Superconductivity and Normal-State Properties of Kagome Metal RbV₃Sb₅ Single Crystals

Two-dimensional (2D) kagome lattice composed of corner-sharing triangles and hexagons is one of the most studied systems in the last decades due to its unique structural feature. On the one hand, if only the spin degree of freedom is considered, insulating magnetic kagome materials can host some exotic magnetism ground states, like quantum spin liquid state, because of the nature of strongly geometrical frustration for kagome lattice.^[1–6] On the other hand, when the charge degree of freedom becomes dominant (partial filling), the band topology starts to manifest its features in kagome metals, such as nontrivial Dirac points and flat band in the band structure.^[7–9] More interestingly, when both of spin and charge degrees of freedom exist, many of exotic phenomena appear in the correlated magnetic kagome metals. For example, in ferromagnetic Fe₃Sn₂ and TbMn₆Sn₆ kagome metals, due to the spin orbital coupling and breaking of time reversal symmetry, the Chern gap can be opened at the Dirac point, leading to large anomalous Hall effect (AHE), topological edge state and large magnetic-field tunability.^[10–12] Moreover, antiferromagnetic Mn₃Sn and ferromagnetic Co₃Sn₂S₂ kagome metals exhibit large intrinsic AHE, which is related to the existence of Weyl node points in these materials.^[13–15]

Besides the intensively studied magnetic kagome metals, other correlation effects and ordering states in partially filled kagome lattice have also induced great interests. Theoretical studies suggest that the doped kagome lattice could lead to unconventional superconductivity.^[1,16–19] Especially, when the kagome lattice is filled near van Hove filling, the Fermi surface (FS) is perfectly nested and has saddle points on the M point of Brillouin zone (BZ).^[18] Depending on the variations of on-site Hubbard interaction U and Coulomb interaction on nearest-neighbor bonds V, the system can develop different ground states, including unconventional superconductivity, ferromagnetism, charge bond order and charge density wave (CDW) order and so on.^[18,19] However, the realizations of superconducting and charge ordering states are still scarce in kagome metals.

Very recently, a novel family of kagome metals AV₃Sb₅ (A = K, Rb and Cs) were discovered.^[20] Among them, KV₃Sb₅ and CsV₃Sb₅ exhibit superconductivity with transition temperature T_c = 0.93 and 2.5 K, respectively.^[21,22] The proximity-induced spin-triplet superconductivity was also observed in Nb/KV₃Sb₅ devices.^[23] More importantly, theoretical calculations and angle-resolved photoemission spectroscopy (ARPES) demonstrate that there are several Dirac nodal points near the Fermi energy level (E_F) with a non-zero Z₂ topological invariant in KV₃Sb₅ and CsV₃Sb₅.^[20–22,24] Moreover, AV₃Sb₅ exhibits transport and magnetic anomalies at T^* ∼ 80–110 K.^[20–22] The x-ray diffraction (XRD) and scanning tunnelling microscopy (STM) measurements on KV₃Sb₅ and CsV₃Sb₅ indicate that there is a 2 × 2 superlattice emerging below T^*, i.e., the formation of charge order (CDW-like state).^[21,25] Furthermore, the STM spectra show that this charge order has a chiral anisotropy, which can be tuned by magnetic field and may lead to the AHE at low temperature even KV₃Sb₅ does not exhibit magnetic order or local moments.^[24–26]

Motivated by these studies, in this work, we carried out a comprehensive study on physical properties of RbV₃Sb₅ single crystals. It is found that RbV₃Sb₅ shows a superconducting transition at T_c ∼ 0.92 K, which coexists with the anomalies of properties at T^* ∼ 102–103 K. This could be related to the emergence of charge ordering state. Below T^*, the transport properties change significantly, possibly rooting in the dramatic changes of electronic structure due to the formation of charge order. Furthermore, the analysis of low-temperature quantum oscillations (QOs) indicates that there are small FSs with low effective mass in RbV₃Sb₅, revealing the existence of highly dispersive bands near the E_F.

