Superconducting properties of ${\mathrm{K}}_{1\text{−}x}{\mathrm{Na}}_x{\mathrm{Fe}}_2{\mathrm{As}}_2$ under pressure

The effects of hydrostatic pressure and partial Na substitution on the normal-state properties and the superconducting transition temperature $\left(T_c\right)$ of ${\mathrm{K}}_{1\text{−}x}{\mathrm{Na}}_x{\mathrm{Fe}}_2{\mathrm{As}}_2$ single crystals were investigated. It was found that a partial Na substitution leads to a deviation from the standard $T^2$ Fermi-liquid behavior in the temperature dependence of the normal-state resistivity. It was demonstrated that non-Fermi-liquid like behavior of the resistivity for ${\mathrm{K}}_{1\text{−}x}{\mathrm{Na}}_x{\mathrm{Fe}}_2{\mathrm{As}}_2$ and some ${\mathrm{KFe}}_2{\mathrm{As}}_2$ samples can be explained by a disorder effect in the multiband system with rather different quasiparticle effective masses. Concerning the superconducting state our data support the presence of a shallow minimum around 2 GPa in the pressure dependence of $T_c$ for stoichiometric ${\mathrm{KFe}}_2{\mathrm{As}}_2$. The analysis of $T_c$ in ${\mathrm{K}}_{1\text{−}x}{\mathrm{Na}}_x{\mathrm{Fe}}_2{\mathrm{As}}_2$ at pressures below 1.5 GPa showed that the reduction of $T_c$ with Na substitution follows the Abrikosov-Gor'kov law with the critical temperature $T_{c0}$ of the clean system (without pair breaking), which linearly depends on the pressure. Our observations also suggest that $T_c$ of ${\mathrm{K}}_{1\text{−}x}{\mathrm{Na}}_x{\mathrm{Fe}}_2{\mathrm{As}}_2$ is nearly independent of the lattice compression produced by the Na substitution. Further, we theoretically analyzed the behavior of the band structure under pressure within the generalized gradient approximation (GGA). A qualitative agreement between the calculated and the recently measured—in de Haas–van Alphen experiments [T. Terashima et al., Phys. Rev. B 89, 134520 (2014)]—pressure dependencies of the Fermi-surface cross sections has been found. These calculations also indicate that the observed minimum around 2 GPa in the pressure dependence of $T_c$ may occur without a change of the pairing symmetry.

Figure 1

The $T$ dependence of the resistivity of ${\mathrm{K}}_{1\text{−}x}{\mathrm{Na}}_x{\mathrm{Fe}}_2{\mathrm{As}}_2$ at zero pressure. The data for $x=0.18$ and $x=0.31$ are taken from the supplementary information in Ref. [4]. (a) The temperature range below 10 K. Symbols: experimental data; solid lines: single-band fit Eq. (1); dashed lines: multiband fit Eq. (2). (b) The temperature range up to 300 K. Inset: the dependence of the exponent $n$ vs the residual resistivity ${\rho }_0$; see Eq. (1). The data for ${\mathrm{K}}_{1\text{−}x}{\mathrm{Na}}_x{\mathrm{Fe}}_2{\mathrm{As}}_2$ crystals $\left(x\ne 0\right)$ are taken from the supplementary information in Ref. [4], for $x$ = 0 (open rhombus) from Ref. [23]. The solid line is a guide to the eye.

Figure 2

The $T$ dependence of the magnetization of ${\mathrm{K}}_{1\text{−}x}{\mathrm{Na}}_x{\mathrm{Fe}}_2{\mathrm{As}}_2$ measured in 10 Oe at different applied pressures. The second superconducting transition at higher temperature [panels (a)–(c)] corresponds to a low-temperature pressure gauge such as Sn or Pb. (a) $x=0$ (S1 sample) measured in a Quantum Design pressure cell. (b) $x=0$ (S2 sample) measured in a homemade pressure cell. Inset: zoom to the data at high pressure. (c) Sample with $x=0.31$ measured in a Quantum Design pressure cell. (d) Sample with $x=0.18$ measured in a He gas pressure cell. The data in panels (b) and (d) were corrected for the background signal.

Figure 3

(a) The pressure dependence of $T_c$ for ${\mathrm{K}}_{1\text{−}x}{\mathrm{Na}}_x{\mathrm{Fe}}_2{\mathrm{As}}_2$ samples from this work and those taken from the literature for stoichiometric K-122 [1, 16, 18, 20, 21], Cs122 [21], and Rb122 [26]. Dashed lines: linear fit; and solid lines: results obtained from Eq. (3). (b) The normalized critical temperature $T_c/T_{c0}\left(P\right)$ vs the ratio between the residual resistivity at zero pressure and the critical temperature for clean K-122 ${\rho }_0/T_{c0}\left(P\right)$. The solid line denotes the Abrikosov-Gor'kov formula. The ${\rho }_0$ values for disordered K-122 and Cs-122 samples (in brackets) are taken from Ref. [21] and have been normalized according to procedure described in the text.

Figure 4

The hydrostatic pressure dependence of the lattice parameters for K-122. The experimental data for K-122 are taken from Ref. [21]. The data points for ${\mathrm{K}}_{1\text{−}x}{\mathrm{Na}}_x{\mathrm{Fe}}_2{\mathrm{As}}_2$ correspond to the lattice parameters which have been obtained from x-ray data at atmospheric pressure. The data points are shifted along the pressure axis to fit the values for ${\mathrm{KFe}}_2{\mathrm{As}}_2$ at external hydrostatic pressure. (a) The data obtained at room temperature. Closed symbols are the experimental data for ${\mathrm{K}}_{1\text{−}x}{\mathrm{Na}}_x{\mathrm{Fe}}_2{\mathrm{As}}_2$; open symbols are taken from Ref. [21]. (b) Closed symbols: the experimental data for ${\mathrm{K}}_{1\text{−}x}{\mathrm{Na}}_x{\mathrm{Fe}}_2{\mathrm{As}}_2$ obtained at $T\approx$ 100 K. Solid lines: theoretical calculations corresponding to the low-temperature regime. Inset: single crystalline diffraction pattern of $\left(hk0\right)$, and $\left(h0l\right)$ planes for a Na doped crystal.

Figure 5

${\mathrm{KFe}}_2{\mathrm{As}}_2$ band structure considering different crystal lattice parameters (namely, the experimental and PM ones; see Table 3).

Figure 6

${\mathrm{KFe}}_2{\mathrm{As}}_2$ band structure for the experimental lattice structure (taken as $P=0$ GPa geometry) and lattice parameters extrapolated with compressibility at 1.66 and 3.00 GPa (see Table 4).

Figure 7

Fermi surface cuts along the positive $k_z$ direction for ${\mathrm{KFe}}_2{\mathrm{As}}_2$ for the experimental crystal structure and for the extrapolated lattice parameters using the predicted compressibility. The inner, middle, and outer FSs around the $\Gamma$ point are identified as $\alpha ,\zeta$ and $\beta$, respectively. (For the orbital content of FSs see text.)

Figure 8

The variation of the total density of states (TDOS) under pressure for ${\mathrm{KFe}}_2{\mathrm{As}}_2$ (per one spin). Inset: pressure dependence of the TDOS at the Fermi level.

Figure 9

The variation of the band-resolved DOS at the Fermi level under pressure for ${\mathrm{KFe}}_2{\mathrm{As}}_2$.

Figure 10

The variation of the various Fe 3d orbitals projected density of states under pressure for ${\mathrm{KFe}}_2{\mathrm{As}}_2$ per spin and one Fe atom.