⇦Chinese Physics Letters, 2022, Vol. 39, No. 6, Article code 067101 Nontrivial Topological States in BaSn$_{5}$ Superconductor Probed by de Haas–van Alphen Quantum Oscillations Lixuesong Han (韩李雪松)1†, Xianbiao Shi (石贤彪)2,3†, Jinlong Jiao (焦金龙)4, Zhenhai Yu (于振海)1, Xia Wang (王霞)1,5, Na Yu (余娜)1,5, Zhiqiang Zou (邹志强)1,5, Jie Ma (马杰)4, Weiwei Zhao (赵维巍)2,3, Wei Xia (夏威)1,6*, and Yanfeng Guo (郭艳峰)1,6* Affiliations 1School of Physical Science and Technology, ShanghaiTech University, Shanghai 201210, China 2State Key Laboratory of Advanced Welding & Joining and Flexible Printed Electronics Technology Center, Harbin Institute of Technology, Shenzhen 518055, China 3Shenzhen Key Laboratory of Flexible Printed Electronics Techniology, Harbin Institute of Technology, Shenzhen 518055, China 4Key Laboratory of Artificial Structures and Quantum Control (Ministry of Education), School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China 5Analytical Instrumentation Center, School of Physical Science and Technology, ShanghaiTech University, Shanghai 201210, China 6ShanghaiTech Laboratory for Topological Physics, ShanghaiTech University, Shanghai 201210, China Received 23 February 2022; accepted 24 April 2022; published online 29 May 2022 †Lixuesong Han and Xianbiao Shi contributed equally to this work.
*Corresponding author. Email: xiawei2@shanghaitech.edu.cn; guoyf@shanghaitech.edu.cn Citation Text: Han L X S, Shi X B, Jiao J L et al. 2022 Chin. Phys. Lett. 39 067101    Abstract We report the nontrivial topological states in an intrinsic type-II superconductor BaSn$_{\boldsymbol{5}}$ ($T_{\rm{c}} \sim 4.4$ K) probed by measuring the magnetization, specific heat, de Haas–van Alphen (dHvA) effect, and by performing first-principles calculations. The first-principles calculations reveal a topological nodal ring structure centered at the $H$ point in the $k_{\rm{z}} = \pi$ plane of the Brillouin zone, which could be gapped by spin-orbit coupling (SOC), yielding relatively small gaps below and above the Fermi level of about 0.04 eV and 0.14 eV, respectively. The SOC also results in a pair of Dirac points along the $\varGamma$–$A$ direction, located at $\sim $0.2 eV above the Fermi level. The analysis of the dHvA quantum oscillations supports the calculations by revealing a nontrivial Berry phase originating from the hole and electron pockets related to the bands forming the Dirac cones. Thus, our study provides an excellent avenue for investigating the interplay between superconductivity and nontrivial topological states.
cpl-39-6-067101-fig1.png
cpl-39-6-067101-fig2.png
cpl-39-6-067101-fig3.png
cpl-39-6-067101-fig4.png
cpl-39-6-067101-fig5.png
cpl-39-6-067101-fig6.png
 DOI:10.1088/0256-307X/39/6/067101 © 2022 Chinese Physics Society Article Text The Majorana zero mode, which has been conceived to host potential applications for decoherence topological quantum computation due to its non-Abelian statistics characteristic,[1–6] generally occurs at topological defects such as vortices, boundaries, domain walls, or edges of some effectively spinless superconducting systems with odd-parity triplet pairing symmetry, i.e., topological superconductors (TSCs).[2,5–9] The one-dimensional (1D) p-wave and 2D $p_x \pm ip_y$ superconductors have been proposed for such targets.[9] Unfortunately, the pairing symmetry of a few candidates, such as Sr$_{2}$RuO$_{4}$, Cu$_{x}$Bi$_{2}$Se$_{3}$, and SrxBi$_{2}$Se$_{3}$, is still under hot debate.[10] Other ways to create equivalent $p + ip$ pairing superconductivity including the construction of Bi$_{2}$Te$_{3}$/NbSe$_{2}$ heterostructure[11–13] and chemically doped superconductors, such as FeTe$_{0.55}$Se$_{0.45}$,[14–17] (Li$_{0.84}$Fe$_{0.16}$)OHFeSe,[18,19] CaKFe$_{4}$As$_{4}$,[20] and cobalt-doped LiFeAs,[21] have reported to be effective for inducing topological superconductivity. Among them, FeTe$_{0.55}$Se$_{0.45}$ is most intensively studied. It hosts both topological insulator (TI) and Dirac semimetal (DSM) states, with the latter slightly above the Fermi level ($E_{\rm F}$). However, compared with the heterostructures and doped superconductors, intrinsic TSCs are preferable for studying the Majorana zero mode because they are free of the disturbance from complicated interface physics and doping result detects or disorder. Therefore, intrinsic superconductors hosting nontrivial topological states have been the targets because the interplay of nontrivial electronic band topology and superconductivity could constitute an excellent solid environment for the Majorana zero mode.