Strong correlations, strong coupling, and $s$-wave superconductivity in hole-doped ${\mathrm{BaFe}}_2{\mathrm{As}}_2$ single crystals

We present a comprehensive study of the low-temperature heat capacity and thermal expansion of single crystals of the hole-doped ${\mathrm{Ba}}_{1-x}{\mathrm{K}}_x{\mathrm{Fe}}_2{\mathrm{As}}_2$ series $\left(0<x<1\right)$ and the end-members ${\mathrm{RbFe}}_2{\mathrm{As}}_2$ and ${\mathrm{CsFe}}_2{\mathrm{As}}_2$. A large increase of the Sommerfeld coefficient ${\gamma }_n$ is observed with both decreasing band filling and isovalent substitution (K, Rb, and Cs) revealing a strong enhancement of electron correlations and the possible proximity of these materials to a Mott insulator. This trend is well reproduced theoretically by our density functional theory + slave-spin (DFT+SS) calculations, confirming that 122-iron pnictides are effectively Hund metals, in which sizable Hund's coupling and orbital selectivity are the key ingredients for tuning correlations. We also find direct evidence for the existence of a coherence-incoherence crossover between a low-temperature heavy Fermi liquid and a highly incoherent high-temperature regime similar to heavy fermion systems. In the superconducting state, clear signatures of multiband superconductivity are observed with no evidence for nodes in the energy gaps, ruling out the existence of a doping-induced change of symmetry (from $s$ to $d$ wave). We argue that the disappearance of the electron band in the range $0.4<x<1.0$ is accompanied by a strong-to-weak coupling crossover and that this shallow band remains involved in the superconducting pairing, although its contribution to the normal state fades away. Differences between hole- and electron-doped ${\mathrm{BaFe}}_2{\mathrm{As}}_2$ series are emphasized and discussed in terms of strong pair breaking by potential scatterers beyond the Born limit.

Figure 1

(a) and (b) Temperature dependence of the heat capacity of under- and overdoped ${\mathrm{Ba}}_{1-x}{\mathrm{K}}_x{\mathrm{Fe}}_2{\mathrm{As}}_2$ single crystals, respectively. The dotted and dash-dotted lines represent the lattice contributions of ${\mathrm{KFe}}_2{\mathrm{As}}_2$ and $\mathrm{Ba}({\mathrm{Fe}}_{0.85}{\mathrm{Co}}_{0.15}){}_2{\mathrm{As}}_2$, respectively, derived from Refs. [59] and [6]. The insets show a magnification of the high- and low-temperature regions, respectively. (c) and (d) Temperature dependence of the in-plane thermal expansion of under- and overdoped ${\mathrm{Ba}}_{1-x}{\mathrm{K}}_x{\mathrm{Fe}}_2{\mathrm{As}}_2$ single crystals, respectively. The dashed line is the thermal expansion of $\mathrm{Ba}({\mathrm{Fe}}_{0.67}{\mathrm{Co}}_{0.33}){}_2{\mathrm{As}}_2$ taken from Ref. [60]. The low-temperature thermal-expansion data of ${\mathrm{KFe}}_2{\mathrm{As}}_2$ ($T<4$ K) in 0 and 5 T were taken from Refs. [43] and [44].

Figure 2

(a)–(d) Temperature dependence of the normalized electronic heat capacity, ${\mathrm{C}}_e/{\gamma }_{n}T$, of under- and overdoped ${\mathrm{Ba}}_{1-x}{\mathrm{K}}_x{\mathrm{Fe}}_2{\mathrm{As}}_2$ single crystals, respectively. The solid curves represent the one-band weak-coupling BCS heat capacity for a $s$-wave superconductor. (c) and (d) Magnification of the low-temperature region for under- and overdoped single crystals, respectively. The dotted line indicates a linear behavior expected for a nodal superconductor.

