Understanding Doping, Vacancy, Lattice Stability, and Superconductivity in KxFe2− ySe2

Metal‐intercalated iron selenides are a class of superconductors that have received much attention but are less understood in comparison with their FeAs‐based counterparts. Here, the controversial issues such as Fe vacancy, the real phase responsible for superconductivity, and lattice stability have been addressed based on first‐principles calculations. New insights into the distinct features in terms of carrier doping have been revealed.

The discovery of superconductivity in A x Fe 2− y Se 2 (A = K, Cs, Rb, Tl/Rb, Tl/K) [1][2][3][4][5][6] triggered another surge of research on ironbased superconductors, which were previously only featured by iron pnictides [7][8][9][10][11][12] and β-FeSe. [ 13 ] The new series of superconductors can be regarded as the formation from the intercalation of metals between the FeSe layers of β-FeSe. In comparison to the iron pnictides, metal-intercalated iron selenides are much more complicated in terms of structure, chemical composition, and phases. The nature of the superconducting (SC) phase, for example, is still in debate though considerable progress has been made over the last few years. [14][15][16] One of the most controversial issues is whether the SC phase is Fe vacancy free or not, i.e., the FeSe layers in A x Fe 2− y Se 2 are stoichiometric ( y = 0), or off-stoichiometric due to the existence of considerable Fe vacancies. The origin of this issue is largely due to the phase separation that inevitably occurs in these systems at 500-578 K, [ 17 ] leading to the coexistence of the insulating phase (A 2 Fe 4 Se 5 ) and the SC phase. [18][19][20][21][22][23] The thus-obtained SC phase is not standing freely; instead, it intergrows with the insulating phase in the form of nanostrip and its volume fraction is quite low, 10%-20%, as estimated by various measurements. [24][25][26][27][28][29] This is the main obstacle that prohibits the precise determination of the structure and the composition of the SC phase. Ying et al. [ 30,31 ] verifi ed that the SC phases in the K-intercalated iron selenides are almost no Fe vacancy in the FeSe layers based on their study of the superconductors obtained via a liquid ammonia route. The neutron diffraction some light on understanding this distinct SC family and should also be applicable to other metal-intercalated iron selenides besides K x Fe 2− y Se 2 .
To begin with, we study the formation energy for the process of K intercalation. The process can be expressed as a chemical reaction The formation energy per unit cell can be described as Fe 2 Se 2 and E K 0 Fe 2 Se 2 are the total energies of K x Fe 2 Se 2 and an assumed K 0 Fe 2 Se 2 with a similar bct structure but without any K ion, and E K is the energy of elemental K. The variation of Δ E I versus K content is shown in Figure 1 , which indicates that the intercalation of K into FeSe layers is always energetically favored for the bct structure.
Then we break down Δ E I in terms of energy from the structural constituent units and their interactions. For the K intercalation process, as shown in Figure 1 a, K atoms lose their valence electrons and form K ion layers between adjacent FeSe layers after K entering into the lattice, while FeSe layers are charged and deformed and eventually lead to K x Fe 2 Se 2 with a bct structure. Hence, the following contributions to the total formation energy are considered: (1) The formation of K ion layers: Δ E K ion layers = E (K ion layers) x + − x E K (2) The electron doping in FeSe layers: Δ E e-doping = E (FeSe layers) x− − E FeSe layers (3) The deformation of FeSe layers: Δ E FeSe layer deformation = E FeSe layers − E K0Fe2Se2 (4) The Coulomb attraction between K ion layers and FeSe layers: And E (K ion layers) x + is the energy of K ion layers, E (FeSe layers) x − the energy of charged FeSe layers, and E FeSe layers the energy of FeSe layers in K x Fe 2 Se 2 . Therefore, the total formation energy can be written as the sum of these four energy changes: The variations of these four energy contributions as a function of the K content are calculated and the results are shown in Figure 1 b, respectively. First of all, FeSe layers are prominent in contributing more and more positive energy to Δ E I when they are negatively charged with more and more electrons. This is easily understood since charging a neutral FeSe sheet will incur additional energy like charging a capacitor. Inversely, with the increasing K content, more charges will come into effect in the Coulomb attraction between K ion layers and the charged FeSe layers and thus greatly enhance |Δ E C |, which will suffi ciently offset the increase of energy induced by electron doping into the FeSe layer. In contrast with these two contributions, the other two energy contributions due to the formation of K ion layers and the deformation of FeSe layers are much smaller. Therefore, we conclude that the former two contributions dominate the energy change during the K intercalation.