[9] Thus, examining the nontrivial topological states in superconductors is a crucial intergradient for discovering intrinsic TSCs. A series of intrinsic superconductors hosting nontrivial topological states have been investigated before,[22–29] and some of them have been reported as TSCs. [29] The BaSn$_{5}$ is an already known tin-rich superconductor with about 4.4 K superconducting critical temperature $T_{\rm c}$.[30] As shown in Fig. 1(a), it demonstrates a peculiar crystal structure, in which the Sn atoms are arranged in graphite-like honeycombs. Two such honeycombs form the hexagonal prisms centered by Ba, which could be viewed as a variant of the well-known MgB$_{2}$ family of superconductors. Previous studies suggested the BaSn$_{5}$ as an s-wave multi-band superconductor with an isotropic upper critical field, though its electron–phonon coupling strength is debated.[30–32] More interestingly, van Hove singularity in the density of states arising from the lone pairs was observed, which is usually discussed for high $T_{\rm c}$ cuprate superconductors rather than such a low $T_{\rm c}$ intermetallic superconductor.[32] BaSn$_{5}$ has been less studied despite these interesting properties, particularly considering its electronic band structure. It is a natural inspiration to explore the nontrivial topological states in BaSn$_{5}$, considering that several compounds with close compositions, such as CaSn, CaSn$_{3}$, and BaSn$_{3}$ superconductors,[24–26,33–35] host intriguing nontrivial topological states in the bulk band structures and nontrivial topological surface states. In this study, combining basic characterizations of the superconducting properties, measurements of the dHvA effect, and first-principles calculations, we demonstrated that the s-wave type-II BaSn$_{5}$ superconductor hosts intriguing multiple nontrivial topological states that highly resemble those of FeTe$_{0.55}$Se$_{0.45}$.[14] Single BaSn$_{5}$ crystals were grown using the Sn self-flux method.[31] Figure 1(c) shows an optical picture of a typical crystal. The crystal's crystallographic phase and quality were examined on a Bruker D8 single crystal x-ray diffractometer with Mo $K_{\alpha }$ ($\lambda = 0.71073$ Å) at 300 K. The result indicates a hexagonal structure with the $P6/mmm$ (No. 191) space group and lattice parameters $a=b = 5.37$ Å, $c = 7.097$ Å, $\alpha =\beta = 90^{\circ}$, $\gamma = 120^{\circ}$, which are nicely consistent with the previously reported values.[30] The graphite-like Sn atoms, shown in Figs. 1(a) and 1(b), form a two-dimensional slab of face-sharing hexagonal prisms doubly capped with Ba. The element-resolved energy dispersive spectroscopy (EDS) characterizations on more than ten different areas of several crystal pieces revealed a good stoichiometry of $1\!:\!5$, as shown in Fig. 1(c). The high quality of BaSn$_{5}$ single crystals could be evidenced by the perfect reciprocal space lattice of the single-crystal x-ray diffraction without any impurities, as shown in Figs. 1(d)–1(f). Magnetization and specific heat measurements were performed on magnetic property and physical property measurement systems from Quantum Design.
cpl-39-6-067101-fig1.png
Fig. 1. (a)–(b) Schematic crystal structures of BaSn$_{5}$ viewed along different orientations. (c) The measured compositions using EDS and an optical picture of a typical BaSn$_{5}$ crystal. (d)–(f) Diffraction patterns in the reciprocal space along the ($h0l$), ($0kl$), and ($hk0$) directions.
First-principles calculations were performed within the framework of the projector augmented wave method,[36,37] employing the generalized gradient approximation (GGA)[38] with the Perdew–Burke–Ernzerhof formula,[39] as implemented in the Vienna ab initio simulation package.[40–42] For all calculations, the cutoff energy for the plane-wave basis was set to 500 eV, the Brillouin zone (BZ) sampling was performed with a $\varGamma$-centered Monkhorst–Pack $k$-point mesh of size $11 \times 11 \times 8$, and the total energy difference criterion was defined as 10$^{-8}$ eV for self-consistent convergence. The optimized structural parameters were used in the electronic structure calculations. The temperature dependence of the magnetizations with the magnetic field $B$ of 10 Oe applied perpendicular to the (001) plane of the BaSn$_{5}$ crystal in the zero-field-cooling (ZFC) and field-cooling modes are illustrated in Fig. 2(a), showing a $T_{\rm c}$ of $\sim $4.4 K, consistent with that in the literature.[31] The isothermal magnetizations measured at 2 K are depicted by the inset of Fig. 2(a), indicating a type-II superconductivity by the clear hysteresis. Note that, unlike BaSn$_{5}$, Sn is a type-I superconductor with a $T_{\rm c} \sim 3.6$ K. Also, the bulk superconductivity is demonstrated by the $\sim 76.3$% superconducting shielding volume fraction estimated from the ZFC data at 2 K.
