Pairing interaction in superconducting UCoGe tunable by magnetic field

The mechanism of unconventional superconductivity, such as high-temperature-cuprate, Fe-based, and heavy-fermion superconductors, has been studied as a central issue in condensed-matter physics. Spin fluctuations, instead of phonons, are considered to be responsible for the formation of Cooper pairs, and many efforts have been made to confirm this mechanism experimentally. Although a qualitative consensus seems to have been obtained, experimental confirmation has not yet been achieved. This is owing to a lack of the quantitative comparison between theory and experiments. Here, we show a semiquantitative comparison between the superconducting-transition temperature ($T_{\mathrm{SC}}$) and spin fluctuations derived from the NMR experiment on the ferromagnetic (FM) superconductor UCoGe in which the FM fluctuations and superconductivity are tunable by external fields. The enhancement and abrupt suppression of $T_{\mathrm{SC}}$ by applied fields, as well as the pressure variation of $T_{\mathrm{SC}}$ around the FM criticality are well understood by the change in the FM fluctuations on the basis of the single-band spin-triplet theoretical formalism. The present comparisons strongly support the theoretical formalism of spin-fluctuation-mediated superconductivity, particularly in UCoGe.

Figure 1

Angle dependence of $1/T_1$ of ${}^{59}\mathrm{Co}$ on the $bc$ plane measured at 2.1 K for the determination of the $b$ axis. The peak angle was determined as the $b$ axis.

Figure 2

The AC susceptibility measured at various fields in (a) $H\parallel a$ and (b) $H\parallel b$. The inset of (a) shows the angle dependence of the decrease in ${\chi }_{ac}$ below $T_{\mathrm{SC}}$ measured at 3 T and 0.12 K. The angle with the largest decrease was determined as the $a$ axis. The inset of (b) shows the angle dependence of $T_{\mathrm{SC}}$ measured at 12.5 T. The angle with the highest $T_{\mathrm{SC}}$ was determined as the $b$ axis.

Figure 3

Upper critical field $H_{\mathrm{c}2}$ along each crystal axis. $H_{\mathrm{c}2}$ was determined using the AC susceptibility measurements. The inset shows the zoom of $H_{c2}$ along the $c$ axis.

Figure 4

Magnetic-field ($H$) dependence of (a) $1/T_{1}T$ and (b) $1/T_{2}T$ measured in $H\parallel a$ and $H\parallel b$. (c) The $H$ dependence of $G_b\left(0\right)$ and $G_c\left({\omega }_0\right)$ derived from $1/T_{1}T$ and $1/T_{2}T$ (see the text).

Figure 5

(a) Temperature dependence of $1/T_{1}T$ of ${}^{59}\mathrm{Co}$ measured at various magnetic fields from ${\mu }_{0}H=1.5$ to 23 T in $H\parallel b$. The phenomenological fitting of the experimental $1/T_{1}T$ is shown by the dotted curves with $\theta =2.5$, 0, and $-5\mathrm{K}$. (b) $H$ dependence of $T_{\mathrm{Curie}}$ determined by the peak or maximum of $1/T_{1}T$ measured in $H\parallel a$ and $H\parallel b$ in the present sample. The characteristic temperature $\theta \left(H\right)$ corresponding to the magnetic anomaly in the present sample is also plotted. The $T_{\mathrm{Curie}}$ determined by the bending of the electric resistance is also plotted for comparison.

Figure 6

The $H$ dependence of the difference of $T_{\mathrm{SC}}$ between $H\parallel b$ and $H\parallel a$ normalized by zero-field $T_{\mathrm{SC}}\left(0\right),[T_{\mathrm{SC}}\left(H^b\right)-T_{\mathrm{SC}}\left(H^a\right)]/T_{\mathrm{SC}}\left(0\right)\equiv \delta T_{\mathrm{SC}}\left(H^b\right)$ on the present sample and the $H$ dependence of the normalized difference of $T_{\mathrm{SC}}$ evaluated from the previous reports [9, 16] are also plotted for the left axis, and the vertical scale of the results reported by Wu et al. [16] is reduced to half for the comparison. The $H$ dependence of $[T_1(H\sim 0)T-T_1\left(H^b\right)T]/T_1(H\sim 0)T\equiv \delta T_1\left(H^b\right)T$ measured at 1.5 K on the present sample is plotted on the right axis.

Figure 7

The $H$ dependence of $[T_{\mathrm{SC}}\left(H^c\right)-T_{\mathrm{SC}}\left(0\right)]/T_{\mathrm{SC}}\left(0\right)\equiv \delta T_{\mathrm{SC}}\left(H^c\right)$ on the present sample, and its $H$ dependence evaluated from previous reports is also plotted for the left axis [12, 16]. The $H$ dependence of $[T_1(H\sim 0)T-T_1\left(H^c\right)T]/T_1(H\sim 0)T\equiv \delta T_1\left(H^c\right)T$ measured at 1.5 K reported by our group is plotted on the right axis [12]. The $H^c$ dependence of $1/T_{1}T$ is evaluated from the angle dependence of $1/T_{1}T$ on the $bc$ plane.

Figure 8

The pressure ($P$) dependence of $[T_{\mathrm{SC}}\left(P\right)-T_{\mathrm{SC}}\left(0\right)]/T_{\mathrm{SC}}\left(0\right)\equiv \delta T_{\mathrm{SC}}\left(P\right)$ in the zero field is plotted on the left axis [29, 30]. The $P$ dependence of $[T_1\left(0\right)T-T_1\left(P\right)T]/T_1\left(0\right)T\equiv \delta T_1\left(P\right)T$ measured at 1 K on the same sample of the $P$ dependence of $T_{\mathrm{SC}}$ is plotted on the right axis [29, 30]. The inset shows the $P$ dependence of $1/T_{1}T$ measured at 1 K. Paramagnetic (PM) (FM) means $1/T_{1}T$ in the paramagnetic (ferromagnetic) state.