Superconductivity and Quantum Oscillations
in Single Crystals of the Compensated Semimetal CaSb₂ 

We characterize the superconducting state of CaSb₂ and show bulk superconductivity with sharp superconducting transitions, using magnetic susceptibility, specific heat, and resistivity measurements. The magnetic susceptibility at low temperature is shown in Fig. 2(a). The DC magnetization was measured between 0.4 K-1.9 K using zero-field-cooling (ZFC) processes and field-cooling processes (FC). The volume susceptibility ($\chi_{V}$) vs. temperature measured in a field of 1 mT is shown in Fig. 2(a), where $\chi_{V}$ was corrected for demagnetization effects due to sample shape using the expression for a slab-like geometry in Ref. Brandt (1999). Both ZFC and FC measurements show evidence of bulk superconductivity with a strong diamagnetic signal that has an onset of 1.6 K in a field of 1 mT. The superconducting transition width $\Delta T$, defined as the temperature range over which the diamagnetic signal reaches 90% of its maximum value below $T_{\text{c}}$, is $\sim$0.25 K. The estimated superconducting volume fraction after demagnetization correction in the ZFC data is 100%. We note that demagnetization calculated using a rectangular cuboid in Ref. Prozorov and Kogan (2018) resulted in a volume fraction exceeding 100%. Thus, our data suggests near-perfect shielding in the superconducting state. The difference between the ZFC and FC values of $\chi_{V}$ in our single crystal samples is suggestive of a type-II superconductor with flux pinning. However, the significant flux expulsion in the FC data suggests that pinning of flux vortices is weak.

In order to map out the field-temperature phase diagram, we track the shift in $T_{\text{c}}$ for different applied magnetic fields in DC magnetization and specific heat measurements in Figs. 2(b) and  3(a), respectively. We measured a drop in electrical resistivity at the superconducting transition that is suppressed with applied field, as shown in Fig. 2(d), but due to noise in the data estimating $T_{\text{c}}$ at each applied field is difficult. Also, we measured the AC susceptibility of the CaSb₂ and the real ($\chi$’_AC) and imaginary ($\chi$”_AC) parts are shown in Sup. Fig. 2 in Supplemental Materials Sup . Although we measured a sharp superconducting transition with a peak in $\chi$”_AC at 1.67 K, we find this measurement is sensitive to the onset of superconductivity rather than the bulk $T_{\text{c}}$. From magnetization, $T_{\text{c}}$ is defined as the temperature where the diamagnetic signal reaches 5% volume fraction at different magnetic fields. From specific heat, $T_{\text{c}}$ is defined as the mid point of the transition seen in $C_{\rm{el}}$, inset of Fig. 3(a). The magnetic fields plotted against $T_{\text{c}}$ as extracted from both measurements are shown in Fig. 2(e). To estimate $\mu_{\rm{0}}H_{\rm{c2}}(0)$, we use the Werthamer-Helfand-Hohenberg (WHH) equation Werthamer et al. (1966). We get $\mu_{\rm{0}}H_{\rm{c2}}(0)$ of 21.5 mT from specific heat and 24.4 mT from magnetization. Using the $\mu_{\rm{0}}H_{\rm{c2}}(0)$ of 24.4 mT, we estimate the coherence length $\xi_{\rm{GL}}$ using the equation

To measure $\mu_{\rm{0}}H_{\rm{c1}}(0)$ we measured the field-dependent magnetization ($M$) at different temperatures below the critical temperature, as shown in Fig. 2(c). The low-field linear fit for the 0.4 K measurement is shown with a black line from the origin ($M_{\rm{fit}}$). This fit was subtracted from all the measured curves to construct a $M$-$M_{\rm{fit}}$ plot. The field where the magnetization deviated from linear response, taken as the black line in the inset of Fig. 2(f), is the uncorrected $\mu_{\rm{0}}H^{\rm{*}}_{\rm{c1}}(0)$ for that temperature. All $\mu_{\rm{0}}H^{\rm{*}}_{\rm{c1}}(0)$ are plotted against the corresponding temperature in Fig. 2(f) and are fitted to the equation