cpl-39-6-067101-fig2.png
Fig. 2. (a) Temperature dependence of BaSn$_{5}$ magnetization at 10 Oe. The inset shows the isothermal magnetization at 2 K. (b) Isothermal magnetizations measured at $T=2$–4.3 K with an interval of 0.5 K. (c) $C_{\rm p}/T$ around $T_{\rm c}$ with 0–300 Oe magnetic field. The arrows indicate the positions for the jump. (d) Upper (lower) critical field $H_{\rm c2}$ ($H_{\rm c1}$) versus the normalized temperature $T/T_{\rm c}$. The solid lines denote the fitting results to $H_{\rm c2}$ and $H_{\rm c1}$ using the Ginsburg–Landau (G-L) equation.
The bulk nature of the superconductivity is further confirmed by the evident jump in the specific heat $C_{\rm p}(T)$ at $B = 0$ T. Moreover, the $T_{\rm c}$ observed in the $C_{\rm p}(T)$ is $\sim $4.4 K, nicely consistent with that derived from the magnetization characterization. To evaluate the lower and upper critical field, the temperature-dependent isothermal magnetizations and magnetic-field-dependent low-temperature specific heat of BaSn$_{5}$ are presented in Figs. 2(b) and 2(c), respectively. The lower critical magnetic fields $H_{\rm c1}$ were determined using the magnetic field values at which the magnetizations start to deviate from the linear evolution, as shown by the arrows in the inset to Fig. 2(b). This allows the extraction of the $H_{\rm c1}$(0) via fitting these values by employing the G-L equation $H_{\rm c1}(T)=H_{\rm c1}(0)\frac{1-t^{2}}{1+t^{2}}$, where $t=T/T_{\rm c}$ is the reduced temperature. This yields $H_{\rm c1}(0)=16.53$ mT. Next, the superconducting jump in $C_{\rm p}(T)$ exhibits an apparent suppression to low temperatures upon the external magnetic field, allowing the $H_{\rm c2}$–$T$ diagram to be constructed. By employing the G-L equation $H_{\rm c2}(T)=H_{\rm c2}(0)\frac{1-t^{2}}{1+t^{2}}$,[34] the solid line in Fig. 2(d) shows the fitting result. The upper critical magnetic field $H_{\rm c2}(0)$ is 67.48 mT, comparable with previous studies.[31] The electronic band structure of BaSn$_{5}$ was studied by performing the first-principles calculations to understand its transport properties. As shown in Fig. 3(a), the BaSn$_{5}$ band structure, without considering the SOC, manifests a metallic character with several bands crossing the $E_{\rm F}$. The enlarged view in Fig. 3(b) shows that two band-crossing points at $-$0.04 eV along the $L$–$H$ path and at 0.14 eV along the $H$–$A$ line are visible in the BZ. The fat band structure shows that the contribution to these crossing bands comes mainly from the Sn 5$p$ states, forming an inverted band structure at the $H$ point. Figure 3(c) shows that the coupling of band-crossing points is not isolated but belong to a nodal ring centered at the $H$ point in the $k_{z}=\pi$ plane. Therefore, BaSn$_{5}$ superconductor could be classified into a topological metal with a Dirac nodal ring structure when the SOC is ignored. When the SOC is considered, as presented in Fig. 3(d), the BaSn$_{5}$ nodal line is gapped, showing a relatively small gap with a maximum size of about 20 meV along the $L$–$H$ path of the BZ. This is much smaller than the energy dispersion of the nodal ring (about 180 meV). Moreover, the SOC induces a pair of Dirac points (DPs) along the $\varGamma$–$A$ direction, as shown in Fig. 3(f). The DPs located at about 0.2 eV above the $E_{\rm F}$ can contribute to the transport behavior since the Dirac cone bands cross the $E_{\rm F}$. Hence, the BaSn$_{5}$ superconductor hosts both the DSM and TI states when the SOC is considered.
cpl-39-6-067101-fig3.png
Fig. 3. (a) Electronic band structure of BaSn$_{5}$ without considering the SOC. (b) Enlarged band structure along the $L$–$H$ and $H$–$A$ paths. The symbol size in (b) corresponds to the projected weight of the Bloch states onto the Sn $p_{x}$ (red), Sn $p_{y}$ (green), and Sn $p_{z}$ (blue) orbits. (c) Schematic illustration of the nodal ring centered at the $H$ point in the $k_{z}=\pi$ plane. (d) Electronic band structure of BaSn$_5$ with the SOC considered. (e) The same as (b) but with the SOC included. (f) Enlarged band structure along the $A$–$\varGamma$ path, showing a pair of Dirac points.
