Theoretical study of vertical van der Walls metal-porphyrin and metal free-porphyrin junctions

This paper provides a theoretical study of the thermoelectric properties of a vertical graphene/porphyrins/graphene architecture. It presents the details of calculating the conductance of metal-porphyrins, which is found to be enhanced by manipulating the metal central atom of the organic 125 porphyrin framework over the family Ni,Zn,FeII, and CoII. The results demonstrate that even when there is no direct inter-molecular coupling, indirect inter-molecular interactions mediated by the graphene electrodes produce quantum interference effects in the electronic structure of the molecular junction. These junctions are all observed to be HOMO-dominated, meaning that their Seebeck exhibit the same sign and similar behavior. The resulting single-molecule thermopowers range from almost +50 μV/K for both Zn - and Ni -porphyrin to +77 μV/K and +85 μV/K for Co - and Fe -porphyrin, respectively. For these geometries, the effect of the metal complex with porphyrin on the conductance of the junctions can be seen. An extra resonance appeared in the HOMO-LOMO gap, and it can be disappeared or be shifted closer to Fermi energy to create a new path for the electron transmission. It only depends on the type of metal coordinating at the porphyrin center. Introducing such a new technique for designing high conductance and thermopower opens a door for high thermoelectric performance materials.


Introduction
At a molecular level, molecular-scale devices of single or multiple molecules trapped between two metallic electrodes have been devoted to quantum interference (QI). This is due to their potential to control the charge transfer through the molecular levels of these materials. Nowadays, there is a significant interest in developing and improving organic thermoelectric materials and this is due to their promising properties including, ecofriendly, non-toxic, abundance, and cheap, unlike inorganic materials. Aligning with that direction, there is a specific interest in exploring the room-temperature thermoelectric properties of single-molecule junctions [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. Porphyrin-based molecules are attractive as building blocks for molecular-scale devices because they are highly-conjugated structures with rigid planar geometries, which are chemically stable and form metalprophyrins by coordinating metal ions in the center of their macrocyclic. Recently electrical conductance of porphyrin-based molecules has been subject to many studies [16][17][18][19][20][21][22][23][24][25][26][27][28]. The electrical switching properties of a series of porphyrin molecules with pendant dipoles connected to graphene source and drain were investigated. It is found that the conductance ratio can increase from 100 with one spacer to 200 with four spacers by coupling the dipole of the functionalized porphyrin to an external electric field [24]. Moreover, the thermoelectric properties of metal-porphyrins connected to gold electrodes were studied and it found that by varying the transition metal-center of porphyrin molecule over a range of metallic atoms can be tuned the molecular energy levels relative to the Fermi energy (E F ) of the electrodes leading to tune the thermoelectric properties of metalporphyrin [25], which makes the metal-porphyrin more attractive material for molecular-scale thermoelectric devices. Therefore, a study of their transport properties is expected to reveal how varying the center metal atom can influence the charge transfer distributions and how this effect reflects in their Seebeck coefficients. For Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
further development of such molecular junctions, and in order to find new molecular scale thermoelectric materials, design new thermoelectric devices and focus on the intrinsic parent molecules, I combine the wellconjugated and manipulatable porphyrin molecules with graphene sheets to form Van der Waals heterojunctions. the electric and thermoelectric properties of a set of metal-porphyrin organic derivatives is going to be explored in the current study. As shown in figure 1 below, the porphyrin monomer of interest consists of four pyrrole cores (the inner ring п-system)), which are joined at their carbon atoms by methine bridges (=CH). My aim of this paper is to investigate the effect on the thermoelectric performance of varying the metal atom χ over the series of χ = Ni, Zn, Co, and Fe. This paper presents theoretical work to explore thermoelectrical properties of cross-plane (graphene/ porphyrin/graphene) junctions for potential high-efficiency thermoelectrical devices where the electron is transmitted through the porphyrin plane from graphene electrodes as shown in figure 2(a). The transmission coefficients for electrons with energy E passing through a molecule from the top left electrode (lead 1) to the bottom-right electrode (lead 4) were calculated using density-functional theory (DFT) [29]. The calculation results demonstrate that the electrical and thermoelectrical properties could be tuned by manipulating the metal central atom.
The first stage in the theoretical modeling is to simulate the vertical four-terminal devices (see figures 2(a)-(b)). The optimum geometries of the porphyrin derivatives between graphene electrodes were computed. After obtaining each individual molecule's fully relaxed geometry, the porphyrin derivatives were sandwiched between two parallel graphene electrodes in which the center of the molecule is located at a hollow position relative to the hexagonal lattice of the graphene electrodes as shown in figure 2(b). These two graphene sheets are electronically decoupled, except via the transport path through the porphyrin derivatives from the top to the lower graphene sheet. The structure is assigned periodic boundary conditions in the z and y directions to avoid spurious edge effects. The main aim of this work is to explore these derivatives' electric transport properties and  investigate the possibility of tuning their thermopower by varying the coordinating metal atoms, including Ni, Zn, ( ) Fe II , and ( ) Co II , in their centers. The charge transfer distribution was analyzed in this study. Since 4 out of 5 of the studied porphyrin derivatives include metallic atoms so spin polarisation calculations are required to obtain the correct trend. Table 1 above indicts that all the metal atoms donate a fraction of the electron (except Zn, an electron) to the host porphyrin. This result is expected as the metallic atoms are electron rich.
Furthermore, the number of electrons in the spin-up and spin-down states are equal for Ni and Zn metals indicating that these two metals are not spin-polarized. This is consistent with the transmission coefficient calculation of these two metals, as seen in

