the state and the corrected spin contamination values are given in parenthesis. In our non-relativistic quantum-mechanic calculations, the operator S^{2} commutes with the Hamiltonian that does not depend on the spin, therefore the spin S is a good quantum number.

The presented new results correspond to the isolated cluster. According to

In ^{2} after correction on the spin contamination practically agree with the correct value S(S + 1), except when S = 1⁄2. This indicates that all calculated energies with only one mentioned exception can be accepted as correct.

In

According to

Multiplicity | Embedded Cluster | Isolated Cluster | ||||
---|---|---|---|---|---|---|

Ba_{4}Fe_{5}As_{8} | Energy (a.u.) | S^{2} (ħ^{2}) | Energy (a.u.) | S^{2} (ħ^{2}) | ||

2 | −24,310.14749 | 0.75 (0.75) | −24,292.584548 | 0.75 (0.75) | ||

4 | −24,310.14388 | 3.75 (3.75) | −24,292.564423 | 3.75 (3.77) | ||

6 | −24,310.20111 | 8.75 (8.75) | −24,292.618489 | 8.75 (8.82) | ||

8 | −24,310.10686 | 15.75 (15.75) | −24,292.715159 | 15.75 (15.76) | ||

10 | −24,292.675436 | 24.75 (24.76) | ||||

Ba_{4}Fe_{4}RhAs_{8} | ||||||

1 | −24,141.44594 | 0 (0.00) | −23,140.426635 | 0 (0) | ||

3 | −24,141.42368 | 2 (2.00) | −23,139.851233 | 2 (2.00) | ||

5 | −24,141.30910 | 6 (6.06) | −23,139.941032 | 6 (6.08) | ||

7 | −24,141.27509 | 12 (12.01) | −23,139.917454 | 12 (12.06) | ||

Ba_{4}Fe_{4}PdAs_{8} | ||||||

2 | −23,174.63554 | 0.75 (2.93) | −23,157.175340 | 0.75 (1.07) | ||

4 | −23,174.61090 | 3.75 (3.76) | −23,157.088309 | 3.75 (12.13) | ||

6 | −23,174.58909 | 8.75 (8.76) | −23,157.208473 | 8.75 (8.77) | ||

8 | −23,174.52517 | 15.75 (15.76) | −23,157.117429 | 15.75 (15.87) | ||

Embedded Cluster | Isolated Cluster | |||
---|---|---|---|---|

Atomic Charge | Valence orbital population | Atomic Charge | Valence orbital population | |

Ba_{4}Fe_{5}As_{8} | S = 5 / 2 | 7 / 2 | ||

Fe | 0.08 | 4 s 0.4 3 d 7.45 | 0.73 | 4 s 0.45 3 d 6.65 |

Fe (n.n.)a | 0.76 | 4 s 0.47 3 d 6.64 | 0.78 | 4 s 0.50 3 d 6.57 |

Fe (n.n.)b | 0.49 | 4 s 1.03 3 d 6.36 | 0.73 | 4 s 0.78 3 d 6.39 |

As (n.n.) | −1.49 | 4 s 1.82 4 p 4.53 | −1.50 | 4 s 1.81 4 p 4.51 |

Ba_{4}Fe_{4}RhAs_{8} | S = 1 | S = 2 | ||

Rh | −2.61 | 5 s 0.55 4 d 9.10 5 p 1.72 | −2.77 | 5 s 0.50 4 d 9.00 5 p 2.03 |

Fe (n.n.)a | 0.68 | 4 s 0.45 3 d 6.74 | 0.83 | 4 s 0.55 3 d 6.49 |

Fe (n.n.)b | 0.18 | 4 s 1.38 3 d 6.32 | 0.77 | 4 s 0.81 3 d 6.44 |

As (n.n.) | −0.64 | 4 s 1.68 4 p 3.83 | −0.79 | 4 s 1.65 4 p 4.02 |

Ba_{4}Fe_{4}PdAs_{8} | S = 3 / 2 | S = 5 / 2 | ||

Pd | −1.74 | 5 s 0.46 4 d 9.20 5 p 1.85 | −2.17 | 5 s 0.49 4 d 9.29 5 p 2.12 |

Fe (n.n.)a | 0.84 | 4 s 0.45 3 d 6.59 | 0.58 | 4 s 0.42 3 d 6.88 |

Fe (n.n.)b | 0.24 | 4 s 0.98 3 d 6.70 | 0.81 | 4 s 0.81 3 d 6.30 |

As (n.n.) | −0.9 | 4 s 1.65 4 p 4.10 | −0.94 | 4 s 1.65 4 p 4.15 |

Embedded Cluster | Isolated Cluster | |
---|---|---|

Detailed charge orbital population for 3d (Fe), 4d (Rh, Pd), 5p (Rh, Pd) and 4p (As) | Detailed charge orbital population for 3d (Fe), 4d (Rh, Pd), 5p (Rh, Pd) and 4p (As) | |