Figure 3

(a) Phase diagram of ${\mathrm{Ba}}_{1-x}{\mathrm{K}}_x{\mathrm{Fe}}_2{\mathrm{As}}_2$ and $\mathrm{Ba}({\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x){}_2{\mathrm{As}}_2$ derived from our specific-heat and thermal-expansion measurements [45, 63]. $T_c$ of $A{\mathrm{Fe}}_2{\mathrm{As}}_2$ ($A=\text{Rb}$ and Cs) is also shown. $T_1$ and $T_2$ indicate the magnetic phase transitions $\mathrm{SDW}\text{−}\mathrm{C}2 \to$ SDW-C4 and $\mathrm{SDW}\text{−}\mathrm{C}4 \to$ SDW-C2 from Ref. [45]. (b) Sommerfeld coefficient ${\gamma }_n$ (blue symbols) and the residual electronic density of states, ${\gamma }_r={lim}_{T\to 0}\frac{C_e}{T}$, in superconducting samples (green symbols) for ${\mathrm{Ba}}_{1-x}{\mathrm{K}}_x{\mathrm{Fe}}_2{\mathrm{As}}_2,\mathrm{Ba}({\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x){}_2{\mathrm{As}}_2$ (from Ref. [63]) and $A{\mathrm{Fe}}_2{\mathrm{As}}_2$ ($A=\text{Rb}$ and Cs). The red stars represents density functional theory + slave spin (DFT+SS) calculations for the tetragonal paramagnetic phase. The magenta area indicate the loss of density of states due to the reconstruction of the Fermi surface in the SDW phase. Lines are guide to the eyes.

Figure 4

(a) Low-temperature specific heat of K-, Rb-, and ${\mathrm{CsFe}}_2{\mathrm{As}}_2$ in 0 and 5 T. (b) Temperature dependence of the uniaxial thermal expansion of K-, Rb-, and ${\mathrm{CsFe}}_2{\mathrm{As}}_2$. The arrows indicate the coherence temperature $T^{*}$. The low-temperature thermal-expansion data of ${\mathrm{KFe}}_2{\mathrm{As}}_2$ ($T<4$ K) in 0 and 5 T were taken from Refs. [43] and [44]. (c) and (d) Evolution of $d{\gamma }_n/d{p}_a$ and the Grüneisen parameter ${\mathrm{\Gamma }}_a\propto -\frac{1}{{\gamma }_n}\frac{d{\gamma }_n}{d{p}_a}$ for ${\mathrm{Ba}}_{1-x}{\mathrm{K}}_x{\mathrm{Fe}}_2{\mathrm{As}}_2 \left(0<x<1\right)$ and $A{\mathrm{Fe}}_2{\mathrm{As}}_2$ ($A=\text{K}$, Rb, and Cs), respectively.

Figure 5

Electronic specific heat of ${\mathrm{Ba}}_{0.49}{\mathrm{K}}_{0.51}{\mathrm{Fe}}_2{\mathrm{As}}_2$ single crystal. The black curve represents a two-gap fit using the empirical two-band $\alpha$ model. The blue and red curves are the partial specific-heat contributions of the $\{\alpha ,\zeta ,\epsilon ,e\}$ and $\beta$ bands, respectively.

Figure 6

(a) and (b) Three-band analysis (solid line) of the superconducting-state heat capacity of ${\mathrm{Ba}}_{1-x}{\mathrm{K}}_x{\mathrm{Fe}}_2{\mathrm{As}}_2$. For clarity, curves are vertically shifted from each other by 3 in (a) and by 2 in (b). (c) and (d) Inferred gap and individual density-of-states values, respectively.

Figure 7

(a) Phase diagram of ${\mathrm{Ba}}_{1-x}{\mathrm{K}}_x{\mathrm{Fe}}_2{\mathrm{As}}_2$ and $\mathrm{Ba}({\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x){}_2{\mathrm{As}}_2$ derived from our specific-heat and thermal-expansion measurements. (b) Evolution of the specific-heat jump at $T_c$. (c) Larger energy gaps derived from the temperature dependence of $C_e/T$. (d) Evolution of the zero-temperature thermodynamic critical field derived from our specific-heat measurements. Data for $\mathrm{Ba}({\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x){}_2{\mathrm{As}}_2$ are taken from Refs. [60, 63]. The green line indicates the weak-coupling single-band BCS value.

