Single crystal growth and superconductivity in RbNi₂Se₂

Since the discovery of copper oxide superconductors, researchers have extensively searched for superconductivity in materials with transition metalsBednorz et al. (1987); Takada et al. (2003). Significant progress has been made in iron pnictide and chalcogenide compounds, where several structural systems have been identified with the highest $T_{c}$ of 55 K achieved in LaFeAsORen et al. (2008). Superconductivity was also observed in the compounds consisting of Cr and Mn under pressure, such as CrAs, KCrAs, and MnPWu et al. (2014); Mu et al. (2017); Cheng et al. (2015). Among all of them, superconductivity has been found to be in the vicinity of an antiferromagnetic (AF) order, suggesting that spin fluctuations may play an important role in the mechanism of superconductivity. Nickel oxide materials have analogous structures with copper oxide superconductors. Superconductivity with $T_{c}=9-15$ K has also been observed in films of nickel-based compoundsLi et al. (2019); Gu and Wen (2022).

$A$Ni₂Se₂ ($A=$ K, Cs, and Tl) crystalizes in the ThCr₂Si₂ structure and shows metallic behavior and Pauli paramagnetism. At low temperatures, superconductivity emerges with the superconducting (SC) transition temperatures of $T_{c}$ $\approx$ 0.8 K for KNi₂Se₂ polycrystals, 2.7 K for CsNi₂Se₂, and 3.7 K for TlNi₂Se₂ single crystalsNeilson et al. (2012); Chen, Huimin and Yang, Jinhu and Cao, Chao and Li, Lin and Su, Qiping and Chen, Bin and Wang, Hangdong and Mao, Qianhui and Xu, Binjie and Du, Jianhua and Fang, Minghu (2016); Wang et al. (2013). While K_0.95Ni_1.86Se₂ single crystals do not show superconductivity down to 0.3 K, yielding that the superconductivity is sensitive to the stoichiometry of the samplesLei et al. (2014). As a comparison, A_xFe₂Se₂ system exhibits $T_{c}$s ranging from 20 $-$ 30 K. With different amount of iron vacancies, A_xFe_2-δSe₂ exhibits a variety of AF orders and iron vacancy ordersGuo et al. (2010); Fang et al. (2013); Dai (2015); Wang et al. (2016). The replacement of Fe by Co suppresses the superconductivity and induces a ferromagnetic (FM) order in RbCo₂Se₂Yang et al. (2013); Huang et al. (2021). The $A$Ni₂Se₂ superconductors with a formal valence of Ni^1.5+ have been revealed to exhibit remarkable properties. In particular, they usually exhibit a large Sommerfeld coefficient $\gamma$, suggesting a large density of states and unconventional pairing at low temperatureWang et al. (2013); Chen, Huimin and Yang, Jinhu and Cao, Chao and Li, Lin and Su, Qiping and Chen, Bin and Wang, Hangdong and Mao, Qianhui and Xu, Binjie and Du, Jianhua and Fang, Minghu (2016). One possibility that was proposed was that the large Sommerfeld coefficient might be induced by local charge orderNeilson et al. (2012). However, angle resolved photoemission spectroscopy (ARPES) measurements yield weak electronic correlations in KNi₂Se₂ and the origin of the large Sommerfeld coefficient may be driven by the large density states and the Van Hove singularity in the vicinity of the Fermi energyFan et al. (2015).

In this work, we report the successful synthesis and characterization of RbNi₂Se₂ single crystals. The crystal structure, electronic band structure, and transport properties have been investigated. We find that RbNi₂Se₂ is a Pauli paramagnetism and exhibits a SC transition at $T_{c}$ = 1.20 K. Normal-state specific heat measurements suggest an effective electronic mass enhancement with $m^{*}\approx 6m_{e}$. In SC state, a two-gap BCS model can match well with the zero-field electronic specific heat, indicating that RbNi₂Se₂ is a multi-gap superconductor. Comparison with the density functional theory (DFT) calculations and ARPES measurements reveals that RbNi₂Se₂ is a weakly correlated superconductor with multi bands crossing the Fermi level.