Ba_{4}Fe_{5}As_{8} | S = 5 / 2 | S = 7 / 2 |

Fe | d x y 1.72 + d x z 0.66 + d y z 1.14 + d x 2 − y 2 1.96 + d z 2 1.96 | d x y 1.92 + d x z 1.91 + d y z 0.59 + d x 2 − y 2 0.65 + d z 2 1.58 |

Fe (n.n.)a | d x y 1.28 + d x z 1.09 + d y z 0.64 + d x 2 − y 2 1.81 + d z 2 1.82 | d x y 1.18 + d x z 1.03 + d y z 0.80 + d x 2 − y 2 1.64 + d z 2 1.91 |

Fe (n.n.)b | d x y 0.65 + d x z 0.88 + d y z 1.45 + d x 2 − y 2 1.78 + d z 2 1.55 | d x y 1.60 + d x z 1.75 + d y z 0.66 + d x 2 − y 2 0.92 + d z 2 1.44 |

As (n.n.) | p x 1.52 + p y 1.57 + p z 1.44 | p x 1.44 + p y 1.56 + p z 1.50 |

Ba_{4}Fe_{4}RhAs_{8} | S = 1 | S = 2 |

Rh | d x y 1.56 + d x z 1.81 + d y z 2.17 + d x 2 − y 2 1.93 + d z 2 1.63 p x 0.51 + p y 0.61 + p z 0.61 | d x y 1.79 + d x z 1.90 + d y z 1.56 + d x 2 − y 2 1.94 + d z 2 1.80 p x 0.60 + p y 0.72 + p z 0.70 |

Fe (n.n.)a | d x y 1.74 + d x z 0.69 + d y z 0.77 + d x 2 − y 2 1.73 + d z 2 1.81 | d x y 1.45 + d x z 0.74 + d y z 0.72 + d x 2 − y 2 1.65 + d z 2 1.94 |

Fe (n.n.)b | d x y 1.52 + d x z 1.78 + d y z 0.6 + d x 2 − y 2 1.05 + d z 2 1.48 | d x y 1.80 + d x z 1.95 + d y z 1.95 + d x 2 − y 2 0.25 + d z 2 0.39 |

As (n.n.) | p x 1.11 + p y 1.49 + p z 1.23 | p x 1.28 + p y 1.28 + p z 1.46 |

Ba_{4}Fe_{4}PdAs_{8} | S = 3 / 2 | S = 5 / 2 |

Pd | d x y 1.89 + d x z 1.69 + d y z 1.74 + d x 2 − y 2 1.96 + d z 2 1.91 p x 0.58 + p y 0.6 + p z 0.67 | d x y 1.87 + d x z 1.83 + d y z 1.74 + d x 2 − y 2 1.96 + d z 2 1.88 p x 0.65 + p y 0.74 + p z 0.73 |

Fe (n.n.)a | d x y 1.9 + d x z 0.48 + d y z 1.02 + d x 2 − y 2 1.36 + d z 2 1.82 | d x y 1.63 + d x z 0.83 + d y z 0.68 + d x 2 − y 2 1.82 + d z 2 1.91 |

Fe (n.n.)b | d x y 0.56 + d x z 1.92 + d y z 1.91 + d x 2 − y 2 1.96 + d z 2 0.35 | d x y 1.78 + d x z 1.95 + d y z 1.95 + d x 2 − y 2 0.27 + d z 2 0.34 |

As (n.n.) | p x 1.24 + p y 1.39 + p z 1.47 | p x 1.42 + p y 1.37 + p z 1.36 |

It is instructive to compare the obtained valence orbital population for the embedded and isolated pure cluster with the valence orbital population of free atoms: Fe: [Ar] 3d^{6}4s^{2} and As: [Ar] 4s^{2}4p^{3}. According to

Let us return to ^{8}5s^{1} and Pd: [Kr] 4d^{10}, as follows from

In

As follows from _{xy} and d_{yz}, whereas for Pd doping there is the zero-spin density on all orbitals. We would like to mention that for Pd doping, the spin density population does not depend on direction of Fe(n.n.). Also, for Rh doping there is spin density population for the Fe(n.n.)a on the d_{xy} and d_{xz} where α and β-spin density populations are observed.