Table 1. Parameters derived from dHvA oscillations for BaSn$_{5}$, where $k_{\rm F}$ is the Fermi wave vector, $v_{_{\scriptstyle \rm F}}$ denotes the Fermi velocity, $\tau_{_{\scriptstyle \rm Q}}$ is the relaxation time, and $\varPhi_{\rm B}$ is the Berry phase.
$F$ (T) $A$ (nm$^{-2}$) $k_{\rm F}$ (nm$^{-1})$ $v_{_{\scriptstyle \rm F}}$ (m$/$s) $m^{\ast}/m_{\rm e}$ $T_{\rm D}$ (K) $\tau_{_{\scriptstyle \rm Q}}$ (s) $\varPhi_{\rm B}$
$F_{1}$ 21 0.200 0.252 $4.3 \times 10^{5}$ 0.07 1.792 $6.78 \times 10^{-13}$ 0.508 $\pi$
$F_{2}$ 86.4 0.822 0.512 $5.5 \times 10^{5}$ 0.11 6.287 $1.93 \times 10^{-13}$ 1.038 $\pi$
$F_{3}$ 103.3 0.985 0.560 $5.2 \times 10^{5}$ 0.12 9.338 $1.30 \times 10^{-13}$ 1.100 $\pi$
$F_{4}$ 53.3 0.508 0.402 $5.8 \times 10^{5}$ 0.08 3.84 $3.17 \times 10^{-13}$ $-0.214 \pi$
$F_{5}$ 185 1.764 0.750 $1.05 \times 10^{6}$ 0.07 21.99 $5.5 \times 10^{-14}$ 1.440 $\pi$
$F_{6}$ 207 1.973 0.793 $1.09 \times 10^{6}$ 0.07 16.80 $7.2 \times 10^{-14}$ 0.740 $\pi$
$F_{7}$ 561 5.349 1.305 $1.59 \times 10^{6}$ 0.08 20.11 $6.0\times 10^{-14}$ 0.452 $\pi$
The dHvA effect was measured to probe the Fermi surface (FS) of BaSn$_{5}$. As shown in Figs. 4(a) and 4(d), we measured the isothermal magnetization of the BaSn$_{5}$ up to 7 T with $B$//[1$\bar{1}$0] and $B$//[001] at various temperatures. As shown in Figs. 4(b) and 4(e), after subtracting the background, the magnetizations $\Delta M$ ($\Delta M= M - M_{\rm background}$) display striking oscillations in the temperature range 2–30 K, which could be well described by the Lifshitz–Kosevich (L-K) formula[43] $$ \Delta M\propto B^{\lambda }R_{\rm T}R_{\rm D}R_{\rm S}\sin \Big[2\pi \Big(\frac{F}{B}-\gamma -\delta\Big)\Big], $$ where $R_{\rm T}=2\pi^{2}k_{\rm B}T/[\hslash \omega_{\rm c}\sinh (2\pi^{2}k_{\rm B}T/\hslash \omega_{\rm c})]$ with $\omega_{\rm c}={eB} / m^{\ast}$ being the cyclotron frequency and $m^{\ast}$ denoting the effective cyclotron mass. $R_{\rm D}=\exp (-2\pi^{2}k_{\rm B}T_{\rm D}/\hslash \omega_{\rm c})$ with $T_{\rm D}$ being the Dingle temperature and $R_{\rm S}=\cos (\pi gm^{\ast }/2m_{\rm e})$. The index $\lambda$ is determined by the dimensions, with the $\lambda$ denoting 1/2 in the three-dimensional (3D) case and 0 in the 2D case.[44] The phase factor ($-\gamma - \delta$) is used to describe the oscillation part, where $\gamma = 1/2 -\varPhi_{\rm B}/2\pi$, with $\varPhi_{\rm B}$ being the Berry phase. The dimension of the FS determines the phase shifts, which takes the value of 0 for the 2D and $\pm 1/8$ ($-$ corresponding to the electron-like pocket and + to the hole-like pocket) for the 3D case. The first-order differential of $M$ vs $1/B$ allows the analysis of the fast Fourier transformation (FFT), as shown in Figs. 4(c) and 4(f). Multiple fundamental frequencies were obtained when $B$//[1$\bar{1}$0] and $B$// [001], as shown in Figs. 4(c) and 4(f), respectively. For a more accurate fitting, band-pass filtering was performed to separate the low and high-frequency components. Thus, the obtained fundamental frequencies are labeled as $F_{1}$ (21 T), $F_{2}$ (86.4 T), $F_{3}$ (103.3 T), $F_{4}$ (53.3 T), $F_{5}$ (185 T), $F_{6}$ (207 T), and $F_{7}$ (561 T). Here, 2$F_{2}$ and 2$F_{6}$ are the harmonic values indicating the same FS for $F_{2}$ and $F_{6}$. The multiple fundamental frequencies imply that several Fermi pockets are across or close to the $E_{\rm F}$. The areas of the Fermi pockets $A_{\rm F}$ can be calculated using the Onsager relationship $F=A_{\rm F}(\phi /2\pi^{2}$). The effective mass $m^{\ast}$ can be obtained by fitting the temperature-dependent oscillation amplitude to the thermal resistance factor $R_{\rm T}$, as shown in Figs. 5(b) and 5(e). Figures 5(c) and 5(f) show that the obtained Dingle temperatures $T_{\rm D}$ are evaluated for the fundamental frequencies $F_{1} $–$ F_{7}$ by fitting the field-dependent amplitude of the quantum oscillations at 2 K. The other parameters are listed in Table 1.
