Thermal Expansion and Grüneisen Parameters of $\mathrm{Ba}({\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x{)}_2{\mathrm{As}}_2$: A Thermodynamic Quest for Quantum Criticality

Thermal expansion data are used to study the uniaxial pressure dependence of the electronic-magnetic entropy of $\mathrm{Ba}({\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x{)}_2{\mathrm{As}}_2$. Uniaxial pressure is found to be proportional to doping and, thus, also an appropriate tuning parameter in this system. Many of the features predicted to occur for a pressure-tuned quantum critical system, in which superconductivity is an emergent phase hiding the critical point, are observed. The electronic-magnetic Grüneisen parameters associated with the spin-density wave and superconducting transitions further demonstrate an intimate connection between both ordering phenomena.

Figure 1

Thermal expansion coefficients divided by temperature, $\alpha /T$, versus $T$ of $\mathrm{Ba}({\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x{)}_2{\mathrm{As}}_2$ along (a) the tetragonal $a$ axis and (b) $c$ axis for $0<x<33\%$. The colors represent the doping levels (black, underdoped, nonsuperconducting; blue, underdoped, superconducting; red, overdoped, superconducting; green, overdoped, nonsuperconducting).

Figure 2

(a) Normal-state electronic Sommerfeld coefficient , ${\gamma }_N(x)$, versus Co content derived from experiment [17, 24] and theory. (b) Normal-state $\alpha /T$ values at 8 K and associated uniaxial pressure derivatives of ${\gamma }_N(x)$, $\frac{\partial {\gamma }_N}{\partial p_i}=-\frac{{\alpha }_i^{\mathrm{elec}.}}{T}(i=a,c)$. The inset in (a) shows that $\frac{\partial {\gamma }_N}{\partial x}$ is quite similar to $\frac{\partial {\gamma }_N}{\partial p_i}$, implying a close relationship between pressure and doping.

Figure 3

Electronic-magnetic contribution of $\alpha /T$, ${\alpha }^{\mathrm{elec}.}/T$, along (a) the tetragonal $a$ axis and (b) the $c$ axis for $0<x<15\%$ Co. The electronic-magnetic contribution was derived by subtracting the data for $x=33\%$ Co, which consist mostly of a phonon contribution and a nearly zero electronic term [see Fig. 2b]. (c) Expanded view of ${\alpha }_a^{\mathrm{elec}.}/T$ showing clear evidence for NFL behavior above $T_{\mathrm{SDW}}/T_c$. The arrows mark a crossover temperature, $T^{*}$, below which Fermi-liquid behavior is observed.

Figure 4

Electronic uniaxial Grüneisen parameters, ${\Gamma }_{\mathrm{Gr}\ddot{\mathrm{u}}\mathrm{neisen}}^{\mathrm{elec}.}$, for the tetragonal $a$ axis, derived by subtracting ${\alpha }_a^{\mathrm{elec}.}/T$ data from Fig. 3a by the electronic heat capacity, $C_p^{\mathrm{elec}.}/T$, data from Refs. 17, 24. The upper inset shows a phase diagram in which the temperature at which ${\alpha }_a^{\mathrm{elec}.}/T$ changes sign, as well as the Fermi-liquid to non-Fermi-liquid crossover temperature $T^{*}$, are plotted versus $x$. The lower inset provides a comparison of ${\Gamma }_{\mathrm{Gr}\ddot{\mathrm{u}}\mathrm{neisen}}^{\mathrm{elec}.}$ for $x=5.5$ and 6%.