Thermoelectric conductivity of monolayer FeCl2 under LDA+U

We investigated the influence of Coulomb repulsion via the LDA+U method on the thermoelectric conductivity of monolayer 1T-FeCl2. As the spin-orbit coupling is included, the state transition from metallic state to insulating state appears as the Coulomb repulsion increases, in good agreement with previous results. This is also followed by reduction of thermoelectric conductivity. Nevertheless, the magnetic moment remains unchanged as the Coulomb parameter increases, thus keeping the magnetism in monolayer FeCl2. This indicates that the Coulomb repulsion reduces the thermoelectric conductivity but keeps the magnetism in FeCl2.


Introduction
One of the hot topics in the renewable energy is to utilize heat both from waste and environment [1,2].The main reason is that production of energy using the heat source is absolutely clean.This heat can be converted to electricity which can replace fossil-based fuel.So, it can avoid the dependence of nonrenewable energy and reduce air pollution drastically.However, it requires sufficient technological equipment and related concepts.
The theoretical prediction can be realized by controlling electron spin in material.By building the device from this material, a conversion from heat to electricity can be run at a low cost.This is due to a required small energy to manipulate electron spin.In general, there are two quantities that parameterize the efficiency of the conversion in thermoelectric materials, i.e., the Seebeck effect and Nernst effect.While the Seebeck effect is widely used in the non-magnetic material, the Nernst effect holds in the magnetic material [3].
Previous authors reported that two-dimensional materials are prominent for the candidate of thermoelectric materials [4].One of potential candidates of thermoelectric material is FeCl2 monolayer, which is a family member of metal dihalides [5][6][7].From the previous studies [8], FeCl2 is more potential than other metal dihalides for the thermoelectric material since it has a high value of anomalous Nernst coefficient.Based on the band structure, FeCl2 is a half metallic material which is very useful for spintronic devices.So, we will use the magnetism to see the thermoelectric conductivity which is proportional to the anomalous Nernst coefficient.
Syariati et al. [8] shows that thermoelectric conductivity has a maximum value near the critical temperature of FeCl2.Based on their results, we extend the discussion by including Coulomb repulsion U in the framework of LDA+U to see the tendency of thermoelectric conductivity.In general, the inclusion of Coulomb repulsion will change the metallic state to the insulating state or increase the band gap in the insulator.In this study, we will discuss the potentiality of FeCl2 as a thermoelectric material under the U by holding spin-orbit coupling.

Method
For the study, a two-dimensional magnetic 1T-FeCl2 was constructed with the optimized lattice parameter of 3.48 Å as obtained by Syariati et al. [8] with the vacuum distance in the z axis of 17.26 Å, as shown in Fig.Since the out of plane ferromagnetic state is more stable than the in plane one, we fixed the magnetic moment of Fe atom as 0 o in the z direction.We afterwards increase the U to investigate the thermoelectric conductivity.Following Syariati et al. [8], we assigned the Fe atom with the orbitals of 2s, 3p, 3d, and 1f within 6.0 Bohr cutoff radius.Meantime, we assigned the Cl atom with the orbitals of 3s, 3p, and 2d within 7.0 Bohr cutoff radius.However, they did not include the Coulomb parameter, so they will not observe the electronic transition.During the calculations, generalized gradient approximation [9] was utilized to describe the exchange-correlation potential of electrons with the k-point mesh of 20 × 20 × 1.To obtain the thermoelectric conductivity, we include two open-source packages, namely, the OpenMX code for calculating the self-consistent calculations on LDA+U approach [10,11] and Wannier90 for calculating the thermoelectric conductivity [12,13].

Results and Discussions
First, to see the impact of U on the electronic structure, we plot the band energies as found in Fig. 2. It is shown that the metallic property initially emerges without U ( = 0 eV).Nevertheless, the property leads to the insulating/semiconducting state as the energy gap emerges as applying U. Notice that this conversion only occurs as the spin-orbit coupling holds, in good agreement with Botana and Norman [14].On the contrary, without the spin-orbit coupling, it will preserve the metallic property as increasing U [15].In addition, increasing U without the spin-orbit coupling can also generate the low-spin ferromagnetic state in a two-dimensional magnetic 1T-MnCl2 [16], another metal dihalide.
Next, we consider the thermoelectric conductivity by running the Wannier90 code [13] by using the constant relaxation time of 10 fs and a 300 × 300 × 1 k-point mesh.This calculation is based on the  Since the Curie temperature of FeCl2 monolayer is about 109 K [6], we calculate the thermoelectric conductivity at  = 100 K in order to keep magnetism.Figure 3 displays the thermoelectric conductivity as a function of chemical potential for some U.It is shown that the highest peak of thermoelectric conductivity occurs at  = 0 eV.The peak of thermoelectric conductivity then reduces and disappears at the large U.As we check the magnetic moment of Fe atom, there is no significant change of magnetic moment, see table 1.This indicates that the large U will vanish the thermoelectric conductivity but preserves the magnetism.Here the positive/negative peak of thermoelectric conductivity is caused by the slope of tensor of anomalous hall conductivity evaluated at a certain chemical potential.Since the unit of thermoelectric conductivity is A/mK, it can be related to the dependence of current on temperature whose peak at a certain chemical potential describes the highest current.This means that increasing U will avoid the flowing current in the material, invaluable for thermoelectric materials.

Conclusions
We have explored the thermoelectric conductivity with respect to Coulomb repulsion U.It is shown that the high Coulomb repulsion will vanish the thermoelectric conductivity.This can be seen by the reduction of the peak of thermoelectric conductivity by increasing U. Furthermore, it is also shown that the metallic state is replaced by the insulating state as U is taken into account.
As we examine the magnetic property, we find that the magnetism still keeps by observing the nearly constant magnetic moment of Fe atom.This deduces that even though the high U avoids the thermoelectric conductivity, but the magnetism is still exploitable.

Figure 1 .
Figure 1.Top view of crystal structure of monolayer FeCl2.
transport which is related to the self-consistent calculation via the Wannier function.Here, the thermoelectric conductivity is an x-y component of the tensor of Mott's formula   Eq. (1),   , , T,   , , and  describe the Boltzmann constant, electron charge, temperature, tensor of anomalous hall conductivity, energy, and chemical potential.

Table 1 .
Calculated thermoelectric conductivity and magnetic moment (M) as the function of U.