In Fig. 3(b), the transition seen in $C_{\rm{el}}$ with a $T_{\text{c}}$ of 1.52 K in zero field and a $\Delta T$ of 0.15 K provides further evidence for the bulk nature of the superconductivity. Magnetic fields push superconducting transitions seen in specific heat $C_{P}$ to lower temperatures (Fig. 3(a)). The phonon contribution to $C_{P}$ was determined by fitting the data measured in a field of 50 mT, where no transition is observed, between 0.55 K-4.0 K to the expression

The result of this fit is $\gamma_{n}=1.778$ mJ mol^-1 K^-2 and $\beta=0.471$ mJ mol^-1 K^-4. The phonon contribution represented by the cubic term $\beta T^{3}$ was subtracted from the total $C_{P}$ to evaluate the electronic part of the specific heat $C_{\rm{el}}$. The electronic contribution to specific heat $C_{\rm{el}}$ divided by temperature is plotted as a function of temperature in Fig. 3(b). The flat part of the graph above 1.55 K corresponds to a constant $\gamma$ above the transition temperature consistent with metallic behavior. At $T_{\text{c}}$ a jump is observed corresponding to a second-order phase transition into the superconducting state, and the narrow transition further confirms the high-quality of the crystals. We observed the superconducting transition in $\chi_{V}$ and $C_{\rm{el}}$ measurements on multiple crystals, grown in different batches, where the $T_{\text{c}}$ as defined above differs by less than 1%.

The Debye model is used to calculate the Deybe temperature $\Theta_{D}$

The magnitude of the specific heat jump $\Delta C/C_{en}\sim 0.95$, $C_{en}$ being the electronic specific heat in the normal state, is about 34% smaller than the value of 1.43 expected for a BCS model in the weak-coupling limit. In Fig. 3(b) we show the BCS expectation of a jump at $T_{\text{c}}$ and the exponential decay of $C_{\rm{el}}$ below $T_{\text{c}}$ with a black dotted line, and it is clear that the data in CaSb₂ are not well explained with the BCS model. As we have near-perfect shielding from the Meissner signal in the magnetization measurements for multiple samples, we exclude the possibility of sample inhomogeneity as the origin of this smaller jump in the specific heat. We attempt a fit to the data based on the alpha model Padamsee et al. (1973); Johnston (2013), and we find that experimental data deviates from this model for $\alpha=1.425~{}\&~{}1.50$, as shown in Sup. Fig. 3. More importantly, the decay in $C_{\rm{el}}$ $/T$ with decreasing temperature is weaker than that expected for a BCS superconductor. Even at the lowest temperature of 0.22 K the value of $C_{\rm{el}}$ is not negligible and continues to be temperature-dependent. This unusual behaviour in $C_{\rm{el}}$ suggests the presence of a small gap component, due perhaps to extreme gap anisotropy, differing gaps on different bands, or gaps with nodes.

We consider a model at the extreme end of possible gap anisotropy, one that includes line nodes in the superconducting gap Won and Maki (1994), shown in red in Fig. 3(b). In this model, the average gap magnitude is smaller than that in the fully gapped case, and this is reflected in a reduced specific heat jump. Also, at low temperature the $T$ dependence of $C_{\rm{el}}$ $/T$ changes from exponential to power law Maeno et al. (2011), because the existence of nodes allows for the thermal excitation of quasiparticles down to very low temperatures. This nodal-gap function was recently suggested for a Bi₂Se₃ related material, namely Cu-intercalated (PbSe)₅(Bi₂Se₃)₆ Sasaki et al. (2014), considered to potentially be topological superconductors. In CaSb₂ this model fits better to the superconducting jump at $T_{\text{c}}$, but the low temperature experimental data has lower values than expected in this model. The data clearly do not fit the BCS values or the simplest line-node model, which suggests strong anisotropy in the gap function. This may be related to one of the Fermi surfaces in our calculation being dominated by Sb chains in the distorted square-net (Sb1 site), discussed in Sec. III.2. We note the recent report of ^121/123Sb-nuclear quadrupole resonance (NQR) measurement in a polycrystalline sample of CaSb₂, which suggests $s$-wave superconductivity, in part based on observing a coherence peak below $T_{\text{c}}$ Takahashi et al. (2021). The suggestion of exponentially-activated behaviour in those NQR data at low temperatures is not clear, as is also the case for our specific heat data. A coherence peak was observed in nuclear magnetic resonance (NMR) measurements of the unconventional superconductors $\rm{CePt}_{3}\rm{Si}$ Yogi et al. (2006a, b) and $\rm{Li}_{2}(\rm{Pd}_{1-x}\rm{Pt}_{x})_{3}\rm{B}$ Harada et al. (2010); Shimamura et al. (2007), which is composition dependent in the latter. Ultimately the resolution of the gap structure in CaSb₂ will require lower temperature measurements on single crystal samples, and a variety of spectroscopic probes.

