Anisotropic thermodynamic and transport properties of single-crystalline ${\mathrm{CaKFe}}_4{\mathrm{As}}_4$

Single-crystalline, single-phase ${\mathrm{CaKFe}}_4{\mathrm{As}}_4$ has been grown out of a high-temperature, quaternary melt. Temperature-dependent measurements of x-ray diffraction, anisotropic electrical resistivity, elastoresistivity, thermoelectric power, Hall effect, magnetization, and specific heat, combined with field-dependent measurements of electrical resistivity and field and pressure-dependent measurements of magnetization indicate that ${\mathrm{CaKFe}}_4{\mathrm{As}}_4$ is an ordered, stoichiometric, Fe-based superconductor with a superconducting critical temperature, $T_c=35.0\pm 0.2$ K. Other than superconductivity, there is no indication of any other phase transition for $1.8\text{K}\le T\le 300$ K. All of these thermodynamic and transport data reveal striking similarities to those found for optimally or slightly overdoped $\left({\mathrm{Ba}}_{1-x}{\mathrm{K}}_x\right){\mathrm{Fe}}_2{\mathrm{As}}_2$, suggesting that stoichiometric ${\mathrm{CaKFe}}_4{\mathrm{As}}_4$ is intrinsically close to what is referred to as “optimal-doped” on a generalized, Fe-based superconductor, phase diagram. The anisotropic superconducting upper critical field, $H_{c\text{2}}\left(T\right)$, of ${\mathrm{CaKFe}}_4{\mathrm{As}}_4$ was determined up to 630 kOe. The anisotropy parameter $\gamma \left(T\right)=H_{c\text{2}}^{\perp }/H_{c\text{2}}^{\parallel }$, for $H$ applied perpendicular and parallel to the $c$ axis, decreases from $\simeq 2.5$ at $T_c$ to $\simeq 1.5$ at 25 K, which can be explained by interplay of paramagnetic pair breaking and orbital effects. The slopes of $d{H}_{c\text{2}}^{\parallel }/dT\simeq -44$ kOe/K and $d{H}_{c\text{2}}^{\perp }/dT\simeq -109$ kOe/K at $T_c$ yield an electron mass anisotropy of $m_{\perp }/m_{\parallel }\simeq 1/6$ and short Ginzburg-Landau coherence lengths ${\xi }_{\parallel }\left(0\right)\simeq 5.8\text{Å}$ and ${\xi }_{\perp }\left(0\right)\simeq 14.3\text{Å}$. The value of $H_{c\text{2}}^{\perp }\left(0\right)$ can be extrapolated to $\simeq 920$ kOe, well above the BCS paramagnetic limit.

Figure 1

X-ray diffraction data showing $\left(00\ell \right)$ diffraction peaks from in-lab diffraction measurements on a single-crystalline plate (upper data set) and high-energy x-ray diffraction measurements taken at APS (lower data set). Note that $\ell$ = odd $\left(00\ell \right)$ lines are consistent with the ordered ${\mathrm{CaKFe}}_4{\mathrm{As}}_4$ structure and are formally forbidden in a $\left({\mathrm{Ca}}_{1-x}{\mathrm{K}}_x\right){\mathrm{Fe}}_2{\mathrm{As}}_2$ structure [15].

Figure 2

Temperature dependence of ${\mathrm{CaKFe}}_4{\mathrm{As}}_4 a$- and $c$-lattice parameters as well as unit cell volume as determined from (200) and (009) diffraction lines measured via high-energy x-ray diffraction.

Figure 3

Temperature-dependent in-plane, ${\rho }_a\left(T\right)$, and interplane, ${\rho }_c\left(T\right)$, resistivity of ${\mathrm{CaKFe}}_4{\mathrm{As}}_4$, plotted using normalized resistivity scales, $\rho \left(T\right)/\rho \left(300\text{K}\right)$. At 300 K, ${\rho }_a\sim 300 \mu \mathrm{\Omega }\text{cm}$ and ${\rho }_c\sim 1000$–$2000 \mu \mathrm{\Omega }$ċm. (Inset) Picture of a ${\mathrm{CaKFe}}_4{\mathrm{As}}_4$ single crystal shown over a millimeter grid.

Figure 4

Anisotropic, temperature-dependent magnetization divided by applied field $\left[M\right(T)/H]$ of ${\mathrm{CaKFe}}_4{\mathrm{As}}_4$ taken for $H=50$ kOe applied along the crystallographic $c$ axis and perpendicular to the crystallographic $c$ axis. Due to $T_c$ at 35 K, data shown are for $40\text{K}<T<300$ K.

Figure 5

Temperature-dependent Hall resistivity divided by field, ${\rho }_{\mathrm{H}}\left(T\right)/H$, (Hall coefficient) of ${\mathrm{CaKFe}}_4{\mathrm{As}}_4$ with $H=140$ kOe applied along the crystallographic $c$ axis. Inset shows field-dependent Hall resistivity ${\rho }_{\mathrm{H}}$ at 40, 100, and 200 K.

