1 Introduction

High-temperature superconductivity in iron-based materials was first observed in 2008 [1]. This discovery opened a period of intensive experimental studies of this class of compounds [2,3,4]. The unique properties caused by the layered structure [2] and multi-band electronic structure [5, 6] were observed. A typical iron-based superconductor (IBSC) exhibits the multi-gap [6, 7] unconventional superconductivity with the s±-wave pairing symmetry [8], i.e., the superconducting gap at different pockets of the Fermi surfaces has opposite signs [9]. In contrast to the other IBSC, the KFe2As2 compound (K122) possesses some unusual features. For example, the d-wave gap with the nodal line is expected in the superconducting phase [10,11,12,13,14,15,16]. Moreover, K122 is the Pauli limit superconductor [17], what is in contrast to majority of IBSC [18].

KFe2As2 is the end-member of the Ba1−xKxFe2As2 series. The maximum of critical temperature Tc = 38 K was found at optimal doping x ≈ 0.4 [19]. However, for a clean K122, Tc is estimated to be approximately 3.5 K [10, 20]. In a contrary to the parent Ba122 compound, K122 does not possess the low-temperature orthorhombic phase and does not show any magnetic order [21].

Usually, the IBSCs exhibit a structural phase transition between the tetragonal and orthorhombic phases by lowering temperature or inducing chemical pressure [22]. K122 is very special because it crystallizes in the tetragonal structure, even at very low temperatures. What is more interesting, with increasing pressure it exhibits the isostructural phase transition from the tetragonal (T) phase to the collapsed tetragonal (cT) phase [23, 24]. This transition is associated with a strong modification of the lattice parameters without lost of the crystal symmetry. The experimental evidence of the T–cT phase transition has been reported also for other 122 IBSC representatives, e.g., Ca122 [25], Ba122 [26, 27], or Eu122 [28].

Superconducting properties of K122 can be changed by increasing the external pressure. In particular, Tc shows nonmonotonic dependence on pressure [23, 24, 29,30,31,32], with a minimum near 1.8 GPa [32]. Moreover, Tc exhibits an universal behavior of the V-shape dependence in the weak pressure regime [33]. It has been reported not only in a case of K122 [32] but also for Rb122 [33] and Cs122 [34].

In the present work, we discuss the effect of the pressure on the superconducting phase of K122. We present it in the context of electronic properties obtained by the ab initio density functional theory (DFT) method. The paper is organized as follows. In Section 2, we show the numerical results and describe the influence of the external hydrostatic pressure on physical properties of the system related to the isostructural phase transition: main structural properties (Section 2.1), electronic properties (Section 2.2), and superconducting properties (Section 2.3). Additionally, in Section 2.4, we discuss our results in the context of other high-Tc superconductors. Finally, Section 3 is devoted to conclusions.

2 Numerical Results and Discussion

The DFT calculations were performed using the PBE GGA [36] exchange-correlation functional implemented in VASP [37, 38]. They were carried out for conventional cell (cf. Fig. 1a) using the 16 × 16 × 8 Monkhorst–Pack grid of k-points [39] and the energy cut-off of 450 eV. The crystal structure was optimized using the conjugate gradient technique with the energy convergence criteria set at 10− 7 eV and 10− 5 eV for the electronic and ionic iterations, respectively. During calculations, the lattice constants a and c as well as the position of As (zAs) were optimized. The optimization of the crystal structure was repeated for several hydrostatic pressures and the obtained results are presented in Fig. 1b and c.

Fig. 1
figure 1

(a) Tetragonal (I4/mmm) structure of KFe2As2. The image was rendered using VESTA software [35]. Pressure dependencies of (b) lattice constants a and c and (c) angle α between two Fe–As bonds [indicated in panel (a)] and distance hAs of the As atom from the Fe plane

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2.1 Isostructural Phase Transition

K122 crystallizes in the tetragonal I4/mmm structure (space group: 139) (Fig. 1a). Under normal conditions, the structural parameters of the K122 tetragonal phase were specified as a = b = 3.842 Å, c = 13.861 Å, and zAs = 0.3525 [42]. Imposed hydrostatic pressure leads to the isostructural phase transition. With increasing pressure, the lattice constant c decreases monotonically with the largest reduction of its value around the critical pressure pc ≈ 14 GPa (Fig. 1b). In contrast, the lattice constant a starts to increase at pressure around 10 GPa and it is suddenly enlarged around pc. At the same time, the dramatic changes of the angle α, defined in Fig. 1a, and the distance of the As atoms from the Fe plane (hAs) are also observed. Above pc, α and hAs reach the lowest and highest value, respectively. The obtained dependencies of the lattice parameters are in agreement with the experimental data [23].