Single crystals of RbNi₂Se₂ were grown by the self-flux method. First, the precursor NiSe was prepared by heating Ni powders and Se pellets at $500^{\circ}$C. Then, NiSe powders and Rb were put into alumina crucibles according to stoichiometry and sealed in an evacuated silica tube. The mixture was kept at $150^{\circ}$C for 5 h, then heated to $760^{\circ}$C in 40 h and kept for 5 h, after that heated to $1050^{\circ}$C in 40 h and held for 5 h. Finally, the temperature was cooled down to $700^{\circ}$C at a rate of $3.5^{\circ}$C/h. To prevent the reaction of Rb with water and air, all of the processes were conducted in an argon-filled glove box. Shiny plate-like single crystals with a typical size of 4 $\times$ 5 $\times$ 1 mm³ were grown as shown in the inset of Fig. 1(b).

Single crystal x-ray diffraction (XRD) were conducted on a SuperNova (Rigaku) x-ray diffractometer. The sample was blowed by N₂ during the data collection to avoid exposure to air. The elemental analysis was measured by using an energy-dispersive x-ray spectroscopy (EDS) (EVO, Zeiss). Electrical transport, magnetic and specific heat measurements were performed on a physical property measurement system (PPMS, Quantum Design). The in-plane resistivity $\rho_{ab}(T)$ was measured using the standard four-probe method on a rectangular sheet crystal to keep current flowing in the ab-plane. The Vienna Ab initio Simulation Package (VASP) was employed for the DFT calculationsKresse and Hafner (1993). ARPES measurements were performed on a helium-lamp based system with a DA30 electron analyzer. Single crystals were cleaved in-situ in ultra-high vacuum with a base pressure better than $5\times 10^{-11}$ Torr at 30 K. Energy and angular resolutions were better than 20 meV and $0.1^{\circ}$, respectively.

All peaks from single crystal XRD can be indexed with the ThCr₂Si₂-type structure (space group: I4/mmm), which is illustrated in the inset of Fig. 1(a). The determined lattice parameters are $a=b=3.9272(3)$, and $\textit{c}=13.8650(5)$ Å at 150 K with the volume of unit cell between that of KNi₂Se₂ and CsNi₂Se₂. Details of the atom coordinates and other key information are shown in Table 1. To show the quality of the samples, we present a $\theta$$-$2$\theta$ scan of a single crystal along the $(H=0,K=0,L)$ in Fig. 1(a) and an XRD pattern on a powder sample in Fig. 1(b), where $(H,K,L)$ are Miller indices in reciprocal lattice units. No peaks from impurity could be identified. The EDS results for several single crystals are rather homogenous and the determined average atomic ratios are Rb:Ni:Se = 1.16(4):2.04(3):2.00(6) when the content of Se is normalized to be 2, close to the stoichiometry of RbNi₂Se₂.

Temperature dependence of the resistivity for RbNi₂Se₂ is shown in Fig. 2(a). The electric current is applied in the ab-plane. The value of $\rho_{ab}$ is about 92.1 $\mu\Omega\cdot$cm at 300 K and only about 1.5 $\mu\Omega\cdot$cm at 1.8 K. The residual resistivity ratio (RRR) of 61.4 [$\rho_{ab}(300\ \text{K})/\rho_{ab}(1.8\ \text{K})$] suggests remarkable metallicity and high quality of the single crystalsLei et al. (2014); Ehrlich and Rivier (1968); Böhmer et al. (2016). The resistivity measured at 0, 0.15, and 11 T shows a metallic behavior without any anomaly or strong magnetic field dependence. The resistivity $\rho_{ab}$ below 40 K can be well described by the equation $\rho_{ab}($T$)=\rho_{0}+A$T² as shown in the inset of Fig. 2(a), where $\rho_{0}$ = 1.211 $\mu\Omega\cdot$cm and A = 0.012 $\mu\Omega\cdot$cm/K², revealing a paramagnetic Fermi liquid behaviorAnalytis et al. (2014). The magnetic susceptibility is nearly independent of temperature, yielding a Pauli paramagnetic behaviour as shown in Fig. 2(b). A weak FM sign is revealed from the hysteresis loop shown in the inset of Fig. 2(b), which could be ascribed to a small amount of Ni impurityLi et al. (2020). The specific heat from 2 to 200 K shown in Fig. 2(c) also suggests that no phase transition occurs in this temperature range.

