Bulk Superconductivity and Role of Fluctuations in the Iron-Based Superconductor FeSe at High Pressures

The iron-based superconductor FeSe offers a unique possibility to study the interplay of superconductivity with purely nematic as well magnetic-nematic order by pressure ($p$) tuning. By measuring specific heat under $p$ up to 2.36 GPa, we study the multiple phases in FeSe using a thermodynamic probe. We conclude that superconductivity is bulk across the entire $p$ range and competes with magnetism. In addition, whenever magnetism is present, fluctuations exist over a wide temperature range above both the bulk superconducting and the magnetic transitions. Whereas the magnetic fluctuations are likely temporal, the superconducting fluctuations may be either temporal or spatial. These observations highlight similarities between FeSe and underdoped cuprate superconductors.

Figure 1

(a) Schematic temperature-pressure phase diagram of FeSe, showing the extent of tetragonal (tet), orthorhombic (o), paramagnetic (pm), magnetic (m), and superconducting (sc) states and the two characteristic pressures $p_1$ and $p_2$ (see main text). (b) Selected specific heat data sets, $C/T$ vs $T$, at different pressures. Light grey regions indicate the position of the various anomalies detected by $C/T$, related to the structural (circles), the superconduting (squares), and the magnetic transition (triangles). The inset illustrates schematically the measurement configuration [56] to measure the specific heat under $p$.

Figure 2

(a) Specific heat anomaly of the magnetic transition at $T_M$, $\mathrm{\Delta }C/T$, which is present for $p\ge 0.91\mathrm{GPa}$ ($\sim p_1$) and obtained by subtracting a background from $C/T$ data. Data are offset for clarity. Faint, grey triangles indicate the position of $T_M$ in each data set. $t_l$, $t_m$, and $t_r$ are used to estimate asymmetry and width of the specific heat peak. (b) Hysteresis $\mathrm{\Delta }T$ of $T_M$ between warming and cooling. Inset shows $d(C/T)/dT$ at 2.1 GPa upon warming and cooling. (c) Asymmetry (left axis) and width (right axis) of the specific heat peak at $T_M$. Dashed and dotted lines are guides to the eye, the purple bar indicates the position of the critical pressure range $p_2$.

Figure 3

(a)–(c) Estimate of the specific heat anomaly in FeSe at the superconducting transition, $\mathrm{\Delta }C/T$, in the pressure regimes $0\mathrm{GPa}\le p\le 0.84\mathrm{GPa}$ [$p<p_1$, (a)], $0.91\mathrm{GPa}\le p\le 1.58\mathrm{GPa}$ [$p_1<p<p_2$, (b)], and $1.72\mathrm{GPa}\le p\le 2.36\mathrm{GPa}$ [$p>p_2$, (c)]. The inset of (c) shows a blow-up of the data set in the main panel. The dotted lines in the inset of (a) indicate exemplarily the equal-area construction in $\mathrm{\Delta }C/T$ used to determine the superconducting jump size $\mathrm{\Delta }{C}_{sc}/T_c$ and the critical temperature $T_c$. (d) Evolution of $\mathrm{\Delta }{C}_{sc}/T_c$ (left axis) as well as superconducting transition width (right axis; see Fig. S10) as a function of $p$. Purple bars indicate the position of critical pressures $p_1$ and $p_2$. (e) $\mathrm{\Delta }{C}_{sc}/T_c$ as a function of the ratio $T_c/T_M$. Black circles (grey triangles) correspond to data in the pressure regime $p_1<p<p_2$ ($p>p_2$).

Figure 4

(a) Temperature-pressure phase diagram of FeSe, determined from specific heat measurements $C(T,p)$ (full squares). Red symbols correspond to the structural transition temperature $T_s$, black symbols to the superconducting transition temperature $T_c$ and blue symbols to the magnetic transition temperature $T_M$. The phase regions are labeled by t/pm (tetragonal/paramagnetic; light yellow), o/pm (orthorhombic/paramagnetic; red), o/m (orthorhombic/magnetic; blue) and sc (superconducting; brown/grey). Purple dotted vertical lines mark two characteristic pressures, $p_1$ and $p_2$. The error in the determination of $p_2$ is indicated by the light purple bar. The specific heat data are contrasted with data from various other techniques from literature, i.e., x-ray scattering [35], NMR [37], resistance, and magnetization ($R\text{−}1$ [38], $R\text{−}2$ [39], and $M\text{−}2$ [39], $R\text{−}3$ [40], $R\text{−}4$ [41]), and $\mu \mathrm{SR}$ [43, 44]. (b) Comparison of $C/T$ data at 1.58 GPa to $M$ [39] data, x-ray data of the orthorhombic distortion [35], and $R$ [38] ($R$) data at similar nominal pressures.