Figure 6

Temperature-dependent thermoelectric power $\left[S\right(T\left)\right]$ for ${\mathrm{CaKFe}}_4{\mathrm{As}}_4$ for a temperature gradient applied perpendicular to the crystallographic $c$ axis.

Figure 7

Thermodynamic and transport data taken on ${\mathrm{CaKFe}}_4{\mathrm{As}}_4$ near $T_c$: (a) normalized electrical resistivity. Inset shows the magneto-optic image on a ${\mathrm{CaKFe}}_4{\mathrm{As}}_4$ single crystal (see text for details). (b) FC and ZFC magnetization for $H=50$ Oe for $H$ applied perpendicular to the crystallographic $c$ axis (note the FC susceptibility data is multiplied by 25 for clarity), (c) zero-field specific heat $C_p\left(T\right)/T$.

Figure 8

The superconducting critical temperature $T_c$ of ${\mathrm{CaKFe}}_4{\mathrm{As}}_4$ as a function of applied pressure. Open square symbols from piston cylinder cell and filled symbols from moissanite anvil cell. (Inset) $M\left(T\right)$ measured in a moissanite anvil cell for $p=0$, 1.95, and 3.9 GPa.

Figure 9

Magnetization as a function of magnetic field applied perpendicular to the crystallographic $c$ axis of ${\mathrm{CaKFe}}_4{\mathrm{As}}_4$ for $T=1.85$ K. (Inset) Low-field extended view and solid line showing ideal $\chi =-1/4\pi$.

Figure 10

Temperature-dependent electrical resistance of ${\mathrm{CaKFe}}_4{\mathrm{As}}_4$ for $H$ applied parallel and perpendicular to the crystallographic $c$ axis for representative fields $H\le 140$ kOe. Onset and offset criteria for $T_c$ are shown by dashed lines in the panel.

Figure 11

Field-dependent resistance measured in a 630 kOe pulsed magnet at different temperatures with (a) $H\parallel c$ and (b) $H\perp c$. A temperature-independent background signal was subtracted for clarity (see text). Dotted line and arrows indicate different criteria for determining $H_{c\text{2}}$ (see text).

Figure 12

(a) Anisotropic $H_{c2}\left(T\right)$ data for ${\mathrm{CaKFe}}_4{\mathrm{As}}_4$ inferred from the temperature-dependent electrical resistivity data presented in Fig. 10. The $H_{c2}\left(T\right)$ data for $H\parallel c$ inferred from temperature and field-dependent specific heat measurements (Fig. 13), using an equientropic midpoint criterion, are also shown. (b) Anisotropic $H_{c\text{2}}\left(T\right)$ up to 630 kOe, including the data shown in (a) for field below 140 kOe. Black diamonds represent $H_{c\text{2}}^{\perp }\left(T\right)$. Red circles represent $H_{c\text{2}}^{\parallel }\left(T\right)$. Open and filled symbols indicate offset and onset criteria as described in the text. Black and red solid lines in the main figure are theoretically fitted curves to the onset criteria (see text). The inset shows the anisotropic parameter $\gamma \left(T\right)=H_{c\text{2}}^{\perp }/H_{c\text{2}}^{\parallel }$ together with the theoretically fitted curve (black solid line).

Figure 13

Temperature-dependent specific heat data for ${\mathrm{CaKFe}}_4{\mathrm{As}}_4$ taken for $H\parallel c=0$, 50, 90, 140 kOe. The data for finite $H$ have been normalized to those for $H=0$ kOe in the normal state above $T_c$.

Figure 14

Elastoresistivity coefficients of $2m_{66}$ and $m_{11}-m_{12}$ of ${\mathrm{CaKFe}}_4{\mathrm{As}}_4$ (open and filled circles) measured using crossed samples glued to a piezostack, shown schematically in the right inset. The $2m_{66}$ coefficient data of optimally doped K-doped ${\mathrm{BaFe}}_2{\mathrm{As}}_2$ (grey +'s) from Ref. [55] are plotted for comparison.

Figure 15

Log-log plot of $\mathrm{\Delta }{C}_p$ jump at $T_c$ vs $T_c$ (BNC plot [Refs. 6, 12, 56]). Note that data for $\left({\mathrm{Ba}}_{1-x}{\mathrm{K}}_x\right){\mathrm{Fe}}_2{\mathrm{As}}_2$ are plotted for $x<0.8$. For $\mathrm{Ba}({\mathrm{Fe}}_{1-x}{TM}_x){}_2{\mathrm{As}}_2,TM=\text{Ni}$, Co, Rh, Pd, Pt (Ref. [12]). Dashed line has a slope corresponding to $\mathrm{\Delta }{\mathrm{C}}_p\sim T_c^3$ and is a guide for the eyes.