cpl-39-6-067101-fig4.png
Fig. 4. [(a), (d)] Magnetizations of BaSn$_{5}$ vs $B$ for $B$//[1$\bar{1}$0] and $B$//[001], respectively, at various temperatures. [(b), (e)] The quantum oscillations for $B$//[1$\bar{1}$0] and $B$//[001] vs 1/$B,$ respectively, at various temperatures. The inset in (b) shows the three filtered oscillatory parts of $\Delta M$. The orange line represents the raw dHvA oscillatory signal of the BaSn$_{5}$ at 2 K. [(c), (f)] Fast Fourier transform (FFT) spectra of $\Delta M$ at various temperatures.
cpl-39-6-067101-fig5.png
Fig. 5. [(a), (d)] Landau indexes $n=N - \frac{1}{4}$ against 1/$B$ at 2 K for $B$//[1$\bar{1}$0] and $B$//[001], respectively. Each inset enlarges the fitting intercept. [(b), (e)] Temperature dependence of relative FFT amplitudes of the oscillations for $B$//[1$\bar{1}$0] and $B//[001]$, respectively. [(c), (f)] Dingle plots of dHvA oscillations at 2 K for $B$//[1$\bar{1}$0] and $B//[001]$, respectively.
The Landau level (LL) fan diagrams were established to test the Berry phase of the BaSn$_{5}$ accumulated along with the cyclotron orbit. The LL phase diagrams for $F_{1}$–$F_{3}$ and $F_{4}$–$F_{7}$ are shown in Figs. 5(a) and 5(d), respectively, where the valley positions of $\Delta M$ correspond to the Landau indices of $n=N-\frac{1}{4}$, and the peak positions of $\Delta M$ correspond to the Landau indices of $n=N + \frac{1}{2}-\frac{1}{4}$.[44] The linear fitting intercepts are 0.379, 0.394, 0.425, 0.018, 0.596, 0.245, and 0.101, corresponding to fundamental frequencies from $F_{1}$ to $F_{7}$, respectively. The data ambiguously unveil the 3D FS of BaSn$_{5}$. As shown in Fig. 6(b), these angle-dependent frequencies $F_{1}$–$F_{7}$ are consistent with the calculated values for bands 1, 2, 3, and 4, respectively, as illustrated in Fig. 6(c). The angle-dependent dHvA oscillations could provide further information about the shape of the FS. Figure 6(a) shows the amplitudes of the oscillations versus 1/$B$ at various $\theta$ and 2 K. For the BaSn$_{5}$, band 1 (red) contributes hole pockets, whereas bands 2 (blue), 3 (orange), and 4 (olive green) contribute the electron pockets. By carefully comparing the experimental and theoretical values, $F_{4}$ and $F_{1}$ are assigned to the electron pockets contributed by band 4, while $F_{5}$, $F_{2}$, $F_{6}$, $F_{3,}$ and $F_{7}$ are assigned to the hole pockets contributed by band 1. Thus, the Berry phases of $F_{1}$–$F_{7}$ are $2 \times (0.379-0.125)\pi$, $2 \times (0.394 + 0.125)\pi$, $2\times (0.425+0.125)\pi$, $2 \times(0.018-0.125)\pi$, $2 \times(0.596+0.125)\pi$, $2 \times(0.245 +0.125)\pi$, and $2 \times(0.101+ 0.125)\pi$, respectively. The $F_{2}$ and $F_{3}$ Berry phases are close to $\pi$, indicating the nontrivial topological nature of band 1, consistent with the first-principle calculations. The Berry phases for $F_{5}$, $F_{6}$, and $F_{7}$, which are 1.44$\pi$, 0.74$\pi$, and 0.456$\pi$, respectively, deviate clearly from $\pi$, which may be due to their $N$ to be too large, consequently increasing the uncertainty of the fits. However, since $F_{5}$, $F_{2}$, $F_{6}$, $F_{3}$, and $F_{7}$ are assigned to the hole pockets contributed by band 1, which forms the Dirac nodal ring, the Berry phases for the three fundamental frequencies derived from the quantum oscillations measured at a higher magnetic field should be close to $\pi$.