To explore the possible superconductivity, we show measurements down to 50 mK in Fig. 3. The magnetic susceptibility at low temperatures is shown in Fig. 3(a). A clear diamagnetic response appears below 1.20 K under zero field, indicating a SC transition. The transition shifts to lower temperatures with an increase of the $dc$ bias field, consistent with the Meissner effect of superconductivity. With knowing $T_{c}$ $\approx$ 0.8 K for KNi₂Se₂Neilson et al. (2012) and $T_{c}\approx$ 2.7 K for CsNi₂Se₂Chen, Huimin and Yang, Jinhu and Cao, Chao and Li, Lin and Su, Qiping and Chen, Bin and Wang, Hangdong and Mao, Qianhui and Xu, Binjie and Du, Jianhua and Fang, Minghu (2016), the diamagnetic response at 1.20 K should correspond to the SC transition of RbNi₂Se₂. We plot the onset SC transition temperatures at various magnetic fields and fit the upper critical field $\mu_{0}H_{c2}(0)$ using the Ginzburg-Landau theory with the formula $\mu_{0}H_{c2}(T)=\mu_{0}H_{c2}(0)\times(1-t^{2})/(1+t^{2})$, where $t$ is the reduced temperature $t=T/T_{c}$. As shown in Fig. 3(b), the resultant $\mu_{0}H_{c2}(0)$ = 1.38 T is within the Pauli limit, $\mu_{0}H^{p}_{c2}(0)$ = 2.37 T, indicating a weak coupling behaviorBao et al. (2015).

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Figure 3(c) displays the specific heat measured at low temperatures. In normal state, a fit to the specific heat $C_{p}$ from 1.5 to 3.8 K using $C_{p}/T$ = $\gamma_{N}$+$\beta T^{2}$ results in the Sommerfeld coefficient $\gamma_{N}$ = 30.30 mJ$\cdot$mol^-1K^-2 and $\beta$ = 3.67 mJ$\cdot$mol^-1K^-4, as shown in the inset. The Debye temperature $\Theta_{D}$ is estimated to be of 167 K from the equation $\Theta_{D}$ = $(12\pi^{4}NR/5\beta)^{1/3}$, where $N=5$ is the atomic number in each formula unit and $R$ is the ideal gas constant. The $\gamma_{N}$ for RbNi₂Se₂ is comparable with that of KNi₂Se₂ ($\sim$44 mJ$\cdot$mol^-1K^-2)Neilson et al. (2012), CsNi₂Se₂ ($\sim$77 mJ$\cdot$mol^-1K^-2)Chen, Huimin and Yang, Jinhu and Cao, Chao and Li, Lin and Su, Qiping and Chen, Bin and Wang, Hangdong and Mao, Qianhui and Xu, Binjie and Du, Jianhua and Fang, Minghu (2016), and TlNi₂Se₂ ($\sim$40 mJ$\cdot$mol^-1K^-2)Wang et al. (2013).The effective mass of electrons $m^{*}$ can be estimated through Eq. 1DeLong et al. (1985):

where $k_{B}$ is the Boltzmann constant and the carrier density $n$ is calculated by the number of electrons ($Z$) per cell volume ($V$). Using a spherical Fermi surface approximation, the Fermi wave vector can be estimated by $k_{F}$ = $(3\pi^{2}n)^{1/3}$. Assuming that Ni contributes 1.5 electrons ($Z$ = 6), we obtain $k_{F}$ = $9.4\times 10^{9}$ m^-1. The estimated effective mass of electrons $m^{*}/m_{e}$ = 6 for RbNi₂Se₂ is significantly enhanced compared with the bare electron mass $m_{e}$. Combining the parameters from fitting the electronic specific heat and the quadratic temperature dependent regime of resistivity, the Kadowaki-Woods ratio, $A$/$\gamma_{N}^{2}$, is calculated to be 0.94 $\times$ $10^{-5}\mu\Omega\cdot$cm(mol$\cdot$K²mJ)², close to $\sim$10${}^{-5}\mu\Omega\cdot$cm(mol$\cdot$K²mJ)² of heavy fermion systemsKadowaki and Woods (1986). This scaling relation yields RbNi₂Se₂ with the heavy-fermion behavior.

