Superconductivity induced by doping holes in the nodal-line semimetal NaAlGe

The nodal-line semimetals NaAlSi and NaAlGe have significantly different ground states despite having similar electronic structures: NaAlSi exhibits superconductivity $<7\mathrm{K}$, while NaAlGe exhibits semiconductive electrical conductivity at low temperatures, indicating the formation of a pseudogap at $\sim 100\mathrm{K}$. The origin of the pseudogap in NaAlGe is unknown but may be associated with excitonic instability. We investigated hole-doping effects on the ground state in the solid solution $\mathrm{Na}\left({\mathrm{Al}}_{1–x}{\mathrm{Zn}}_x\right)\mathrm{Ge}$ and discovered that the pseudogap is suppressed continuously with increasing Zn content, followed by the appearance of a superconducting dome with the highest transition temperature of 2.8 K. This superconductivity most likely results from excitonic fluctuations.

Figure 1

(a) Relationship between the actual amount of Zn substitution $x$ as determined by wavelength-dispersive x-ray (WDX) measurements and the nominal substitution $x_{\mathrm{n}}$ in preparations. The slope of the solid line is 0.35(1). (b) Powder x-ray diffraction (XRD) patterns of $\mathrm{Na}\left({\mathrm{Al}}_{1_{-}x}{\mathrm{Zn}}_x\right)\mathrm{Ge}$. (c) Composition dependences of the lattice constants. The solid lines are linear fits to the data points: $a=-0.25\left(2\right)x+4.166\left(1\right)$ and $c=0.56\left(3\right)x+7.413\left(1\right)$.

Figure 2

Temperature dependence of electrical resistivity for two samples of NaAlGe. The black filled circles represent data for a sample covered with liquid paraffin (left axis), whereas the red open triangles represent data for a sample without liquid paraffin (right axis). The inset shows a magnified view of the low-temperature region, where a slight decrease due to partial superconductivity is likely observed only for the sample without paraffin coating.

Figure 3

(a) Evolution of the temperature-dependent electrical resistivity ρ, normalized to $T=3.5\mathrm{K}$, as a function of the Zn content. (b) Temperature dependences of magnetic susceptibility χ. Most data curves depicted by filled circles were recorded in a magnetic field of $H=5\mathrm{Oe}$ after cooling to 1.8 K in zero magnetic field, while the red open circles depicted field-cooled data for the $2.8\%$ sample in the same field. The inset magnifies the datasets $<2.5\mathrm{K}$. (c) Temperature dependence of $C_{\mathrm{e}}/\gamma T$ for the 1.5, 2.8, and $3.7\%$ substituted samples, where $C_{\mathrm{e}}$ and γ are, respectively, the electronic part of the heat capacity and the Sommerfeld coefficient. The broken line shows the approximate, ideal variation expected for the $2.8\%$ curve, considering transition broadening and entropy balance.

Figure 4

$C/T$ vs $T^2$ plots for $0\%$ (black circles), $1.5\%$ (magenta squares), $3.7\%$ (blue triangles), $5.2\%$ (green pentagons) polycrystalline samples, and $0\%$ single crystal (filled black circles) [34]. The straight lines are fits to the equation $C\left(T\right)=\gamma T+\beta T^3$.

Figure 5

Hole-doping dependence of $T_{\mathrm{c}}$ derived from electrical resistivity (blue circles), magnetic susceptibility (magenta triangles), and heat capacity (green diamonds) measurements (Fig. 3). The samples for which superconductivity was not observed $>1.8\mathrm{K}$ are represented by open symbols with arrows pointing downward. The solid lines serve as visual aids.

Figure 6

(a) Temperature dependence of electrical resistivity for all composition samples from 2 to 300 K. (b) Temperature dependence of $E_{\rho }$, defined as $k_{\mathrm{B}}{T}^2\left(\partial l\mathrm{n}\rho /\partial T\right)$, for $0–2.8\%$ substituted samples. The temperatures depicted by the arrows correspond to the pseudogap temperature $T^{*}\left(\rho \right)$. In both (a) and (b), only the data from the pure sample are plotted along the right vertical axis.

Figure 7

(a) Temperature dependence of magnetic susceptibility measured at $B=7\mathrm{T}$ after the subtraction of nuclear diamagnetic contributions for samples of various compositions. (b) Temperature derivative of magnetic susceptibility dχ/$\text{d}T$ for $0–2.8\%$ substituted samples. Offsets were added for clarification. The maximum temperature indicated by the arrow is the pseudogap temperature $T^{*}\left(\chi \right)$.

Figure 8

$T-x$ phase diagram of $\mathrm{Na}\left({\mathrm{Al}}_{1_{-}x}{\mathrm{Zn}}_x\right)\mathrm{Ge}$. $T^{*}$ and $T_{\mathrm{c}}$ (multiplied by a factor of 5) from electrical resistivity (circles), magnetic susceptibility (triangles), and heat capacity measurements (diamonds) are plotted as a function of $x$. The pseudogap and superconducting phases are denoted by PG and SC, respectively.

Figure 9

(a) Doping dependences of the magnetic susceptibility at $T=2\mathrm{K}$ after subtraction of the nuclear diamagnetic contribution ${\chi }_{\mathrm{core}}$ and the Sommerfeld coefficient ${\gamma }_{\exp }$ on the left vertical and right vertical axes, respectively. The ${\chi }_{\mathrm{band}}$ and ${\gamma }_{\mathrm{band}}$ obtained from the band calculations are shown by the red and blue dotted lines, respectively. (b) Doping dependence of the Wilson ratio $R_{\mathrm{W}}$.