cpl-39-6-067101-fig6.png
Fig. 6. (a) FFT spectra of $\Delta M$ at different $\theta$ from $B$//[001] to $B$//[1$\bar{1}$0]. (b) Comparison between experimental and theoretical values of the fundamental frequencies with various $\theta$. (c) The calculated FS, in which band 1 contributes to the hole pocket, whereas bands 2, 3, and 4 contribute to the electron pockets.
In summary, we have demonstrated the nontrivial topological states in the BaSn$_{5}$ superconductor. The characterization by magnetization and specific heat measurements unveils the type-II superconductor nature. The dHvA quantum oscillations and first-principles calculations suggest a topological nodal-line structure in BaSn$_{5}$, which can be gapped by considering the SOC. Alternatively, a pair of DPs appears. Thus, it is interesting that BaSn$_{5}$ hosts both the TI and DSM states in the electronic band structure, highly resembling the case in FeTe$_{0.55}$Se$_{0.45}$ and the cobalt-doped LiFeAs.[14,21] If the FS formed by the Dirac Fermi arc can corporate with the bulk superconductivity, it could form an effective proximity effect in momentum space, serving as the solid environment for the Majorana zero mode. Comparing the BaSn$_{3}$ and CaSn$_{3}$,[24,35] we can realize that the DPs are closer to the $E_{\rm F}$ in the BaSn$_{5}$. This provides an excellent example for studying the correlation between the nontrivial topological states and conventional superconductivity. Acknowledgments. This work was supported by the National Natural Science Foundation of China (Grant Nos. 92065201, 11774223, and U2032213), the Open Project of Key Laboratory of Artificial Structures and Quantum Control (Ministry of Education), Shanghai Jiao Tong University (Grant No. 2020–04). W.W.Z. was supported by the Shenzhen Peacock Team Plan (Grant No. KQTD20170809110344233) and Bureau of Industry and Information Technology of Shenzhen through the Graphene Manufacturing Innovation Center (Grant No. 201901161514). The authors thank the support from Analytical Instrumentation Center, SPST, ShanghaiTech University (Grant No. SPST-AIC10112914). References Majorana returnsTopological superconductors: a reviewNon-Abelian states of matterNon-Abelian anyons and topological quantum computationTopological insulators and superconductorsIntroduction to topological superconductivity and Majorana fermionsMutual friction in superfluid He 3 : Effects of bound states in the vortex corePaired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum Hall effectSuperconducting Proximity Effect and Majorana Fermions at the Surface of a Topological InsulatorThe superconductivity of Sr 2 RuO 4 and the physics of spin-triplet pairingThe Coexistence of Superconductivity and Topological Order in the Bi 2 Se 3 Thin FilmsExperimental Detection of a Majorana Mode in the core of a Magnetic Vortex inside a Topological Insulator-Superconductor Bi 2 Te 3 / NbSe 2 HeterostructureMajorana Zero Mode Detected with Spin Selective Andreev Reflection in the Vortex of a Topological SuperconductorObservation of topological superconductivity on the surface of an iron-based superconductorTopological nature of the FeSe 0.5 Te 0.5 superconductorHalf-integer level shift of vortex bound states in an iron-based superconductorEvidence for Majorana bound states in an iron-based superconductorQuantized Conductance of Majorana Zero Mode in the Vortex of the Topological Superconductor (Li0.84 Fe0.16 )OHFeSeRobust and Clean Majorana