Structural, electronic, and dynamical properties of the tetragonal and collapsed tetragonal phases of ${\mathrm{KFe}}_2{\mathrm{As}}_2$

Compounds with a tetragonal ${\mathrm{ThCr}}_2{\mathrm{Si}}_2$-type structure are characterized by the possibility of the isostructural phase transition from the tetragonal phase to the collapsed tetragonal phase induced by the external pressure. An example of a compound with such a phase transition is ${\mathrm{KFe}}_2{\mathrm{As}}_2$, which belongs to the family of the iron-based superconductors. In this paper, we investigate the effects of the phase transition on the structural, electronic, and dynamical properties of this compound. Performing the ab initio calculations, we reproduce the dependence of the lattice constants on pressure and analyze the changes of the interatomic distances in the tetragonal and collapsed tetragonal phases. Using the tight binding model with maximally localized Wannier orbitals, we calculate and discuss the influence of pressure on the electronic band structure as well as on the shape of the Fermi surface. We found a precursor of the phase transition in the form of enhancement of overlapping between two Wannier orbitals of As atoms. In order to better understand the superconducting properties of ${\mathrm{KFe}}_2{\mathrm{As}}_2$, we study the orbital-projected Cooper pairs susceptibility as a function of pressure. We found a decrease of susceptibility with the increasing pressure in a good qualitative agreement with experimental observation. The structural transition also influences the phonon spectrum of ${\mathrm{KFe}}_2{\mathrm{As}}_2$, which exhibits pronounced changes induced by pressure. Some phonon modes related with the vibrations of Fe and As atoms show an anomalous, nonmonotonic dependence on pressure close to the phase transition.

Figure 1

(a) The conventional cell of the tetragonal (I4/mmm) structure of ${\mathrm{KFe}}_2{\mathrm{As}}_2$. The image was rendered using vesta software [38]. Pressure dependencies of the following structural parameters: (b) Fe-As, Fe-Fe, K-As, and As-As (as labeled) interatomic distances, (c) angle $\alpha$ between two Fe-As bonds [shown in (a)] and distance ${\mathrm{h}}_{\text{As}}$ of As atom from Fe plane, (d) lattice constants $a$ and $c$, and (e) volume $V$ of the K122 crystal cell (in a conventional standard) and relative volume $V_{\text{FeAs}}/V$ of Fe-As layer [shown in (a)].

Figure 2

Contour maps of the spatial distribution of charge density in ${\mathrm{KFe}}_2{\mathrm{As}}_2$ in plane passing through interlayer As atoms calculated for four hydrostatic pressures. The densities from $\text{min}=0$ to $\text{max}=0.345\mathrm{e}/{\AA }^3$ are presented using contours plotted in logarithmic mode.

Figure 3

$p_z$-like maximally localized Wannier orbital localized around As atoms. For the simplicity only orbital originating from one As atom is shown. The second $p_z$-like orbital can by obtained by the mirror point symmetry. Red and blue colors of the orbital correspond to different signs of the wave function. The primitive unit cell of K122 is shown for a comparison.

Figure 4

A comparison of the band structures of K122 obtained from DFT calculations (red solid lines) and tight binding model in maximally localized Wannier orbitals (blue dotted lines). Results in the absence of the hydrostatic pressure.

Figure 5

Comparison of experimental ARPES results and DFT calculations. (a) ARPES band structure along the $\mathrm{\Gamma }-X$ direction around the Fermi level. (b) Comparison of the DFT results (blue line) with ARPES spectra from panel (a) (background and red line). The DFT band structure is shifted with respect to experimental results by approximately 75 meV to higher energy (cf. blue and red lines). The green line denotes the Fermi level obtained from the ARPES. (c) The Fermi surface at $\mathrm{\Gamma }-X$ plane obtained by ARPES. (d) Comparison of the calculated spectral function $\mathcal{A}(\mathit{k},\omega )$ at the effective (shifted) Fermi level $\omega =E_F^{\mathrm{eff}}$ (background) with ARPES data (red dots). ARPES data are adopted from Ref. [58].

Figure 6

(a) The first Brillouin zone of the tetragonal structure (I4/mmm) and its high-symmetry points. The Fermi surface of K122 at ambient pressure (b), before the structural transition (c), and after the structural phase transition (d). The image was rendered using xcrysden software [62].

Figure 7

Electron band projections on Fe $d$ orbitals (left column) and As $p$ orbitals (right column). Sizes of dots correspond to the value of the contribution of a given orbital, while their colors correspond to their type (as labeled). The results obtained below (top row, $p=13.5$ GPa) and above (bottom row, $p=15.0$ GPa) the structural phase transition (in the T and cT phases, respectively).

Figure 8

(a) Total electronic density of states (black solid line and the shadow) and partial electronic DOS projected onto orbitals of Fe and As atoms (color labeled lines as indicated) at $p=0$ GPa. (b) Contributions of given types of the orbitals to the total DOS at the Fermi level as a function of the external pressure. $t_{2g}$ denotes the contribution of $d_{xz}, d_{yz}$, and $d_{xy}$ orbitals of the Fe atoms together, $e_g$ denotes the contribution of $d_{z^2}$ and $d_{x^2-y^2}$ orbitals of Fe atoms together, whereas $p$ is the joint contribution of all $p$-like orbitals of As atoms.

Figure 9

The pressure dependence of the superconducting intraband susceptibility originating from particular groups of the orbitals [as labeled, denotations as in Fig. 8].

Figure 10

Phonon dispersion curves of K122 along the high-symmetry directions of the first Brillouin zone at $p=0$ GPa.

Figure 11

Total phonon DOS (shaded grey area) and partial phonon DOS (red, blue, green lines correspond to the K, Fe, and As atoms, respectively) for several values of the hydrostatic pressure.

Figure 12

The phonon band structure projections on atoms (columns as labeled) and directions (line colors) below (top row, $p=13.5$ GPa) and above (bottom row, $p=15.0$ GPa) the structural phase transition. Sizes of dots correspond to the value of the contribution of oscillations of a given atom, while their colors determines the direction of that oscillations. Red, green, and blue colors correspond to oscillations in $x, y$, and $z$ directions, respectively.

Figure 13

Schematic illustration of Raman and infrared active modes in ${\mathrm{KFe}}_2{\mathrm{As}}_2$. In the parenthesis the frequency of each mode is given.

Figure 14

Evolution of Raman active (a) and infrared active modes (b) frequencies with pressure.