In SC state, the specific heat data reveals a clear $\lambda$-anomaly under zero field with a maximum at 0.94 K [Fig. 3(c)], suggesting bulk superconductivity. Applying an external magnetic field, the SC transition moves quickly to lower temperatures. As shown in the inset of Fig. 3(c), an upturn appears on $C_{p}/T$ below $T^{2}<0.1$ K². The small upturn could be described by the Schottky anomaly for paramagnetic impurity spinsNeilson et al. (2012), which can be well fitted by $C_{N}$ = $D(H)T^{-2}$ with $D(H=0)$ = 0.07 mJ$\cdot$K$\cdot$mol^-1.

In order to get information about the SC gap, we extract the electronic specific heat $C_{es}$ by subtracting the phonon contribution $C_{lattice}$ and Schottky anomaly $C_{N}$ from the total $C_{p}$, $C_{es}=C_{p}-C_{lattice}-C_{N}$. The electronic specific heat data $\textit{C}_{es}(\gamma_{N}\textit{T})^{-1}$ against $T/T_{c}$ are shown in Fig. 3(d), where $\gamma_{N}$ has been revealed in the normal state. The $T_{c}$ is determined to be 0.94 K using an equal-area entropy construction. The heat jump at transition is 0.63, smaller than that of the theoretical value of 1.43 in the BCS weak-coupling scenarioDaams and Carbotte (1981). We employ both one-gap and two-gap BCS models $C_{es}=\sum C_{i}exp(-\Delta_{i}/k_{B}T$) to fit the electronic specific heat, where $\Delta_{i}$ is the size of the $i^{th}$ SC gap at 0 K. The one-gap model reveals $\Delta_{0}/k_{B}T_{c}$ = 0.30 that is smaller than the value of 1.76 in the BCS theoryBardeen et al. (1957). In the two-gap model, the total specific heat can be considered as the sum of electronic contributions from two bands. Our fitting yields the sizes of two gaps of $\Delta_{1}/k_{B}T_{c}$ = 2.51 and $\Delta_{2}/k_{B}T_{c}$ = 0.25, respectively. The ratio of the contributions from the two gaps is $\sim 1.5:1$ as presented by the dashed lines in Fig. 3(d). The better fitting using the two-gap model indicates that RbNi₂Se₂ may be a multi-gap superconductor.

The electron-phonon coupling strength is also calculated by employing the inverted McMillan formulaMcMillan (1968):

where $\mu^{*}$ represents the Coulomb repulsion pseudopotential, which we adopt $\mu^{*}$ = 0.13 for this systemAmon et al. (2018). Generally, the $\lambda_{ep}$ for strongly coupled superconductors are close to 1, and $\lambda_{ep}\rightarrow$ 0.5 is viewed as weak coupled superconductorsMcMillan (1968). The $\lambda_{ep}$ for RbNi₂Se₂ is 0.49, suggesting that RbNi₂Se₂ is a weakly coupled superconductor, consistent with the specific heat analysis.

To check the multi-band character, we conducted DFT calculations and ARPES measurements on the electronic band structure. Figures 4(a) and 4(b) show the calculated electronic bands of Ni ions and the integrated density of states of Ni, Rb, and Se. The electronic states near the Fermi surface are governed by the $3d$ orbitals of Ni ions. In Fig 4(c), the measured electronic structure of RbNi₂Se₂ is compared with the band structure obtained from the DFT calculations. The calculated curves (black lines) are scaled and overlaid onto the photoemission intensity along the $\Gamma-M$ direction. The theoretical and experimental data match qualitatively with a renormalization factor of about 1.8. This factor is relatively small compared to other iron-based superconductorsYi et al. (2017), indicating that the electronic correlations are moderate. The moderate correlations suggest that the superconductivity is most likely originated from the electron-phonon coupling, similar to the two-gap superconductor MgB₂Bouquet et al. (2001a, b). Figure 4(d) shows the photoemission intensity map of the Fermi surface, overlaid with the DFT calculated Fermi surfaces. The multiple Fermi pockets observed are consistent with the multi-band behavior of this compound, reinforcing the condition for the two-gap feature in the SC state. The band structure is reminiscent of the closely-related compound RbCo₂Se₂Huang et al. (2021). However, the $3d_{x^{2}+y^{2}}$ flat band that induces itinerant ferromagnetism in RbCo₂Se₂ is observed here to be well below the Fermi level, due to the electron doping resulted from the replacement of Co by Ni.