Zero Mode in the Vortex Core of High-Temperature Superconductor ( Li 0.84 Fe 0.16 ) OHFeSe A new Majorana platform in an Fe-As bilayer superconductorMajorana zero modes in impurity-assisted vortex of LiFeAs superconductorBulk Fermi surface of the layered superconductor TaS e 3 with three-dimensional strong topological stateObservation of Topological Electronic Structure in Quasi-1D Superconductor TaSe3de Haas–van Alphen Quantum Oscillations in BaSn$_{3}$ Superconductor with Multiple Dirac FermionsPhotoemission Spectroscopic Evidence of Multiple Dirac Cones in Superconducting BaSn3Observation of topological Dirac fermions and surface states in superconducting Ba Sn 3 Evidence of a topological edge state in a superconducting nonsymmorphic nodal-line semimetalObservation of electronic structure and electron-boson coupling in the low-dimensional superconductor Ta 4 Pd 3 Te 16 Evidence of anisotropic Majorana bound states in 2M-WS2Novel Tin Structure Motives in Superconducting BaSn5 – The Role of Lone Pairs in Intermetallic Compounds [1]Physical properties of single crystalline BaSn5Electron–phonon superconductivity in BaSn5Fermi surfaces of the topological semimetal CaSn$_{3}$ probed through de Haas van Alphen oscillationsNematic superconductivity in topoogical semimetal CaSn$_{3}$Superconductivity in CaSn 3 single crystals with a AuCu 3 -type structureProjector augmented-wave methodOrbital-free density functional theory implementation with the projector augmented-wave methodGeneralized Gradient Approximation Made SimpleAccurate and simple analytic representation of the electron-gas correlation energyAb initio molecular dynamics for liquid metalsEfficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis setEfficient iterative schemes for ab initio total-energy calculations using a plane-wave basis setManifestation of Berry's Phase in Metal PhysicsEvidence of Topological Nodal-Line Fermions in ZrSiSe and ZrSiTe
[1] Wilczek F 2009 Nat. Phys. 5 614
[2] Sato M and Ando Y 2017 Rep. Prog. Phys. 80 076501
[3] Stern A 2010 Nature 464 187
[4] Nayak C, Simon S H, Stern A, Freedman M, and Das S S 2008 Rev. Mod. Phys. 80 1083
[5] Qi X L and Zhang S C 2011 Rev. Mod. Phys. 83 1057
[6] Leijnse M and Flensberg K 2012 Semicond. Sci. Technol. 27 124003
[7] Kopnin N B and Salomaa M M 1991 Phys. Rev. B 44 9667
[8] Read N and Green D 2000 Phys. Rev. B 61 10267
[9] Fu L and Kane C L 2008 Phys. Rev. Lett. 100 096407
[10] Mackenzie A P and Maeno Y 2003 Rev. Mod. Phys. 75 657
[11] Wang M X, Liu C, Xu J P, Yang F, Miao L, Yao M Y, Gao C L, Shen C, Ma X, Chen X, Xu Z A, Liu Y, Zhang S C, Qian D, Jia J F, and Xue Q K 2012 Science 336 52
[12] Xu J P, Wang M X, Liu Z L, Ge J F, Yang X, Liu C, Xu Z A, Guan D, Gao C L, Qian D, Liu Y, Wang Q H, Zhang F C, Xue Q K, and Jia J F 2015 Phys. Rev. Lett. 114 017001
[13] Sun H H, Zhang K W, Hu L H, Li C, Wang G Y, Ma H Y, Xu Z A, Gao C L, Guan D D, Li Y Y, Liu C, Qian D, Zhou Y, Fu L, Li S C, Zhang F C, and Jia J F 2016 Phys. Rev. Lett. 116 257003
[14] Zhang P, Yaji K, Hashimoto T, Ota Y, Kondo T, Okazaki K, Wang Z, Wen J, Gu G D, Ding H, and Shin S 2018 Science 360 182
[15] Wang Z, Zhang P, Xu G, Zeng L K, Miao H, Xu X, Qian T, Weng H, Richard P, Fedorov A V, Ding H, Dai X, and Fang Z 2015 Phys. Rev. B 92 115119
[16] Kong L, Zhu S, Papaj M, Chen H, Cao L, Isobe H, Xing Y, Liu W, Wang D, Fan P, Sun Y, Du S, Schneeloch J, Zhong R, Gu G, Fu L, Gao H J, and Ding H 2019 Nat. Phys. 15 1181