Figure 8

(a) Schematics of the two-band model. The hole band (in green) is a deep band. $E_g$ measures the distance of the bottom of the incipient electron band (in blue) from the Fermi energy. ${\omega }_c$ is the high-energy cutoff of the pairing interaction. (b) and (c) Evolution of $T_c$ and the zero-temperature energy gaps as a function of $E_g$. Both are normalized to $T_{c0}$, the transition temperature for the one-band case, i.e., for $E_g/{\omega }_c>1$. (d) Zero-temperature energy gaps normalized by $T_c$ as a function of $E_g$.

Figure 9

(a) $\mathrm{\Delta }C$ as a function of $T_c$ in the log-log Bud'ko-Ni-Canfield representation for our under- (open symbols) and overdoped (closed symbols) single crystals of $\mathrm{Ba}({\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x){}_2{\mathrm{As}}_2$ and ${\mathrm{Ba}}_{1-x}{\mathrm{K}}_x{\mathrm{Fe}}_2{\mathrm{As}}_2$. (b) $\mathrm{\Delta }C/{\gamma }_n{T}_c$ as function of $T_c$. The dashed line indicates the weak-coupling single-band BCS value. Data for $\mathrm{Ba}({\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x){}_2{\mathrm{As}}_2$ are taken from Refs. [60, 63]. Solid lines are guides to the eye.

Figure 10

Electronic specific heat of our optimally doped $\mathrm{Ba}({\mathrm{Fe}}_{0.938}{\mathrm{Co}}_{0.062}){}_2{\mathrm{As}}_2$ single crystal. The black curve is a calculation for an $s\pm$ superconductor (with equal densities of states on the hole and electron Fermi surfaces, ${\gamma }_1={\gamma }_2$) in the self-consistent $T$-matrix approximation (taken from Ref. [100]). The fitting parameters are the Friedel phase shift $\delta$ and the inter- to intraband scattering amplitudes ratio.

Figure 11

The density of states of quasiparticle excitations for an $s\pm$ superconductor $(N_1\left(0\right)=N_2\left(0\right)=N\left(0\right),{\mathrm{\Delta }}_1=-{\mathrm{\Delta }}_2=\mathrm{\Delta }$) with nonmagnetic impurities as a function of the quasiparticle energy, for different values of the scattering rate $\mathrm{\Gamma }/{\mathrm{\Gamma }}_c$. All curves were calculated with $\epsilon =0.46$ (or equivalently $\delta ={60}^{\circ }$). Gaplessness occurs for $\mathrm{\Gamma }/{\mathrm{\Gamma }}_c>0.34$. Here $\mathrm{\Delta }$ stands for $\mathrm{\Delta }(\mathrm{\Gamma },T)$.

Figure 12

(a) Evolution of $T_c$ on the scattering rate $\mathrm{\Gamma }$ in $\mathrm{Ba}({\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x){}_2{\mathrm{As}}_2$. Data are taken from Ref. [63]. ${\mathrm{\Gamma }}_c$ is the critical scattering rate that corresponds to $T_c=0$. Dependence of $\frac{\mathrm{\Delta }C}{{\gamma }_n{T}_c}$ (b) and the residual density of states ${\gamma }_r$ (c) on $T_c$. Open and closed symbols represent under- and overdoped concentrations, respectively. Magenta lines are fit to the data for an $s\pm$ superconducting state (with $N_1\left(0\right)=N_2\left(0\right)=N\left(0\right)$, and ${\mathrm{\Delta }}_1=-{\mathrm{\Delta }}_2=\mathrm{\Delta }$) in the intermediate impurity-scattering strength, i.e., for $\epsilon =0.46$.

Figure 13

(a) Temperature dependence of the heat capacity of ${\mathrm{Ba}}_{0.17}{\mathrm{K}}_{0.83}{\mathrm{Fe}}_2{\mathrm{As}}_2$ derived in the 4-band isotropic BCS model (black line). Individual-band contributions are also shown. The inset shows a magnification of the low-temperature region, $T/T_c<0.1$. (b) Temperature dependence of the individual gaps obtained self-consistently within this model.