The spin distribution obtained in _{xz} and d_{yz} of Fe(n.n.)a, and d_{yz} and d x 2 − y 2 of Fe(n.n.)b are practically occupied by one electron with zero spin population. For the Pd doping, on the orbital d_{yz} of Fe(n.n.)a there is also one electron with zero spin population. The spinless electron resembles the spinless holons proposed by Anderson in his RVB model of high T_{c}-SC [

Embedded Cluster | Isolated Cluster | |||
---|---|---|---|---|

Spin (ħ) | Valence orbital spin population | Spin (ħ) | Valence orbital spin population | |

Ba_{4}Fe_{5}As_{8} | S = 5 / 2 | S = 7 / 2 | ||

Fe | 0.32 | 4 s − 0.01 3 d 0.33 | −0.03 | 4 s 0 3 d − 0.04 |

Fe (n.n.)a | 0.07 | 4 s 0.01 3 d 0.06 | 0.08 | 4 s − 0.02 3 d 0.08 |

Fe (n.n.)b | 0.65 | 4 s 0.59 3 d 0.06 | −0.01 | 4 s 0 3 d − 0.03 |

As (n.n.) | 0.23 | 4 s 0.02 4 p 0.21 | −0.02 | 4 s 0.01 4 p − 0.04 |

Ba_{4}Fe_{4}RhAs_{8} | S = 1 | S = 2 | ||

Rh | −0.05 | 5 s 0 4 d − 0.05 5 p 0 | −0.51 | 5 s 0 4 d − 0.84 5 p 0.29 |

Fe (n.n.)a | 0.03 | 4 s 0.01 3 d 0.02 | 0.06 | 4 s 0.11 3 d − 0.03 |

Fe (n.n.)b | 0.04 | 4 s − 0.02 3 d 0.02 | 0.09 | 4 s 0.02 3 d − 0.02 |

As (n.n.) | 0.02 | 4 s 0 4 p 0.02 | 0.33 | 4 s 0 4 p − 0.31 |

Ba_{4}Fe_{4}PdAs_{8} | S = 3 / 2 | S = 5 / 2 | ||

Pd | −0.04 | 5 s 0.02 4 d − 0.08 5 p 0.02 | 0.07 | 5 s 0.01 4 d 0.04 5 p 0.03 |

Fe (n.n.)a | −0.01 | 4 s 0.01 3 d − 0.02 | −0.37 | 4 s − 0.02 3 d − 0.34 |

Fe (n.n.)b | 0.96 | 4 s 0.33 3 d 0.63 | 0.05 | 4 s 0 3 d 0.02 |

As (n.n.) | 0 | 4 s 0 4 p 0 | −0.07 | 4 s 0.01 4 p − 0.05 |

Embedded Cluster | Isolated Cluster | |
---|---|---|

Detailed spin orbital population for 3d (Fe), 4d (Rh, Pd), 5p (Rh, Pd) and 4p (As) | Detailed spin orbital population for 3d (Fe), 4d (Rh, Pd), 5p (Rh, Pd) and 4p (As) | |

Ba_{4}Fe_{5}As_{8} | S = 5 / 2 | S = 7 / 2 |

Fe | d x y − 0.62 + d x z 0.12 + d y z 0.82 + d x 2 − y 2 0 + d z 2 0.01 | d x y − 0.01 + d x z − 0.01 + d y z − 0.01 + d x 2 − y 2 0.01 + d z 2 − 0.02 |

Fe (n.n.)a | d x y 0.1 + d x z − 0.07 + d y z 0.02 + d x 2 − y 2 0.01 + d z 2 0 | d x y − 0.01 + d x z 0.13 + d y z − 0.05 + d x 2 − y 2 − 0.03 + d z 2 0.04 |

Fe (n.n.)b | d x y 0.18 + d x z 0.06 + d y z − 0.07 + d x 2 − y 2 0 + d z 2 − 0.11 | d x y − 0.03 + d x z 00.2 + d y z 0 + d x 2 − y 2 − 0.01 + d z 2 − 0.01 |