[17] Wang D, Kong L, Fan P, Chen H, Zhu S, Liu W, Cao L, Sun Y, Du S, Schneeloch J, Zhong R, Gu G, Fu L, Ding H, and Gao H J 2018 Science 362 333
[18] Chen C, Liu Q, Zhang T Z, Li D, Shen P P, Dong X L, Zhao Z X, Zhang T, and Feng D L 2019 Chin. Phys. Lett. 36 057403
[19] Liu Q, Chen C, Zhang T, Peng R, Yan Y J, Wen C H P, Lou X, Huang Y L, Tian J P, Dong X L, Wang G W, Bao W C, Wang Q H, Yin Z P, Zhao Z X, and Feng D L 2018 Phys. Rev. X 8 041056
[20] Liu W, Cao L, Zhu S, Kong L, Wang G, Papaj M, Zhang P, Liu Y B, Chen H, Li G, Yang F, Kondo T, Du S, Cao G H, Shin S, Fu L, Yin Z, Gao H J, and Ding H 2020 Nat. Commun. 11 5688
[21] Kong L, Cao L, Zhu S, Papaj M, Dai G, Li G, Fan P, Liu W, Yang F, Wang X, Du S, Jin C, Fu L, Gao H J, and Ding H 2021 Nat. Commun. 12 4146
[22] Xia W, Shi X B, Zhang Y, Su H, Wang Q, Ding L C, Chen L M, Wang X, Zou Z Q, Yu N, Pi L, Hao Y F, Li B, Zhu Z W, Zhao W W, Kou X F, and Guo Y F 2020 Phys. Rev. B 101 155117
[23] Chen C, Liang A J, Liu S, Nie S M, Huang J W, Wang M X, Li Y W, Pei D, Yang H F, Zheng H J, Zhang Y, Lu D H, Hashimoto M, Barinov A, Jozwiak C, Bostwick A, Rotenberg E, Kou X F, Yang L X, Guo Y F, Wang Z J, Yuan H T, Liu Z K, and Chen Y L 2020 Matter 3 2055
[24] Zhang G N, Shi X B, Liu X L, Xia W, Su H, Chen L M, Wang X, Yu N, Zou Z Q, Zhao W W, and Guo Y F 2020 Chin. Phys. Lett. 37 087101
[25] Huang Z, Shi X B, Zhang G N, Liu Z T, Soohyun C, Jiang Z C, Liu Z H, Liu J S, Yang Y C, Xia W, Zhao W W, Guo Y F, and Shen D W 2021 Chin. Phys. Lett. 38 107403
[26] Huang K, Luo A Y, Chen C, Zhang G N, Liu X L, Li Y W, Wu F, Cui S T, Sun Z, Jozwiak C, Bostwick A, Rotenberg E, Yang H F, Yang L X, Xu G, Guo Y F, Liu Z K, and Chen Y L 2021 Phys. Rev. B 103 155148
[27] Xu L X, Xia Y Y Y, Liu S, Li Y W, Wei L Y, Wang H Y, Wang C W, Yang H F, Liang A J, Huang K, Deng T, Xia W, Zhang X, Zheng H J, Chen Y J, Yang L X, Wang M X, Guo Y F, Li G, Liu Z K, and Chen Y L 2021 Phys. Rev. B 103 L201109
[28] Yang H F, Liu X L, Nie S M, Shi W J, Huang K, Zheng H J, Zhang J, Li Y W, Liang A J, Wang M X, Yang L X, Guo Y F, L, Z K, and Chen Y L 2021 Phys. Rev. B 104 L220501
[29] Yuan Y, Pan J, Wang X, Fang Y, Song C, Wang L, He K, Ma X, Zhang H, Huang F, Li W, and Xue Q K 2019 Nat. Phys. 15 1046
[30] Fässler T F, Hoffmann S, and Kronseder C 2001 Z. Anorg. Allg. Chem. 627 2486
[31] Lin X, Bud'ko S L, and Canfield P C 2012 Philos. Mag. 92 3006
[32] Billington D, Ernsting D, Millichamp T E, and Dugdale S B 2015 Philos. Mag. 95 1728
[33] Siddiquee K A M H, Munir R, Dissanayake C, Hu X Z, Yadav S, Takano Y, Choi E S, Le D, Rahman T S, and Nakajima Y 2021 arXiv:2103.08039v1 [cond-mat.supr-con]
[34] Siddiquee K A M H, Munir R, Dissanayake C et al. 2019 arXiv:1901.02087v1 [cond-mat.supr-con]
[35] Luo X, Shao D F, Pei Q L, Song J Y, Hu L, Han Y Y, Zhu X B, Song W H, Lu W J, and Sun Y P 2015 J. Mater. Chem. C 3 11432
[36] Blöchl P E 1994 Phys. Rev. B 50 17953
[37] Lehtomäki J, Makkonen I, Caro M A, Harju A, and Lopez-Acevedo O J 2014 Chem. Phys. 141 234102
[38] Perdew J P, Burke K, and Ernzerhof M 1996 Phys. Rev. Lett. 77 3865
[39] Perdew J P and Wang Y 1992 Phys. Rev. B 45 13244
[40] Kresse G and Hafner J 1993 Phys. Rev. B 47 558
[41] Kresse G and Furthmüller J 1996 Comput. Mater. Sci. 6 15
[42] Kresse G and Furthmüller J 1996 Phys. Rev. B 54 11169
[43] Mikitik G P and Sharlai Y V 1999 Phys. Rev. Lett. 82 2147
[44] Hu J, Tang Z, Liu J, Liu X, Zhu Y, Graf D, Myhro K, Tran S, Lau C N, Wei J, and Mao Z 2016 Phys. Rev. Lett. 117 016602