Temperature dependence of the upper critical field of FeSe single crystals

We present a careful study of the resistive superconducting transition in FeSe single crystals down to $T=40$ mK in continuous magnetic fields up to 30 T applied perpendicular and parallel to the $ab$ plane. In the $\mathbf{H}\parallel \mathbf{c}$ geometry the temperature dependence of the resistive upper critical field $H_{c2}^{*}$, determined as the field at which the in-plane resistivity in the transition region is 90$\%$ of the normal state resistivity is down to temperatures $T/T_c<0.006$, is in close agreement with the Werthamer-Helfand-Hohenberg (WHH) theoretical curve which describes the behavior of the upper critical field in conventional type-II superconductors. In contrast, for the $\mathbf{H}\parallel \mathbf{ab}$ geometry, the data depart from the WHH model with increasing applied magnetic field according to the paramagnetic limitation of superconductivity. An anisotropy parameter $\gamma$ in our FeSe crystals decreases with decreasing temperature and FeSe becomes nearly isotropic when the temperature $T\to 0$.

Figure 1

Top: scanning electron micrographs of as-grown FeSe${}_{0.9}$ crystals with different shapes where the composition measurement points are denoted by crosses. Bottom: (a) x-ray diffraction pattern showing the peaks of the tetragonal (T), and hexagonal (H) FeSe phases. X-ray diffraction pattern (b) shows only the (00N) peaks of the tetragonal phase.

Figure 2

Temperature dependence of the in-plane resistivity (${\rho }_{ab}$) as a function of temperature for FeSe samples with different tetragonal phase concentrations. The arrow indicates the tetragonal-to-orthorhombic structural transition. The inset (a) shows an enlarged view of the resistivity near $T_c$. The inset (b) shows the critical temperature for superconductivity ($T_c$) determined from ${\rho }_{ab}$ as a function of the percent concentration of the tetragonal FeSe phase.

Figure 3

The zero-field ${\rho }_{ab}$ and ${\rho }_c$ as a function of the temperature for two samples No. 1 and No. 2. The ${\rho }_c$ data are divided by 3 and 4.

Figure 4

Temperature dependence of the in-plane ${\rho }_{ab}$ resistivity for crystal No. 6 with $T_c$ (midpoint) = 10.3 K for applied magnetic fields up to 9 T with direction perpendicular to the $ab$ plane of the crystal.

Figure 5

Resistive transitions of crystal No. 2 in magnetic field $\mathbf{H}\parallel \mathbf{c}$ (a) and $\mathbf{H}\parallel \mathbf{ab}$ (b) at different temperatures.

Figure 6

Temperature dependence of $H_{c2}^{*}$ for sample No. 2 extracted from the MR curves in Fig. 5a measured in the $\mathbf{H}\parallel \mathbf{c}$ geometry together with the theoretical WHH curves. The “data points” shown by open circles were obtained from Fig. 5a by a linear extrapolation of ${\rho }_{ab}\left(H\right)$ to ${\rho }_{ab}\left(T\right)={\rho }_n$. The red dashed lines and the arrows in the insets show the methods used to define $H_{c2}^{*}$ from the measured ${\rho }_{ab}\left(H\right)$ curves (see the text).

Figure 7

Semilogarithmic plot of ${\rho }_{ab}\left(T\right)$ at different fixed magnetic fields $\mathbf{H}\parallel \mathbf{c}$ for sample No. 2 extracted from the curves in Fig. 5a. Between 0 and 30 T, the applied magnetic field was increased in steps of 2.5 T.

Figure 8

Reduced critical field $h_{c2}=H_{c2}^{*}/{(-d{H}_{c2}^{*}/dT)}_{T_c}{T}_c$ as a function of the reduced temperature $T/T_c$ for two samples No. 2 and No. 6 at field orientation $\mathbf{H}\parallel \mathbf{c}$ together with the theoretical WHH curve (dashed lines). The $H_{c2}^{*}$ values are the fields at which the resistivity of the samples has reached 100$\%$ (a) and 90$\%$ (b) of its normal-state values ${\rho }_n$ and obtained by two methods determination of $H_{c2}^{*}$ (see the insets to Fig. 6).

Figure 9

Temperature dependence of $H_{c2}^{*}$ for sample No. 2 extracted from the MR curves in Fig. 5b measured in the $\mathbf{H}\parallel \mathbf{ab}$ geometry together with the theoretical WHH curves (dashed lines). The “data points” shown by open circles were obtained from Fig. 5b by a linear extrapolation of ${\rho }_{ab}\left(H\right)$ to ${\rho }_{ab}\left(T\right)={\rho }_n$.

Figure 10

Reduced critical field $h_{c2}=H_{c2}^{*}/{(-d{H}_{c2}^{*}/dT)}_{T_c}{T}_c$ as a function of the reduced temperature $T/T_c$ for samples No. 1 and No. 2 at field orientation $\mathbf{H}\parallel \mathbf{ab}$ together with the theoretical WHH curve. The $H_{c2}^{*}$ values are the fields at which the resistivity of the samples has reached 100$\%$ (a) and 90$\%$ (b) of its normal-state values ${\rho }_n$ and obtained by two methods determination of $H_{c2}^{*}$ (see the insets to Fig. 6). The inset in (a) shows the temperature dependence of the anisotropy parameter $\gamma =H_{c2\parallel ab}^{*}/H_{c2\parallel c}^{*}$ for the sample No. 2, obtained from the 100$\%$ and 90$\%$ criteria by the method 1.