Band Structure of Organic-Ion-Intercalated (EMIM)xFeSe Superconductor

The band structure and the Fermi surface of the recently discovered superconductor (EMIM)xFeSe are studied within the density functional theory in the generalized gradient approximation. We show that the bands near the Fermi level are formed primarily by Fe-d orbitals. Although there is no direct contribution of EMIM orbitals to the near-Fermi level states, the presence of organic cations leads to a shift of the chemical potential. It results in the appearance of small electron pockets in the quasi-two-dimensional Fermi surface of (EMIM)xFeSe.


Introduction
Metal-organic compounds are a recent trend in functional materials design because of the combination of molecule flexibility in creating the framework and conducting, semiconducting, and topological features of the metal ions subsystem. For example, intensively studied tetraoxa [8]circulene [1][2][3] with integrated Li or Na ions are suggested to be a conductor and even a superconductor with the Ca ions [4]. Pure organic compounds are usually the low-temperature superconductors with the critical temperature T c of the order of 10 K, whereas metal-organic compounds demonstrate the higher T c s. For example, potassium-doped p-terphenyl exhibits a superconducting transition temperature in the range from 7 to 123 K, depending on the doping level [5][6][7][8]. Alternatively, one can use organic molecules as electron donors and structure stabilizers to control the features of a metallic system. It was initially suggested that the protonation via the ionic-liquid-gating method [9] makes it possible to increase T c in iron selenide [10] due to the H y -FeSe 0.93 S 0.07 formation. The later study [11] uncovered the formation of an organic ion-intercalated phase (EMIM) x FeSe, where EMIM stands for C 6 H 11 N + 2 (see its structure in Figure 1). The EMIM cations were inserted into FeSe during the electrochemical process in the electrolytic cell with two platinum electrodes and EMIM-BF 4 as an ionic liquid. FeSe was placed on the cathode (negatively charged electrode) where the redox reaction takes place using the electrons transferred through an external circuit from the anode. It is not clear yet which of the chemical species gain electrons. Discovered superconductivity with T c about 44 K in this material brings up questions on the mechanism of Cooper pairing and on the role of EMIM molecules. Iron selenide belongs to a broad family of iron-based superconductors [12][13][14][15][16][17][18][19][20][21] that also includes FeSe monolayer with T c above 80 K [22][23][24][25][26][27][28][29].
To make a first step towards understanding the nature of superconductivity in (EMIM) x FeSe, here we calculate its band structure and Fermi surface using density functional theory (DFT). To place the EMIM molecules together with the FeSe lattice in the crystal structure, we construct a supercell corresponding to (EMIM) 2 Fe 18 Se 18 . Bands near the Fermi level originate from the Fe-d orbitals and orbitals of EMIM contribute at energies from 1 to 1.5 eV above the Fermi level. Although band structures of (EMIM) 2 Fe 18 Se 18 and FeSe with the same crystal structure are similar, their Fermi surfaces are different. In particular, small electron pockets around X-point appears in (EMIM) 2 Fe 18 Se 18 . Therefore, we argue that the main role of (EMIM) 2 is to shift the chemical potential that results in the transformation of the Fermi surface.
It is known that the interaction between C-H bond and the π-system (C-H· · · π interaction) is observed in a large number of organic systems containing π-conjugated organic molecules [49]. According to Ref. [50], the EMIM cations are also linked to each other by C-H· · · π interactions between one methyl carbon and the imidazolium ring of another cation. To provide a similar arrangement of EMIM cations between FeSe layers, we have chosen the 3 × 3 supercell of FeSe with two EMIM cations placed there. The formula for the resulting supercell is (EMIM) 2 Fe 18 Se 18 ; its structure is shown in Figure 2.
The formation energy E f of (EMIM) 2 Fe 18 Se 18 was calculated using the following equation, where E (EMI M) 2 Fe 18 Se 18 is the total energy of a (EMIM) 2 Fe 18 Se 18 system, E Fe 18 Se 18 and E EMI M + are total energies of 3 × 3 supercell of bulk FeSe and EMIM cation, respectively, and ε F is the Fermi energy in the 3 × 3 supercell of a bulk FeSe. Charge density difference induced by the interaction between the FeSe and EMIM cations is calculated as the difference in total charge densities between (EMIM) 2 Fe 18 Se 18 and superpositions of total charge densities of Fe 18 Se 18 and (EMIM) 2 , located at the same positions as in the (EMIM) 2 Fe 18 Se 18 . These calculations were done for the same unit cell and with the same calculation conditions. Mulliken population analysis was used to estimate the charges on the atoms.

