Anomalous metallic state and anisotropic multiband superconductivity in Nb${}_3$Pd${}_{0.7}$Se${}_7$

We report the discovery of superconductivity in Nb${}_3$Pd${}_x$Se${}_7$ with an $x$-dependent superconducting transition temperature as high as $T_c\simeq 2.1$ K for $x\simeq 0.7$ (middle point of the resistive transition). Needlelike single crystals display anisotropic upper-critical fields with an anisotropy $\gamma =H_{c2}^b/H_{c2}^a$ as large as 6 between fields applied along their needle axis (or $b$ axis) or along the $a$ axis. As for the Fe based superconductors $\gamma$ is temperature-dependent, suggesting that Nb${}_3$Pd${}_{0.7}$Se${}_7$ is a multiband superconductor. This is supported by band structure calculations which reveal a Fermi surface composed of quasi-one-dimensional and quasi-two-dimensional sheets of hole character, as well as three-dimensional sheets of both hole and electron character. Remarkably, $H_{c2}^b$ is observed to saturate at $H_{c2}^b(T\to 0\text{K})\simeq 14.1$ T which is $4.26\times H_p$ where $H_p$ is the Pauli-limiting field in the weak-coupling regime. The synthesis procedure yields additional crystals belonging to the Nb${}_2$Pd${}_x$Se${}_5$ phase which also becomes superconducting when the fraction of Pd is varied. For both phases we find that superconductivity condenses out of an anomalous metallic state, i.e., displaying $\partial \rho /\partial T<0$ above $T_c$ similarly to what is observed in the pseudogap phase of the underdoped cuprates. An anomalous metallic state, low-dimensionality, multiband character, extremely high and anisotropic $H_{c2}$'s are all ingredients for unconventional superconductivity.

Figure 1

(a) Resistivity $\rho$ normalized by its value at room temperature $\rho \left(300\text{K}\right)$ as a function of the temperature $T$ for three Nb${}_3$Pd${}_x$Se${}_7$ single crystals, i.e., $x=0.676\left(2\right)$ (blue line), 0.68(1) (green line), and $\simeq 0.7$ (gray line), respectively. The dimensions of the superconducting crystal are $\sim 1\times {10}^{-4}\times 8.5\times {10}^{-3}\times 0.2$ cm${}^{-3}$, leading to a room temperature resistivity of $\sim 70$ $\mu \Omega$cm (for a contacts separation of $\sim 0.05$ cm). Inset: $\rho /\rho \left(300\text{K}\right)$ as a function of $T$ in a linear temperature scale. Red lines are fits to the power law $\rho ={\rho }_0+A{T}^n$ within the range $150\le T\le 300$ K. The superconducting compound displays Fermi-liquid-like behavior or $n=2$. (b) $\rho /\rho \left(300\text{K}\right)$ after subtraction of the aforementioned power law. Notice a sharp anomaly at $T^{☆}\sim 110$ K which is independent on the Pd content $x$, but whose amplitude is seen to decrease as $x$ increases. This anomaly of unknown origin is followed by a minimum in the resistivity at $T_{\text{min}}\sim 24$ K, suggesting electronic localization preceding the superconducting transition. (c) and (d) Resistivity $\rho /\rho \left(300\text{K}\right)$ as a function of the temperature $T$ for two Nb${}_2$Pd${}_x$Se${}_5$ single crystals, i.e., $x=0.7$ and 0.67, respectively. Notice the observation of an upturn in the resistivity at a Pd-dependent temperature, indicating again an interplay with some form of electron localization. The characteristic temperature $T_{\text{min}}$ below which $\rho$ becomes nonmetallic decreases as the Pd content decreases. The dimensions of the Nb${}_2$Pd${}_{0.67}$Se${}_5$ single crystal are $\sim 1.75\times {10}^{-4}\times 3.4\times {10}^{-3}\times 0.235$ cm${}^{-3}$, leading to a room temperature resistivity of $\sim 225$ $\mu \Omega$cm. Red lines are fits to the power-law behavior observed in the metallic state, which yield non-Fermi-liquid-like exponents $n$ ranging from $\sim 1$ (nonsuperconducting) to $\sim 0.4$ (in superconducting samples).

Figure 2

(a) Magnetic susceptibility $\chi$ as a function of the temperature for hundreds of single-crystals from a typical synthesis batch. Vertical arrow indicates the deviation from linearity at $T\sim T^{☆}$ where an anomaly is observed in the resistivity of Nb${}_3$Pd${}_x$S${}_7$ single crystals. Inset: ${\chi }^{-1}$ as a function of $T$ at lower temperatures. (b) Heat capacity normalized by $T$ and as a function of $T$ for hundreds of single-crystals from a typical synthesis batch containing both phases. One observes a broad anomaly centered around $T=0.75$ K, which disappears with the application of a field of 9 T, indicating bulk superconductivity.

Figure 3

(a) From left to right, distinct perspectives of the crystallographic structure of Nb${}_3$Pd${}_{0.7}$Se${}_7$: along the $a$, $b$, and nearly along the $c$ axis, respectively. The Nb(1), Nb(2), and Nb(3) atoms are depicted in turquoise, gray, and magenta colors, respectively. Pd(1) and Pd(2) atoms are depicted in maroon and dark blue colors, whereas all the Se atoms are depicted in clear blue color, respectively.

Figure 4

(a) Resistivity $\rho$ as a function of the field $H$ applied along the $a$ axis of a Nb${}_3$Pd${}_{0.7}$Se${}_7$ single crystal for several temperatures. (b) Same as in (a) but for fields along the $c$ axis and at pumped ${}^3$He temperatures. (c) Same as in (a) but for fields along the $b$ axis.

Figure 5

Upper panel: phase boundary between superconducting and metallic states in the temperature $\left(T\right)$–magnetic field $\left(H\right)$ plane for Nb${}_3$Pd${}_{0.7}$Se${}_7$. Blue markers depict the phase boundary for fields applied along the needle, or the $b$ axis of the crystal. Brown and green markers depict the phase boundary for fields applied along the $c$ axis and the $a$ axis, respectively. Lower panel: anisotropy in upper-critical fields $\gamma =H_{c2}^b/H_{c2}^a$ as a function of temperature. Notice that $\gamma$ is temperature dependent as previously observed in multiband superconductors.

Figure 6

Upper panel: $H_{c2}$ for Nb${}_3$Pd${}_{0.7}$Se${}_7$ as a function of $1-t$, where $t=T/T_c$, for fields applied along the $b$ axis (blue dots), $c$ axis (brown dots), and $a$ axis (green dots), respectively. Red lines are fits to Eq. (1). Lower panel: $H_{c2}^b$ for fields along the $b$ axis normalized by the linear slope at $T_c$ and as a function of $t$. Red line is a fit to the WHH formalism.

Figure 7

(a) Perspective of the geometry of the Fermi surface of Nb${}_3$Pd${}_{0.7}$Se${}_7$ in a $2\times 2\times 2$ conventional reciprocal unit cell as obtained through DFT calculations. (b) Same as in (a) but through a lateral perspective. Maroon and blue FS sheets are holelike, while the gold intersecting cylinders are electronlike in one direction and holelike in the other. Notice that the FS is composed of both quasi-two-dimensional nearly cylindrical sheets as well as highly corrugated quasi-one-dimensional ones. The Q1D surfaces are extremely sensitive to the Fermi energy, which is in turn sensitive to the exact Pd content.