To understand the tendency for appearance of Fe vacancy in the K-intercalated iron selenides, we go further to consider the energy change Δ E Fe vacancy ( is the formation energy per unit cell in the Fe defi ciency structure and µ Fe is the Fe chemical potential.) by removal of an Fe atom from the unit lattice of K x Fe 2 Se 2 and its dependence on the K content. It is found that Δ E Fe vacancy only fl uctuates above and below zero when x ≤ 0.6 as shown in Figure 2 a. When x > 0.6, Δ E Fe vacancy rapidly drops, revealing that Fe vacancies are favored at high levels of K intercalation. This is easily understood since K has a smaller electronegativity than Fe, when the doped electrons due to K are over a limit that Fe Se bond can accommodate, the surplus electrons will prefer to transfer into Fe 2+ . In this way, Fe is repelled out of the lattice in the form of an element. The fi nal compound with Fe vacancy is more energetically a favored state. The limit is around at x = 0.6. The K 2 Fe 4 Se 5 phase contains 20% Fe vacancy corresponding to x = 0.8 is a manifestation of this argument. This tendency of appearance of Fe vacancy with K content is supported by the experimental results. [30][31][32] Besides the above considerations on the formation energy, the infl uence of lattice dynamics on the structural stability should also be accounted for. This is, in particular, true for the compounds with low levels of intercalated K, which can induce lattice instability. Figure 2 b shows the density of phonon states (DOPS) of K 0.2 Fe 2 Se 2 , where negative frequencies appear, meaning the structure is unstable. Partial DOPS further indicate that it is mainly because of the considerable amount of highly unstable Se atoms, which are unbonded due to the  Now we are focused on the trend of structural change upon K intercalation in the range of 0.25 ≤ x ≤ 0.6. Both electronic and size effects of K will be taken into account. In order to better understand the role of electron doping in the structural change, we charge K 0 Fe 2 Se 2 with various electron concentrations, which allows us to explore the effect of electron doping while excluding the size infl uence of K. As shown in Figure 3 a,b, the changes in the lattice constant a , the Fe Se bond, and the Se Fe Se angle clearly indicate that FeSe layers are stretched along the ab plane upon electron doping alone. (Note the results of the highly electron-doped are extrapolated from the trend of the positively charged ones simply because the stable negatively charged K 0 Fe 2 Se 2 cannot always be obtained during the iterative calculations.) For a more realistic case, other factors such as the lattice mismatch and the Coulomb attraction must be included. Figure 3 c shows the lattice parameter variations with K intercalation, which accounts for all these contributions together with the electron doping. We see that the overall effect of K intercalation increases the lattice constant a , stretches FeSe layers, and reduces the lattice constant c . For instance, the lattice constant a of K 0.25 Fe 2 Se 2 and K 0.5 Fe 2 Se 2 expands from 3.65 Å to 3.69 Å and c shrinks from 14.42 Å to 13.92 Å. The predicted lattice constant c of K 0.25 Fe 2 Se 2 , 14.42 Å agrees with 14.28(4) Å of K 0.3 Fe 2 Se 2 having T c of 44 K. [ 31 ] And the predicted lattice constants a and c for K 0.5 Fe 2 Se 2 , 3.69 Å and 13.92 Å are consistent with the periods observed using scanning tunneling microscope along [110] and [001], 5.5 Å (√2a) and 14.1 Å, [ 26 ] respectively, considering the calculation accuracy. It is worth noting that a similar trend of the changes in lattice parameters was observed in other electron-doped ThCr 2 Si 2 structures, such as KFe 2 As 2 [ 9,44 ] and CaFe 2 As 2.