The main source of the isostructural phase transition in 122 compounds is the formation of the As–As bonds between initially non-bonded As atoms caused by the overlap of the 4pz orbitals due to the imposed pressure [43,44,45]. Our findings are in a good agreement with this scenario. An emergence of the As–As bonding with increasing pressure is illustrated in Fig. 2, where the isocharge surfaces around the As atoms for several pressures are presented. The large As–As distance of non-bonded atoms is easily shortened by the pressure up to the value for which the overlap of orbitals is possible. When it happens, the pz orbitals of the As atoms cover partially the pz orbitals of the As atoms from neighboring FeAs layers [46]. It means that the initially separated isosurfaces bring together creating “hourglass” shapes as it is shown in Fig. 2. Consequently, the Fe–As bonds become weaker and the lattice parameter a increases, whereas c decreases at the phase transition.

Fig. 2
figure 2

Formation of the As–As bond with the increasing pressure visualized by the evolution of the isocharge surface (yellow one) corresponding to As atoms (cf. with Fig. 1(a)). The tetragonal K122 conventional cell in the same scale is shown for a comparison. The image was rendered using xCrysDen software [40]

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2.2 Electronic Properties

The spectra measured by the angle resolved photoemission spectroscopy (ARPES) showed the existence of three hole and four electron pockets centered at the Γ and the X point, respectively [20, 47,48,49]. To achieve a consistency of these results with our DFT calculations, it is necessary to shift the theoretical value of the Fermi level of approximately 75 meV [46]. Such a small shift corresponds to the change of the average number of particles by δn ≈ 0.075 [50], what does not change qualitatively the obtained electronic structure.

The resulting Fermi surfaces (FS) are shown in Fig. 3. In the absence of pressure, the FS of K122 have a characteristic cylindrical shape (Fig. 3a), also observed for other IBSC [5, 51,52,53]. Additionally, the FS calculated for p = 0 GPa very well reproduces the experimental results obtained by ARPES [20, 49]. The increasing pressure causes the significant modification of the FS around the Brillouin zone boundary (cf. Fig. 3b, c). The pocket observed close to the X point at p = 0 GPa is shifted to the zone boundary and changes its character from the hole-like to the electron-like at higher pressures [23].

Fig. 3
figure 3

Fermi surface of the K122 for several pressures (as labeled). Color corresponds to the Fermi velocity. On panel (a), Γ- and X- points of the Brillouin zone are indicated. The image was rendered using FermiSurfer software [41]

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Between 13.5 and 15 GPa, the topology of the FS alters indicating that the T–cT structural phase transition is accompanied by the Lifshitz transition [54]. Typically, it is generated by doping [14, 20, 55,56,57,58,59,60,61] or external magnetic field [62,63,64]. In the case of K122, the Lifshitz transition is induced by the pressure, which leads to the strong modification of the band structure around pc [46]. The FS shape is also strongly changed as it is shown in Fig. 3b and c. The largest modification of the FS is observed at a location of the smallest Fermi velocity (\(\textbf {v}_{F} \sim \partial E_{ k}/\partial { k}\)).

The Lifshitz transition is correlated with the modification of the FS topology and also with the alteration of orbitals contribution to the FS. A perfect nesting between electron and hole pockets of the FS [65] reported in many IBSC [3, 66,67,68,69,70,71] is usually observed in the absence of the pressure. In the case of K122, by contrast, the perfect nesting is not observed for pressures lower than pc. Thus, one cannot expect a realization of any magnetic order or nematic phases in this compound [72, 73]. This can be changed by the external pressure, high enough to transform system into the cT phase. For p > pc, the FS shape [23, 74] is very similar to those in the magnetic YFe2Ge2 [75] or YRu2Ge2 [76] compounds. Thus, in the cT phase of K122, one can also expect a realization of a magnetic order. It should be underlined that, in general, the magnetism and superconductivity are competitive phenomena.

2.3 Superconducting Properties

At present, it is believed that the antiferromagnetic spin fluctuations [7, 22, 77,78,79,80] can be a source of the s± superconductivity [80,81,82,83] in the IBSC. On the other hand, the experiments performed for K122 at ambient pressure (p = 0) show that this material is characterized by the nodal d-wave-type gap [10,11,12,13,14,15,16, 84,85,86]. However, with increasing pressure, the gap symmetry can be changed from d-wave to s±-wave [87].

