2 II. MODEL t$_{1}$ = − 2 ( b 2 − a 2 + g $^{2}$) / Δ$_{pd}$ t$_{2}$ = 2 b 2 a 2 − g / Δ$_{pd}$ t$_{3}$ = − ( b 2 + a 2 g $^{2}$) / Δ$_{pd}$ t$_{4}$ = ab − g $^{2}$) / Δ$_{pd}$ , (2) H = − ∑ j,σ,γ,γ ′ ,λ,λ ′ t λ,λ ′ γ,γ ′ ( d † $_{j,γσ,λ}$d$_{j}$$_{+1}$$_{,γ}$ ′ σ,λ ′ + H . c . ) − ∑ j,σ,γ,γ ′ ˜ t$_{γ,γ}$ ′ ( d † j,γσ, $_{1}$d$_{j,γ}$ ′ σ, 2 + H . c . ) + U ∑ j,γ,λ ρ$_{j,γ}$$_{↑}$$_{,λ}$ρ$_{j,γ}$$_{↓}$$_{,λ}$ + ( U ′ − J/ 2) ∑ j,λ ρ$_{j,x,λ}$ρ$_{j,y,λ}$ − 2 J ∑ j,λ S$_{j,x,λ}$ · S$_{j,y,λ}$ + J ∑ j,λ ( d † j,x ↑ $_{,λ}$d † j,x ↓ $_{,λ}$d$_{j,y}$$_{↓}$$_{,λ}$ d$_{j,y}$$_{↑}$$_{,λ}$ + H.c ) , (1) In these studies, we have considered the following Hamiltonian defined on a two-leg ladder geometry two-dimensional system can in principle be obtained by considering the limits of both N and L sent to infinity, although in practice this is difficult to do. This ladder- based procedure has been used to investigate models for the high temperature superconductors 15 and for the HF systems. 4,16 Some important results were obtained with this method. For example, research based on microscopic models for the high T$_{c}$ superconductors, 5 as well as re- search on HF models, 4 indicate that superconductivity mediated by antiferromagnetic fluctuations can be stabi- lized, in agreement with several experiments. Thus, the use of ladders appears to be an important ingredient to unveil dominant ground state tendencies. Moreover, the hopping amplitudes that will be used in our investiga- tions below include next-nearest-neighbor diagonal hop- pings that are only possible when plaquettes exist in the lattice under consideration. Note that microscopic models that may present su- perconductivity induced by antiferromagnetism, such as the one-orbital Hubbard model and the Kondo Lattice model, have been extensively studied by several au- thors. However, microscopic models for superconductiv- ity in a ferromagnetic spin background have been much less explored, with the exception of studies using one- dimensional chains. 11 This may be caused in part by the perception that superconductivity and ferromagnetism, as opposed to antiferromagnetism, can not coexist. 17 However, this perception has been challenged by the dis- covery of superconductivity and FM in the HF com- pounds UGe$_{2}$ 10 and URhGe. 18 Moreover, SC and FM were also observed 19 in the d -band metal ZrZn$_{2}$. where d j,γσ,λ creates an electron with spin projection σ in the orbital γ = x, y ( d$_{xz}$ and d$_{yz}$ , respectively) at the rung j and leg λ = 1 , 2, S$_{j,γ,λ}$ is the electron spin density operator, ρ$_{j,γσ,λ}$ = d † $_{j,γσ,λ}$d$_{j,γσ,λ}$ , and ρ$_{j,γ,λ}$ = ∑ σ d † $_{j,γσ,λ}$d$_{j,γσ,λ}$ . The hopping amplitudes are: t λ,λ x,x = ˜ t$_{y,y}$ = − t$_{1}$ , t λ,λ y,y = ˜ t$_{x,x}$ = − t$_{2}$ , t 1 , 2 γ,γ = t 2 , 1 γ,γ = − t$_{3}$ , t 1 , 2 γ,γ ′ = − t 2 , 1 γ,γ ′ = − t$_{4}$ , if γ = γ $^{′}$, and zero otherwise. To avoid a proliferation of ex- tra parameters in our analysis, we have decided to fix the values of these hoppings from considerations previously used in the pnictide context. Our use of models originally devised for pnictides is simply based on the pragmatic observation that some pairing tendencies and ferromag- netic regions at large J were already observed in those models. 12,13 It should be clear though, that our research is mainly motivated by heavy fermion phenomenology. Following this strategy, then the hopping amplitudes t$_{1}$, t$_{2}$ , t$_{3}$ , and t$_{4}$ are obtained using the Slater-Koster tight-binding scheme, and they are given by 12,13,25 Motivated by the discovery of superconductivity in the HF compound UGe$_{2}$, a few years ago Karchev et al. pro- posed a one-band model to study the coexistence of su- perconductivity and ferromagnetism. 20 However, other researchers have argued that the treatment used to inves- tigate that model, as well as the model itself, were not ap- propriate to describe the coexistence of these phases. 21–23 Due to the lack of studies of microscopic models for su- perconductivity mediated by ferromagnetic fluctuations beyond one-dimensional chains, 11 in this work we have decided to investigate a microscopic ladder model where the coexistence of superconductivity and ferromagnetism appears possible. In fact, it will be shown below that the two-orbital model that has been originally proposed to describe the low-energy physics of the iron-based superconductors 12,13,24 actually leads to the coexistence of pairing and spin ferromagnetic tendencies. Qualita- tive our results are compatible with those reported using chains. 11 Although the model considered here may not be a proper effective model for superconducting ferromag- nets such as UGe$_{2}$, we believe that the mechanism that bounds together the charges carriers (see section III.B) is so simple and generic that our calculations may also apply in a variety of other models as well. where the Fe-As hopping amplitudes are a = 0 . 324( pdσ ) − 0 . 374( pdπ ), b = 0 . 324( pdσ ) + 0 . 123( pdπ ), and g = 0 . 263( pdσ ) + 0 . 31( pdπ ). Δ$_{pd}$ is the energy difference between the p and d levels. We have set ( pdσ ) $^{2}$/ Δ$_{pd}$ = 1 to fix the energy scale, and we use pdπ/pdσ = − 0 . 2, as previously discussed. 12,13 Regard- ing the couplings U , J , and J $^{′}$, note that they are not independent, but they are assumed to satisfy the rela- tion U = U ′ + 2 J , which is strictly valid within a cubic environment for the full t$_{2g}$ sector. 26,27 For an explicit derivation of this relation in the case of manganites see Ref. 28.