Results and Discussion
The unit cell of (EMIM) 2 Fe 18 Se 18 are shown in Figure 2. The optimized lattice parameters are a = b = 11.321 Å, c = 10.450 Å. The length of c parameter is in excellent agreement with the known experimental data [11]. EMIM cations are localized in the space between the nearest Se atoms of the FeSe layers. The smallest distance H-Se is 2.830 Å. Localized EMIM cations make angles of ∼34 • and ∼38 • with the direction of the c-vector. The distance between the methyl carbon and the center of imidazolium ring is equal to 4.083 Å, which is in the range of values for the C-H· · · π interactions [49]. The structural parameters of EMIM cations are presented in Table 1 and are in good agreement with the previous experimental and theoretical results for EMIM halide ionic liquids [51]. The Fe-Se bond lengths are slightly different and range from 2.325 Å to 2.338 Å. These bond lengths are less than those for the unintercalated bulk FeSe [52] and the bond-length difference ranges from −2.6 to −3.1%. To examine the thermodynamic stability of (EMIM) 2 Fe 18 Se 18 the formation energy was calculated with the corresponding value of −0.36 eV per unit cell (−0.02 eV per FeSe formula unit). Negative values confirm that the formation of (EMIM) 2 Fe 18 Se 18 is energetically feasible. Analysis revealed the positive charge on EMIM cations (+0.93ē per one EMIM cation) and the negative charge on FeSe layers (∼−0.10ē per FeSe formula unit). This means that the FeSe snatches electrons during an electrochemical reaction and becomes an anion. The snatched electrons are distributed between the atoms of FeSe so that the number of added electrons on one Fe atom and one Se atom is equal to ∼0.02ē and ∼0.08ē, respectively. The total charge of the system is equal to zero. Thus, the EMIM cations intercalation through the electrochemical process allows one to perform an electronic doping of FeSe. In Figure 3 we show that the electrons snatched by Se atoms are localized closer to EMIM cations (dark teal wireframe areas). This asymmetric electron density distribution results in a redistribution of the electron density in EMIM cations and on iron ions. A small number of electrons move from H atoms located near FeSe layer to C atoms. Further redistribution occurs by a chain mechanism.
The calculated band structure of (EMIM) 2 Fe 18 Se 18 is shown in Figure 4a. Bands cross the Fermi level in Γ − X, M − Γ, Z − R, and A − Z intervals. Lack of dispersion of near-Fermi level bands in Γ − Z, X − R, and M − A directions leads to a quasi-two-dimensional character of the Fermi surface, see Figure 5a. According to Figure 4a,b, latter consists of two hole pockets around Γ-point, two almost degenerate small electron pockets around X-point, and three large Fermi surface sheets in-between these two points. Table 1. Structural parameters of two EMIM cations located in FeSe. The atomic numbering scheme is shown in Figure 1.     The contribution of iron and EMIM orbitals to the band structure is shown in Figure 6.  Figure 7). The PDOS of EMIM equal to 0.047 arb.units. that significantly smaller then PDOS of iron d states. Therefore, there is no direct effect of EMIM on low-energy physics.

Bond Angles in Degrees
To analyze the overall role of organic cations, we compared the band structures of (EMIM) 2 Figure 4a. The same group is above the Fermi level by about 100 meV in the FeSe system without (EMIM) 2 cations, see Figure 4c.
Next, we derived the low-energy effective model from a set of Fe-3d-like Wannier functions. Wannier functions were generated from the Bloch states within the energy window from −3.00 eV to 2.15 eV. The constructed maximally localized Wannier functions are similar to classical d-orbitals and are centered on Fe atoms. Due to the size of the supercell, there are total of 90 Fe d-orbitals involved (five per iron, two irons in a unit cell, nine unit cells in the 3 × 3 supercell). The interpolated bands are shown in Figure 8 together with the original band structure. There is an excellent matching of interpolated and original bands in the vicinity of the Fermi level and in the valence band, which confirms that they are mainly d-like in character. Small mismatches appear above 0.5 eV in the conduction band, which is caused by the contribution of EMIM states. Some bands in the conduction band are absent, due to the small contributions of Fe-d states to them. These bands are predominantly formed by EMIM orbitals. Thus, this effective model of the band structure based on all Fe-d orbitals describes adequately the band structure of (EMIM) 2 Fe 18 Se 18 in the vicinity of the Fermi level.

Conclusions
We have studied the electronic structure of (EMIM) x FeSe using the state-of-art density functional theory in the generalized gradient approximation. In the absence of direct data on the coordinates of EMIM, we show that the reasonable position of it results in the (EMIM) 2 Fe 18 Se 18 unit cell. Fe-d orbitals form the bands near the Fermi level similar to other Fe-based superconductors. Orbitals of EMIM do not affect the low-energy states directly; however, the presence of EMIM leads to the shift of the chemical potential that results in the transformation of the Fermi surface topology and appearance of small electron pockets around X-point in (EMIM) 2 Fe 18 Se 18 in contrast to FeSe with the similar crystal structure.
The appearance of the small electron pockets in EMIM-intercalated FeSe may play a crucial role in the formation of high-T c superconductivity. Spin-fluctuation theory of pairing [20,53,54] predicts the sensitivity of the gap structure to the variation of the sizes of hole and electron pockets [55][56][57][58][59]. Therefore, change of the Fermi surface topology upon EMIM intercalation may be the most important ingredient in the increase of T c reported in ref. [11].
Other important ingredients to the complete theory of superconductivity may come from the strong electronic correlations. Comparison of the LDA+DMFT (local density approximation + dynamical mean-field theory), a hallmark approach to include correlations, and the bare LDA revealed the weak influence of correlation effects on the electronic structure of the FeSe layer [60]. On the other hand, experiments on monolayer FeSe have shown severe changes in the Fermi surface topology compared to DFT results [61]. To explain the discrepancy, strong electronic correlations, nematicity present in (EMIM) x FeSe [62], or something more exotic is needed. Here we made a first step towards making a theory that would include any of the mentioned mechanisms, i.e., we provide a DFT band structure on top of which the exotic mechanism can be built.

Conflicts of Interest:
The authors declare no conflict of interest.