CaSb₂ seems to fall at the border of type-I and type-II superconductivity. YbSb₂, which crystalizes in a higher symmetry crystal structure ($Cmcm$, S.G.:63), has been reported to be a type-I superconductor Zhao et al. (2012). The different crystal structures of YbSb₂ and CaSb₂ may be important to the nature of the emergent superconducting state in these compounds. Studying the effect of pinning centers on flux-trapping in CaSb₂ and further improving the quality of grown crystals will help us clarify whether it is intrinsically a type-I or type-II superconductor, as well as further probing the presence of gap anisotropy. This will be the subject of future studies. A summary of observed superconducting parameters of CaSb₂ is presented in Table 1. We firmly established and characterised bulk superconductivity in CaSb₂. In the next section we place this superconductivity in the context of an unusual semimetal by examining the consistency of the calculated band structure with our transport measurements and de Haas–van Alphen (dHvA) oscillations.

The resistivity measurements reveal metallic behavior with a longitudinal resistivity $\rho_{\mathrm{xx}}$ of ca. 1 $\mu\Omega\cdot$cm at 2 K (Fig. 4a). The high residual resistivity ratio $\rho_{\mathrm{300~{}\rm{K}}}/\rho_{\mathrm{2~{}\rm{K}}}$ of ca. 100 indicates the good quality of the crystals. At low temperature, the resistivity rises rapidly with increasing field. While the zero field resistivity has the temperature dependence of a good metal, in higher fields the resistivity begins to increase with decreasing temperature as in an insulator. The rise in resistivity with decreasing temperature in an applied field larger than $3~{}\rm{T}$ plateaus for temperatures below $\sim 10~{}\rm{K}$. This behavior in our measurements on single crystals of CaSb₂ is similar with those measurements reported on polycrystalline samples Fu and Kane (2008), as well as other topological semimetals such as WP₂ Wang et al. (2017), NbAs₂ Wang et al. (2016), and ZrP₂ Bannies et al. (2021). As shown in Fig. 4b, the magnetoresistance MR, which is defined as $MR=\rho_{\mathrm{xx}}(B)/\rho_{\mathrm{xx}}(B=0)-1$, remains unsaturated up to 9 T, where it reaches large values of 2500% at 2 K. The field dependence of the MR can be described by a power law $MR=\mathrm{a}+\mathrm{b}\cdot B^{\mathrm{c}}$ with $\mathrm{c}=1.79$. This behavior is reproduced in a second sample (see Fig. 4c inset). The Shubnikov-de Haas oscillations superimposed on the MR background (Fig. 4b) further demonstrate good sample quality but are not analyzed here due to the low number of oscillations observed. Instead we will present de Haas-van Alphen (dHvA) oscillations in detail below. Together, the field-driven metal-to-insulator-like transition and the large and unsaturated MR suggest a semimetallic state in CaSb₂.

To verify this scenario, we measured the Hall resistivity $\rho_{\mathrm{xy}}$ in the same field and temperature ranges as the MR (Fig. 4c). While $\rho_{\mathrm{xy}}(B)$ is roughly linear for all temperatures, it cannot be fit with a semi-classical one-band model because $\rho_{\mathrm{xx}}$ is independent of the field in this model, which is contrast to our data (see Fig. 4b). Instead, the data can be fitted with the semi-classical two-band model Pippard (1989):

and

based on the carrier densities $n_{\mathrm{e,h}}$ and mobilities $\mu_{\mathrm{e,h}}$ for electrons and holes. By simultaneously fitting $\rho_{\mathrm{xx}}$ and $\rho_{\mathrm{xy}}$ to the two-band model (see Sup. Fig. 4 for an exemplary fit), we extracted the carrier densities and mobilities of the charge carriers, which are shown as functions of temperature in Fig. 4d-f (the full dataset is shown in Sup. Fig. 5 in Supplemental Materials Sup ). While electrons are the majority charge carriers at high temperatures, the carrier densities are essentially compensated below 50 K with $n_{\mathrm{e}}/n_{\mathrm{h}}=0.99(1)$ at 2 K. The carrier mobilities increase monotonically with decreasing temperature, presumably due to reduced scattering of charge carriers by phonons. The high values of approximately 3000 cm²/Vs at 2 K further indicate good crystal quality. Based on the two-band model fits, we conclude that CaSb₂ is a compensated semimetal in which large MR arises at low temperature from the combination of charge carrier compensation and high carrier mobility.