As (n.n.) | p x 0.07 + p y 0 + p z 0.14 | p x − 0.04 + p y − 0.04 + p z 0 |

Ba_{4}Fe_{4}RhAs_{8} | S = 1 | S = 2 |

Rh | d x y − 0.01 + d x z 0.01 + d y z − 0.06 + d x 2 − y 2 0 + d z 2 0.01 p x 0 + p y 0 + p z 0 | d x y − 0.01 + d x z − 0.36 + d y z − 0.46 + d x 2 − y 2 0 + d z 2 − 0.01 p x 0.04 + p y 0.25 + p z 0 |

Fe (n.n.)a | d x y 0 + d x z 0.01 + d y z 0.04 + d x 2 − y 2 − 0.02 + d z 2 − 0.01 | d x y 0.38 + d x z − 0.38 + d y z 0.01 + d x 2 − y 2 − 0.09 + d z 2 0.05 |

Fe (n.n.)b | d x y − 0.02 + d x z 0 + d y z 0.01 + d x 2 − y 2 0.03 + d z 2 0 | d x y 0.01 + d x z 0.01 + d y z − 0.01 + d x 2 − y 2 − 0.03 + d z 2 0 |

As (n.n.) | p x 0.01 + p y 0.01 + p z 0 | p x 0.11 + p y − 0.21 + p z − 0.21 |

Ba_{4}Fe_{4}PdAs_{8} | S = 3 / 2 | S = 5 / 2 |

Pd | d x y − 0.01 + d x z − 0.04 + d y z − 0.02 + d x 2 − y 2 0 + d z 2 − 0.01 p x 0.01 + p y 0 + p z 0.03 | d x y − 0.01 + d x z 0.05 + d y z 0 + d x 2 − y 2 0 + d z 2 0 p x − 0.05 + p y 0.08 + p z 0 |

Fe (n.n.)a | d x y − 0.02 + d x z 0.01 + d y z − 0.03 + d x 2 − y 2 0.02 + d z 2 0 | d x y − 0.07 + d x z − 0.15 + d y z − 0.16 + d x 2 − y 2 0.03 + d z 2 0.01 |

Fe (n.n.)b | d x y 0.47 + d x z 0.01 + d y z 0.01 + d x 2 − y 2 − 0.01 + d z 2 0.15 | d x y 0 + d x z 0.01 + d y z 0.01 + d x 2 − y 2 0 + d z 2 0 |

As (n.n.) | p x 0 + p y 0 + p z 0 | p x − 0.01 + p y − 0.02 + p z − 0.02 |

As follows from the discussion of our calculations by unrestricted open shell ECM-MP2, the ground state for the isolated cluster is characterized by a different multiplicity than the ground state of the embedded cluster. The background charges modify the energy of the cluster and the valence orbital population. It is also revealed that the calculation by unrestricted open shell MP2 method leads in some cases to high spin contamination of the state.

For the isolated cluster doped by Rh and Pd, we obtained a decrease in population of some valence orbitals. The orbital population for Fe(n.n.) depends on direction, this is in agreement with experiments. For the doped isolated cluster, a charge transfer from the As(n.n.) atoms to the central atom was observed, as in the case of the embedded cluster. Thus, for the embedded and isolated clusters for Rh and Pd doping, the charge transfers from nearest neighbor atoms to the dopants, whereas only for the isolated cluster doped by Rh we obtained spin transfer.

It is important to mention that for both dopants, the spin disappears on the dopants and the charge and spin transfer are completely independent. Thus, obtained in our calculations charge and spin orbital distributions, are in agreement with the spinless electrons proposed by Anderson (Anderson’s holon). This indicates the possibility of the superconductivity mechanism in this material proposed by Anderson in his RVB theory.

The authors thank the DGTIC computer staff for providing access to the MITZLI cluster of Universidad Nacional Autónoma de México. This work was partly supported by grants from DGAPA PAPIT IN111519. We also gratitude Lic. Alejandro Pompa-García and Tec. Cain González for their technical support.

The authors declare no conflicts of interest regarding the publication of this paper.

Columbié-Leyva, R., Miranda, U., López-Vivas, A., Soullard, J. and Kaplan, I.G. (2021) Quantum Mechanical Calculations of High-Tc Fe-Superconductors. Journal of Quantum Information Science, 11, 84-98. https://doi.org/10.4236/jqis.2021.112007