Nonmonotonic pressure evolution of the upper critical field in superconducting FeSe

The pressure dependence of the upper critical field, $H_{\mathrm{c}2,\mathrm{c}}$, of single crystalline FeSe was studied using measurements of the interplane resistivity, ${\rho }_{\text{c}}$, in magnetic fields parallel to tetragonal $c$ axis. $H_{\mathrm{c}2,\mathrm{c}}\left(T\right)$ curves obtained under hydrostatic pressures up to $1.56$ GPa, the range over which the superconducting transition temperature, $T_{\mathrm{c}}$, of FeSe exhibits a nonmonotonic dependence with local maximum at $p_1\approx 0.8$ GPa and local minimum at $p_2\approx 1.2$ GPa. The slope of the upper critical field at $T_{\mathrm{c}},{\left(d{H}_{\text{c2,c}}/dT\right)}_{T_{\mathrm{c}}}$, also exhibits a nonmonotonic pressure dependence with distinct changes at $p_1$ and $p_2$. For $p<p_1$ the slope can be described within a multiband orbital model. For both $p_1<p<p_2$ and $p>p_2$ the slope is in good semiquantitative agreement with a single band, orbital Helfand-Werthamer theory with Fermi velocities determined from Shubnikov–de Haas measurements. This finding indicates that Fermi surface changes are responsible for the local minimum of $T_{\mathrm{c}}\left(p\right)$ at $p_2\approx 1.2$ GPa.

Figure 1

Evolution of the temperature dependence of the $c$-axis resistivity, ${\rho }_{\text{c}}$, with hydrostatic pressure. The inset presents the derivative of ${\rho }_{\text{c}}\left(T\right)$ taken at $1.28$ GPa, with arrows indicating the temperatures of the structural transition, $T_{\text{s}}$, and of the “unknown” transition, $T_{\text{u}}$, which is presumably associated with magnetic ordering.

Figure 2

Temperature dependence of the interplane resistivity, ${\rho }_{\text{c}}\left(T\right)$, taken in magnetic fields of the 0–9 T range in the $H\parallel c$ configuration at pressures of 0.48 GPa (a), 0.98 GPa (b), and 1.56 GPa (c). The upper critical field $H_{\mathrm{c}2,\mathrm{c}}$ is determined from the offset, $10\%, 50\%$, and $90\%$ of ${\rho }_{\text{c}}$ as shown schematically by the arrows in the bottom panel for the 9 T curve and unless otherwise stated, the offset criterion is used in this paper. The temperature of the unknown transition, $T_{\mathrm{u}}$, manifests a very weak dependence on external magnetic fields up to 9 T.

Figure 3

The temperature dependence of the upper critical field $H_{\mathrm{c}2,\mathrm{c}}\left(T\right)$ measured in $H\parallel c$ configuration under various pressures. The data are vertically offset to avoid overlapping, with horizontal lines showing $H=0$. Dashed lines represent upper critical field slopes calculated based on the Fermi velocity, $v_{\text{F}}$, from Ref. [30] for a cylindrical Fermi surface [39] (see text). At 0 GPa, two dotted and dashed lines represent calculated slopes for the largest ($\delta$), smallest ($\beta$), and average $v_{\text{F}}$ [$(\alpha ,\gamma )$ and $(\beta ,\delta )$]. Black dotted lines connect the 0 and 3 T points for various pressures.

Figure 4

(a) Temperature - pressure phase diagram of FeSe as determined from inter-plane resistivity measurements (full symbols) and from previous in-plane resistivity measurements (open symbols) [22, 30, 38]. The circles, triangles and squares represent the tetragonal/orthorhombic, unknown, most likely magnetic and superconducting phase transitions, respectively. The orange solid line is a guide for the eye. Vertical dotted lines show the pressures corresponding to the local maxima and minima of $T_{\text{c}}\left(p\right)$. (b) Pressure dependence of the normalized slope $(-1/T_{\mathrm{c}})(d{H}_{\text{c2},c}/dT){|}_{T_{\mathrm{c}}}$. An abrupt change of slope is observed near 0.8 GPa and 1.2 GPa, corresponding to the maximum and a minimum in $T_{\text{c}}\left(p\right)$ (left axis). For reference we show $v_{\mathrm{F}}^{-2}$ calculated for individual orbits in Shubnikov-deHaas effect measurements of Ref. [30] (right axis). There is clear proportionality between the normalized slope and $v_{\mathrm{F}}^{-2}$ as found in our data analysis for $p>p_1$, despite both slope and $v_{\mathrm{F}}$ showing nonmonotonic changes at $p_2$. Multiple orbits found for $p<p_1$ clearly illustrate the difficulty of $H_{\mathrm{c}2,\mathrm{c}}$ slope calculation in any simple model. (c) Pressure dependence of the normalized slope $(-1/T_{\mathrm{c}})(d{H}_{\text{c2},c}/dT){|}_{T_{\mathrm{c}}}$ for four different criteria; offset, $10\%, 50\%$ and $90\%$ of the resistivity. (d) The slope of the $H_{\mathrm{c}2,\mathrm{c}}$ line plotted vs $T_{\mathrm{c}}$ with pressure as a implicit parameter. The plot clearly reveals the nonmonotonic dependence with three ranges separated at pressures $p_1$ and $p_2$. Arrows indicate the direction of pressure increase.