## 3 Results and Discussion
### Transmission Characteristics  
Organo-metallic complexes such as metallocene and metal benzene possess a distinctive sandwich structure. These complexes typically exhibit half-metallic properties that can be utilized as spin filters in future nanodevices [42; 45; 46; 47]. Additionally, as with some  
Figure 4: Relative energies of PTMCp SCO complexes between two spin states along reaction coordinates (\(X\)). (**a**) PMnCp: HS-IS, (**b**) PFeCp: IS-HS, (**c**) PFeCp: HS-LS, (**d**) PCoCp: LS-IS, (**e**) PCoCp: LS-HS, (**f**) PCoCp: IS-HS. Initially relaxed configurations of spin states with lower energy at \(X=0\), higher energy at \(X=1\), \(C_{i}\) (\(i\) = LS, IS, HS) represents initial configuration in \(i\). metalloporphyrins [48], these complexes demonstrate remarkable giant magnetoresistance (GMR) effects [47] which offer potential applications in magnetic memory. Given that PTMCp also contains several sandwich structures resembling MtCp in terms of geometry and orbital coupling, we are curious to investigate whether they manifest the GMR effect. To examine this possibility further, we will explore their transport properties under magnetic electrodes. Among them, the ground-state sandwich molecules are PScCp (singlet), PTiCp (doublet), PVCp (triplet), and PFeCp (sextet), which will be the intermediate molecules of the device. To match the possible experimental setup, we take four Ni nanowires of different radii as magnetic electrodes, which are connected to the axial direction of the molecule (as shown in Figure 5). In our model, the magnetization direction of the left electrode is fixed, and the magnetization direction of the right electrode is set to be either parallel (P) or anti-parallel (AP), the molecular spin is going in the same orientation of the left electrode. The transmission spectrum at zero bias voltage is calculated. The conductance (G) and the magnetoresistance (MR) are calculated by \(G_{\sigma}=\frac{c}{\hbar}T_{\sigma}(E_{F}),\ \sigma=\uparrow,\downarrow,\ G=G_{ \uparrow}+G_{\downarrow}\) and \(MR=\frac{G_{P}-G_{AP}}{G_{P}+G_{AP}}\), and the spin filtering efficiency (SFE) is given by \(SFE=\frac{T_{\uparrow}(E_{F})-T_{\downarrow}(E_{F})}{T_{\uparrow}(E_{F})+T_{ \downarrow}(E_{F})}\). The conductance and MR values of the device for P are provided in Table 2. All devices exhibit MR values above 80%, regardless of whether the electrode is a Ni monoatomic wire or 2 A Ni nanowires. Notably, when using the latter, the MR value reaches 100%. Furthermore, the device demonstrates a perfect spin-filtering efficiency (SFE) of 100% with a P-type electrode. These ground-states of sandwich PTMCp complexes possess exceptional giant magnetoresistance (GMR) effects and perfect SFE, making them suitable for applications in magnetic storage and logic devices. However, the GMR effect almost disappears and the SFE decreases greatly while the electrodes are made of 3.5 A and 4.5 A Ni nanowires, although it has been reported that the GMR effect can be regulated by electrode contact [10]. Here, we analyze the transport properties of PVCp with four electrodes, and try to explain the difference of the MR with the transmission eigenstates. Figure 5: PTMCp molecular devices with four kinds of electrodes. (**a**) Ni monoatomic wire; (**b**) Ni nanowires of radius 2 Å; (**c**) Ni nanowires of radius 3.5 Å; (**d**) Ni nanowires of radius 4.5 Å. White sphere: H atom; gray sphere: C atom; green sphere: Ni atom; blue sphere: N atom; gold sphere: Sc/Ti/V/Fe atom. The transmission spectra of PVCp for P and AP and the transmission eigenstates at Fermi level are shown in Figure 6. Interestingly, the generation of SFE seems to have little to do with the metal, as non-magnetic PScCp also exhibits perfect SFE in small-sized electrodes. We consider that this phenomenon stems from the way the molecules of this type of structure are coupled to the electrodes, which seems to allow electrons of one spin to pass and forbid the other.