As it was mentioned previously, the lattice parameters change under external pressure (cf. Fig. 1). In particular, the pressure modifies the distance between the As atoms and the Fe layer (hAs at Fig. 1c). For known IBSC, the apparent correlation between the anion (As, Se, and others) height from the iron layer and the superconducting critical temperature was already reported [88]. Figure 4 shows the critical temperature Tc for several IBSC in relation to the distance between anions and Fe layers using data presented in Ref. [88]. We supplemented this plot by the new data for K122 at zero and 30 GPa pressures (blue open circles in Fig. 4). The maximal value of Tc is observed for the anion—the Fe layer distance of approximately 1.4 Å. The FeSe layered material is a nice example of the beneficial influence of the external pressure on the critical temperature (red line in Fig. 4). In this case, a pressure increased from 0 to 4 GPa leads to a decrease of hSe and an increase of Tc. Similar effect is observed for the (Ba,K)122 family (dashed blue line in Fig. 4) but the source of hAs modification is different. The highest Tc was found when K doping leads to hAs ≈ 1.4 Å (pink diamond in Fig. 4).

Fig. 4
figure 4

Relation between the anion height from the iron layer and the critical temperature. Discussed results for KFe2As2 under pressure are shown by solid blue line. Compounds from Ba1−xKxFe2As2 family are marked by yellow frames, while the optimal doping is denoted by pink diamond. Experimental data are adopted from Ref. [88]

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To compare the K122 compound with other IBSC, we analyzed the dependence of hAs on pressure (Fig. 1c) and the experimental value of Tc [23, 24, 29,30,31,32]. Additionally, we assume that the superconductivity vanishes in the high-pressure phase what is in accordance with the predicted occurrence of magnetic order in the cT phase. The obtained solid blue line shown in Fig. 4 diverges from the solid green line determined for other materials due to the fact that K122 has much smaller Tc then the other IBSC with similar hAs. Nevertheless, the rate of Tc change with hAs is similar as in other IBSC.

The above considerations indicate that a question about a source of pairing in the K122 compound is still open and its explanation is necessary to the proper description of K122 properties.

2.4 Comparison with Other High-Tc Superconductors

The isostructural phase transition as well as a disappearance of the superconductivity under pressure should be connected with a modification of the Fermi surface topology, i.e., the Lifshitz transition. This connection was also observed in other high-Tc superconductors. During the Lifshitz transition, an appearing band has energy of the order of the cut-off energy of interaction and of the energy gap [89, 90]. In this regime, the superconducting phase is characterized by a superconducting dome (superstripes) with the critical temperature of the optimally doped phase due to a quantum resonance called the Fano resonance (or the Feshbach resonance) [91,92,93] (for a review see also, e.g., Ref. [94]).

Theoretical calculations for the multi-band superconductors show that the change of the character of the electrons from 3D to 2D behavior (which can be observed as a modification of a shape of the Fermi surface, e.g., from cilidrical-like to spherical-like) leads to a stabilization of the superconductivity [95,96,97] due to the emergence of the narrow bands [98]. In such a case, the van Hove singularity plays an important role in tuning of the superconductivity [99, 100]. The manifestation of this behavior is, for instance, an appearance of the superconducting phase in the magic-angle graphene superlattices [101, 102], width-dependence of Tc of nanofilms [103, 104], or room-Tc (under external pressure) hydride superconductors [105, 106].

The influence of the dimensionality of the system onto superconductivity was also studied in the case of MgB2 [107]. Here, it is worth to mention that in some cases, a reduction of the system dimensionality can lead to stabilization of the unconventional superconducting phases [108]. In the studied case of the K122 under pressure, we observed the opposite behavior to the discussed above. Namely, the pressure leads to increase of the dimensionality of the system and to a disappearance of the superconductivity (cf. Section 2.2).

3 Summary

In this work, we discuss the untypical properties of KFe2As2 according to the results obtained from the ab initio DFT calculations. We showed that the increasing pressure leads to the isostructural phase transition at some critical pressure pc between 13.5 and 15 GPa. Around pc the new strong bonding between As atoms emerges. As a consequence, the strong modification of the lattice parameters without lost of the symmetry is observed. We accurately reproduced the experimentally observed lattice constants dependence on pressure. We demonstrated also that the modification of structural parameters leads to the changes of the electronic structure that yield the Lifshitz transition. Additionally, the analysis of the Fermi surface of the cT phase leads us to the opinion that occurrence of some sort of magnetic order in the collapsed tetragonal phase of K122 is possible.

We examined also the relation between the anion height from Fe layers and the superconducting critical temperature in KFe2As2 compared with the other IBSC. The distance of 1.34 Å at zero-pressure should locate K122 compound among the superconductors with Tc higher than 20 K. However, KFe2As2 has much smaller Tc. An imposed pressure decreases both the superconducting critical temperature (to Tc= 0 K in cT phase) and the distance from the As atom to Fe layers. The resulting Tc dependence on the anion height from Fe layers agrees with the relation observed for other iron-based superconductors but the temperature range is different. The described untypical properties together with the observation of the gap function with nodal lines (i.e., with d-wave symmetry) make KFe2As2 a very attractive compound for future studies. In particular, the very important question of the nature of the superconductivity in this compound is still open.