We confirmed our experimental findings by band structure calculations based on density functional theory (DFT). The resulting band structure of CaSb₂ is shown in Fig. 5(a). The Fermi surface is composed of one hole-like pocket centered in the $k_{\mathrm{y}}=0$ plane and two electron-like pockets at the edge of the Brillouin zone (BZ), see Fig. 5c. While the hole pocket exhibits a complex three-dimensional shape, the two electron-like bands have weak dispersion along $k_{\mathrm{z}}$, giving rise to a quasi-two-dimensional character. The presence of two electron pockets in the calculations seemingly contradicts our analysis of the magnetotransport. However, the volume of the smaller electron pocket is only 1/8 of the volume of the bigger electron pocket. We argue that the two electrons pockets can be effectively described as one combined electron pocket if they have similar average carrier mobilities. This assumption is validated by the good quality of the two-band model fits to the experimental data. Attempts to fit the data with a three-band model containing two electron pockets and one hole pocket did not improve the quality of the fit and were therefore discarded.

To complement our findings about the Fermi surface based on electrical transport experiments, we have studied the magnetization in the normal state at low temperatures. At 1.8 K, we observe de Haas-van Alphen (dHvA) oscillations in fields larger than 2.5 T for $B\parallel c$. In Fig. 6(a), we show the oscillatory component of the dHvA oscillations as a function of inverse magnetic field at selected temperatures between 1.8 K and 10 K. The fast Fourier transform (FFT) spectrum of the data at 1.8 K, shown in Fig. 6b, reveals three peaks at $F_{\alpha}=(70\pm 4)$ T, $F_{\beta}=(78\pm 4)$ T and $F_{\gamma}=(116\pm 4)$ T. Given the limited number of periods collected, and the presence of closely-overlapping frequencies, it is advantageous to fit the dHvA oscillations to a standard model for a Fermi liquid in order to avoid limitations of the FFT method Ramshaw et al. (2011). In this procedure, we fitted the oscillatory component $\Delta M$ of the dHvA oscillations as a function of magnetic field and temperature to the Lifshitz-Kosevich formula Shoenberg (2009):

with $R_{\mathrm{T}}=\frac{XT}{\sinh(XT)}$, $R_{\mathrm{D}}=\exp{(-XT_{\mathrm{D}})}$, and $R_{\mathrm{S}}=\cos{\left(\pi g\frac{m^{*}}{2m_{0}}\right)}$, where $X=2\pi^{2}k_{\mathrm{B}}m^{*}/(\hbar eB)$. $T_{\mathrm{D}}$ is the Dingle temperature and $\phi$ is a phase factor that depends on the dimensionality of the Fermi surface and the Berry phase among other factors.

Accordingly, the temperature and magnetic field dependent damping of the oscillations are described by $R_{\mathrm{T}}$ and $R_{\mathrm{D}}$. Fitting $\Delta M(B,T)$ to eq. 11 therefore gives access to the effective masses and Dingle temperatures. Furthermore, it allows us to refine the initially imprecise frequencies from the FFT. In Fig. 6c, we show a representative part of the fit at a constant temperature of 1.8 K, which demonstrates the good quality of the fit. The data extracted from the fit at all temperatures are summarized in Tab. 2.

The three fundamental frequencies are 71.5(1) T, 77.5(1) T, and 115.8(1) T. Using the Onsager relation $A_{\mathrm{F}}=\frac{2\pi e}{\hbar}F$ Shoenberg (2009), we find that the extremal orbits associated with these frequencies cover 0.357(5)%, 0.387(4)%, and 0.578(2)% of the BZ projected along [001], respectively. The small values suggest small volumes of these Fermi surface pockets, in line with the semimetallic character. All three orbits show similar low effective masses of ~ 0.16 $m_{e}$ and the quantum mobilities range between ~ 2000 and 2400 cm²/Vs, indicative of high sample quality. Altogether, the small extremal areas, low effective masses and high quantum mobilities are consistent with the semimetallic character implied by the transport measurements.