[ 45 ] Furthermore, we explore the temperature-dependent stability of structures in the K region of interest. To this end, we perform the molecular dynamics (MD) simulations on K 0.25 Fe 2 Se 2 and K 0.5 Fe 2 Se 2 . Both structures can survive for at least 1 picosecond using 2√2 × 2√2 × 1 super cells at temperatures up to 500 K, indicating that they are stable at these temperatures. [ 46 ] It should be worth noting that the atom displacements of   Figure S3, Supporting Information), which suggests that they should have similar properties to KFe 2 Se 2 . Considering stoichiometry, formation energy, stability, and electronic structures, K 0.25 Fe 2 Se 2 and K 0.5 Fe 2 Se 2 (see Figure S2, Supporting Information for their structures) are proposed to be responsible for the observed superconductivity at 44 and 30 K, [ 31 ] respectively. The relative stability difference could also explain the diffi culty to obtain the 44 K phase. Based on the results presented above, we schematize a phase diagram for the K x Fe 2y Se 2 system in Figure 4 . In the K-rich portion with x > 0.6, phases with Fe vacancies tend to exist, agreeing well with the observed antiferromagnetic K 2 Fe 4 Se 5 phase. As for the low level K-intercalated compounds ( x < 0.25), there has been no report on the synthesis of free-standing K x Fe 2 Se 2 . We note that the two identifi ed SC phases K 0.3 Fe 2 Se 2 and K 0.6 Fe 2 Se 2 lie in the region of 0.25 ≤ x ≤ 0.6. [ 31 ] T c is 44 K for the former phase and 30 K for the latter one, suggesting that T c is dependent on the K content or the doped electron concentrations. Although experimentally observed K contents in SC phases by far are discrete, recent reports about carrier concentration tuning of T c from 30 to above 40 K in FeSe thin fl akes [ 47,48 ] implies that their variation of T c with carrier concentration can be continuous, similar to the cases in other hightemperature superconductors. Further improvement of T c can be expected considering the experimental progresses achieved in single-layer FeSe fi lms. [49][50][51][52][53][54] Therefore, we infer that the 30 K phase is electron overdoped and the 44 K phase also might not be optimally tuned. Moreover, in alkali-metal-intercalated FeSe compounds prepared at low temperatures, the synergic effect of NH 3 , NH 2 , or C 2 H 4 (NH 2 ) 2 along with alkali metal can stabilize the structures. [ 31,32,55 ] These offer us an effective strategy to raise T c of bulk iron-selenide-based superconductors by controlling carrier doping while stabilizing the structures by intercalations or effect of substrates. At the moment, tremendous efforts are needed toward this goal. Because of the calculation limit and accuracy, we do not consider the slight off-stoichiometric cases, say, the Fe vacancy concentration is less than 3 at%. It should be pointed out that such slight off-stoichiometry is tolerated in K x Fe 2 Se 2 just like many other materials which may be the reason that superconductivity was observed in previous reports of refs. [ 33,40 ] .
In conclusion, we carefully investigate the energetic change and structural evolution of K x Fe 2y Se 2 as a function of intercalated K content using fi rst-principles calculations. Two factors dominating the formation of the phases and the structural evolution are confi rmed. One is due to the accumulation of negative charge in FeSe layers, the other is due to Coulomb attraction between K ion layers and negatively charged FeSe layers. The structural evolution of this series of phases is summarized: at 0.25 ≤ x ≤ 0.6, the bct lattice is stable and Fe is favored to full occupy its site; at x > 0.6, FeSe layers tend to exclude Fe atoms and create Fe vacancies; and at x < 0.25, the structure will collapse for the dynamic instability. A schematic phase diagram is constructed accordingly and the possible route to further improve T c is suggested. The phases responding to the observed superconductivity are proposed to be K 0.25 Fe 2 Se 2 and K 0.5 Fe 2 Se 2 in terms of stoichiometry, formation energy, stability, and electronic structures. Though based on the study of K x Fe 2− y Se 2 , our results should be meaningful to understand the SC and its related phases in metal-intercalated iron selenides and other similar SC systems.

Supporting Information
Supporting Information is available from the Wiley Online Library or from the author.