Methodology
Spin-polarised DFT calculations were carried out using SIESTA [30] to obtain the theoretical results for the distribution mentioned in table 1. A combination of the quantum transport code Gollum and the density functional theory (DFT) code SIESTA to calculate the probability for electrons injected with an energy E from one electrode to be transmitted through the molecule junction [31]. The mean-field Hamiltonian derived from the converged DFT calculation was combined with Gollum implementation of the non-equilibrium Green's function approach to determine the phase-coherent, elastic scattering features of each system consist of top graphene (source) and bottom graphene (drain) sheets as well as the scattering region [31] (molecule). The transmission coefficient ( ) T E for electrons of energy E spin of 6 = [↑, ↓] (moving from the source to the drain) is determined via the relation [32]:

Results and discussion
The transport properties of 5 porphyrin derivatives were modeled using a combination of density functional theory (DFT) and quantum transport theory. To calculate the electrical transport through molecule/electrode porphyrin derivatives, I modeled the junctions as shown in figure 2(b) above. To have a good understanding of electronic properties, the spin state for each molecular junction was initially examined. The d-orbital of the metallic atoms could be partially filled. Therefore, spin-polarised calculations were carried out to obtain the electron and spin transport properties.  A resonance in the vicinity of the Fermi level is essential in designing high-performance thermoelectric materials as it enhances both the electrical conductance G and Seebeck coefficient S, which leads to a high thermoelectric efficiency (characterized by a high dimensionless thermoelectric figure of merit ZT ). These simulations suggest a new strategy for designing high thermoelectric performance materials by tuning the metal atomic type. High thermoelectric performance materials are made possible by the introduction of a new method for creating high conductance and thermopower materials.
After computing ( ) T E , equations (2) and (3) are used to calculate the electrical conductance and the Seebeck coefficient for each junction. The results of corresponding room-temperature conductance and Seebeck coefficient versus a range of Fermi energy E F relative to the DFT-predicated Fermi energy (E F DFT ) are shown in figures 4(a)-(b). These figures indicate clearly that by varying the metal atom residing in the core of the organic porphyrin framework the conductance and the Seebeck increased to reach the maximum value of all at −3 eV and +85 μV/K in the presence of ( ) Fe II . Figure 4(b) shows that the thermopower of all junctions have the same sign. As discussed in the previous section these junctions exhibit HOMO -dominated transmission coefficients at the DFT Fermi energy and the slope of the all transmission curves are negative. Consequently, as expected the thermopower of all these junctions have similar behavior.

Conclusion
In summary, This paper presents the details of a theoretical investigation of porphyrin-based molecules sandwiched between two graphene electrodes by using density functional theory combined with Green's function scattering techniques. It shows the strategy used to study the behavior of metal -porphyrin over a series of chosen metals. The results predict that when Ni or Zn metals are induced in the center of porphyrin, the spinup state overlaps with the spin-down state. Therefore, these simulations classify as non-spin-polarised junctions. When the metal was replaced by ( ) Fe II or ( ) Co II , two different electron transmission paths have been reported. This suggests that ( ) Fe II -porphyrin and ( ) Co II -porphyrin to be spin-polarised molecules. In addition to that as a result of coordinating ( ) Ni Co II , , and ( ) Fe II metals at the center of porphyrin, new features are observed, where a new extra resonance emerged in the HOMO-LUMO gap. This resonance vanished when replacing the center of porphyrin by Ni or Zn metal. As all these junctions are observed to be HOMO-dominated, they have similar signs of Seebeck. However, the thermoelectric performance can be decreased or increased it depends on the type of metal.

Acknowledgments
I thank Taibah University, Saudi Arabia for the support, and Dr. Ali Ismael for his support and assistance throughout all aspects of this study.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files). Data will be available from 12 June 2023.