Next, we compare the extremal orbits from dHvA oscillations with those predicted by DFT calculations. We do not find a quantitative match between theory and experiment. The mismatch can be caused by two possibly concomitant reasons: (i) the as-grown single crystals are self-doped, resulting in an experimental Fermi energy different from the theoretical one or (ii) the functional used in DFT calculations does not describe the semimetallicity accurately, resulting in a wrong Fermi surface. Without studies of the angle-dependence of the dHvA oscillations, it is impossible to deduce the reason for the mismatch. Future studies should therefore aim at obtaining a 3D Fermi surface from quantum oscillations. Importantly, our experimental results confirm the semimetallicity independent of the DFT calculations and thus remain valid despite the mismatch with the DFT calculations.

The bands around the Fermi level are dominated by contributions from Sb states, as shown by the fat band representations of Sb1 and Sb2 atoms in Fig. 6(d) and (e), respectively. The two electron bands crossing the Fermi level along the $\Gamma$-Z line close to the Z point have strong Sb1 character, whereas the hole band centered around the $\Gamma$ point is dominated by Sb2 character. The Ca contribution to these bands is very small as shown in Sup. Fig. 6 in Supplemental Materials Sup . The electron bands crossing the Fermi level along C-Z form nodal lines in the $k_{y}=\pi$ plane in the presence of SOC (see Sup. Fig. 7 in Supplemental Materials Sup ), in agreement with previous calculations Funada et al. (2019). These electron bands have a strong Sb1 character near the Fermi level and significant Sb2 character at higher energy; the Sb1 site forms the distorted square-net in CaSb₂. The similarities in the crystal structure between the $M$Sb₂ distorted square net family and the $M$SbTe square net family may result in similarities in the band structure of these materials. We highlight that the periodicity of atoms compared with the periodicity of the unit cell in a particular direction results in non-symmorphic symmetry, which is important for the nodal-line semimetal state in both cases, rather than the perfect square-net of Sb in $M$SbTe. The connection between the nodal-line semimetal states in both classes of materials warrants further theoretical and experimental investigation. The combined evidence for semimetallic character from electrical transport and dHvA oscillations support our calculated band structure, which includes non-trivial band topology along the C-Z direction at the $k_{y}=\pi$ plane.

Interestingly, CaSb₂ is one of only a handful of materials that displays at ambient pressure both a large MR (i.e. semimetallic character) and superconductivity, see Tab. 3. These materials include MoTe₂ Qi et al. (2016), TaSe₃ Sambongi et al. (1977); Saleheen et al. (2020), and $\alpha$-Ga Chen et al. (2018). Topological states were experimentally observed by angle-resolved photoemission spectroscopy (ARPES) in MoTe₂ Deng et al. (2016) and TaSe₃ Lin et al. (2021). Within superconducting semimetals, CaSb₂ offers a good balance between a moderately high $T_{\text{c}}$ of 1.6 K and large MR of 25 at 2 K and 9 T, similar to TaSe₃. In contrast, the low $T_{\text{c}}$ of MoTe₂ is difficult to reach, posing a significant disadvantage for applications. Among the materials listed in Tab. 3, $\alpha$-Ga has superior characteristics both in terms of $T_{\text{c}}$ and MR, but Ga is difficult to handle due to its low melting point of 29.8 ^∘C Sostman (1977). The combination of $T_{\text{c}}$, MR, and stability of CaSb₂ make it an excellent candidate for exploring the interplay between superconductivity and semimetallicity.

In theoretical work, it was suggested that the nodal-line structure of the normal state Fermi surface of CaSb₂ could survive in the superconducting state when weak-coupling is present Ono et al. (2021). This allows for realizing topological superconductivity in CaSb₂. We find experimentally that superconductivity in single crystals of CaSb₂ is in the weak-moderate coupling limit with $\lambda_{ep}=0.456$ and the decay in the electronic part of the specific heat $C_{\rm{el}}$ in the superconducting state down to 0.22 K is slower and does not follow the exponential decay expected for a conventional BCS superconductor. Future experiments will help us clarify the extent of influence of topology in the normal state of CaSb₂ on the superconducting properties.