As $A$Ni₂Se₂ ($A=$ K, Cs, and Tl) superconductors, RbNi₂Se₂ also exhibits an enhanced effective electron mass with a large Sommerfeld coefficient from specific heat. However, ARPES data reveal that the electronic correlation is not strong. To understand the reason of the large Sommerfeld coefficient, $\gamma$ is estimated with $\lambda_{ep}$ and $N$(0) at the Fermi level through the relationshipXiao et al. (2021):

From the DOS in Fig. 4(b), $N(E_{F})$ is estimated to be 11.68 states/eV per formula, resulting in $\gamma$ = 40.98 mJ$\cdot$mol^-1K^-2 that is close to the experimental value of 30.30 mJ$\cdot$mol^-1K^-2. Therefore, the result suggests that the large $\gamma_{N}$ is related to the large DOS at the Fermi level as proposed in KNi₂Se₂Fan et al. (2015), instead of the heavy fermion state. For $A$Ni₂Se₂ ($A$ = K, Rb, and Cs), the increase of the atomic radiuses of alkali metals works as applying a pressure to the Ni-Se layers. The DOS of Ni $3d$ orbitals at the Fermi surface, the Sommerfeld coefficient $\gamma_{N}$, and the electronic correlations are all enhanced. In the framework of the BCS theory, the SC transition temperature $T_{c}$ increases accordingly, as the experimental observations in KNi₂Se₂, RbNi₂Se₂, and CsNi₂Se₂Neilson et al. (2012); Chen, Huimin and Yang, Jinhu and Cao, Chao and Li, Lin and Su, Qiping and Chen, Bin and Wang, Hangdong and Mao, Qianhui and Xu, Binjie and Du, Jianhua and Fang, Minghu (2016); Wang et al. (2013).

In summary, we have successfully synthesized single crystals of RbNi₂Se₂ and characterized the physical properties. RbNi₂Se₂ is found to be a weakly coupled superconductor with $T_{c}=1.2$ K. In normal state, RbNi₂Se₂ exhibits Fermi liquid behavior and Pauli paramagnetism. In the SC state, the zero-field electronic specific heat can be well described with a two-gap BCS model, indicating that RbNi₂Se₂ possesses multi-gap feature. DFT calculations and ARPES measurements demonstrate that multi electronic bands of the $3d$ orbitals of Ni ions cross the Fermi level. Our analyses reveal that the large Sommerfeld coefficient of RbNi₂Se₂ is originated from the large DOS at the Fermi surface.

We thank Shiliang Li for fruitful discussions. Work at Sun Yat-Sen University was supported by the National Natural Science Foundation of China (Grants No. 12174454, 11904414, 11904416, 11974432, U2130101), the Guangdong Basic and Applied Basic Research Foundation (No. 2021B1515120015), National Key Research and Development Program of China (No. 2019YFA0705702, 2018YFA0306001, 2017YFA0206203), and GBABRF-2019A1515011337. Work at SLAB was supported by NSF of China with Grant No. 12004270, and GBABR-2019A1515110517. ARPES work at Rice is supported by the Robert A. Welch Foundation Grant No. C-2024 (M. Y.). P. D. is also supported by US Department of Energy, BES under Grant No. DE-SC0012311. Work at Berkeley was funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Materials Sciences and Engineering Division under Contract No. DE-AC02-05-CH11231 (Quantum Materials program KC2202).