Single crystals of RbV₃Sb₅ were grown from Rb ingot (purity 99.75%), V powder (purity 99.9%) and Sb grains (purity 99.999%) using the self-flux method.^[21] The eutectic mixture of RbSb and RbSb₂ is mixed with VSb₂ to form a composition with 50 at.% Rb_x Sb_y and 50 at.% VSb₂ approximately. The mixture was put into an alumina crucible and sealed in a quartz ampoule under partial argon atmosphere. The sealed quartz ampoule was heated to 1273 K for 12 h and soaked there for 24 h. Then it was cooled down to 1173 K at 50 K/h and further to 923 K at a slowly rate. Finally, the ampoule was taken out from the furnace and decanted with a centrifuge to separate RbV₃Sb₅ single crystals from the flux. Except sealing and heat treatment procedures, all of other preparation processes were carried out in an argon-filled glove box in order to prevent the reaction of Rb with air and water. The obtained crystals have a typical size of 2 × 2 × 0.02 mm³. RbV₃Sb₅ single crystals are stable in air. XRD pattern was collected using a Bruker D8 x-ray diffractometer with Cu K_α radiation (λ = 0.15418 nm) at room temperature. The elemental analysis was performed using the energy-dispersive x-ray spectroscopy (EDX). Electrical transport and heat capacity measurements were carried out in a Quantum Design physical property measurement system (PPMS-14T). The ab-plane and Hall electrical resistivity were measured using a five-probe method and the current flows in the ab plane of the crystal. The Hall resistivity was obtained from the difference in the transverse resistivity measured at the positive and negative fields in order to remove the longitudinal resistivity contribution due to the voltage probe misalignment, i.e., ρ_yx (μ₀ H) = [ρ_yx (+μ₀ H) − ρ_yx (−μ₀ H)]/2. The c-axial resistivity was measured by attaching current and voltage wires on the opposite sides of the plate-like crystal. Magnetization measurements were performed in a Quantum Design magnetic property measurement system (MPMS3).

As shown in the left panel of Fig. 1(a), RbV₃Sb₅ has a layered structure with hexagonal symmetry (space group P6/mmm, No. 191). It consists of Rb layer and V-Sb slab stacking along c axis alternatively, isostructural to KV₃Sb₅ and CsV₃Sb₅.^[20] The key structural ingredient of this material is the 2D kagome layer formed by the V atoms in the V-Sb slab [right panel of Fig. 1(a)]. There are two kinds of Sb sites and the Sb atoms at Sb1 site occupy the centers of V hexagons when another Sb atoms at Sb2 site locate below and above the centers of V triangles, forming graphene-like hexagon layers. The XRD pattern of a RbV₃Sb₅ single crystal [Fig. 1(b)] reveals that the crystal surface is parallel to the (00l)-plane. The estimated c-axial lattice constant is about 9.114 Å, close to the previously reported values.^[20] The thin-plate-like crystals [inset of Fig. 1(b)] are also consistent with the layered structure of RbV₃Sb₅. The measurement of EDX by examination of multiple points on the crystals gives the atomic ratio of Rb:V:Sb = 0.90(6):3:5.07(4) when setting the content of V as 3. The composition of Rb is slightly less than 1, indicating that there may be small amount of Rb deficiencies in the present RbV₃Sb₅ crystals.

Zoom In Zoom Out Reset image size

Figure 2(a) exhibits the temperature dependence of ab-plane resistivity ρ_ab (T) and c-axial resistivity ρ_c (T) of RbV₃Sb₅ single crystal from 2 K to 300 K. The zero-field ρ_ab (T) exhibits a metallic behavior in the measured temperature range and the residual resistivity ratio (RRR), defined as ρ_ab (300 K)/ρ_ab (2 K), is about 44, indicating the high quality of crystals. At T^* ∼ 103 K, the ρ_ab (T) shows an inflection point and it is related to the onset of charge ordering transition.^[21,25] It should be noted that the T^* is higher than those in KV₃Sb₅ and CsV₃Sb₅,^[21,22] implying that the relationship between T^* and the lattice parameters (or ionic radius of alkali metal) is not monotonic. At μ₀ H = 14 T, ρ_ab(T) is insensitive to magnetic field when T > T^* but the significant magnetoresistance (MR) appears gradually below T^*. On the other hand, the ρ_c (T) has a much larger absolute value than the ρ_ab (T). The ratio of ρ_c /ρ_ab is about 7 at 300 K and increases to about 33 when temperature is down to 2 K, manifesting a significant 2D nature of RbV₃Sb₅. However this anisotropy is smaller than that in CsV₃Sb₅, which could be partially ascribed to the smaller interlayer spacing between two V-Sb slabs.^[21] More importantly, in contrast to ρ_ab (T), the ρ_c (T) shows a remarkable upturn starting from T^* with a maximum at about 97 K and this behavior is distinctly different from that in CsV₃Sb₅.^[21] It is suggested that the q _CDW in RbV₃Sb₅ may have a c-axial component, leading to the significantly gapped FS along the k_z direction. Similar behavior has also been observed in PdTeI with CDW vector q _CDW = (0, 0, 0.396)^[27] and GdSi with spin density wave (SDW) vector q _SDW = (0, 0.483, 0.092).^[28] Figure 2(b) exhibits the ρ_ab (T) as a function of temperature below 1.3 K. It can be seen that there is a sharp resistivity drop appearing in the ρ_ab (T) curve at zero field and it corresponds to the superconducting transition. The onset superconducting transition temperature T_{c, onset} determined from the cross point of the two lines extrapolated from the high-temperature normal state and the low-temperature superconducting state is 0.92 K with the transition width ΔT_c = 0.17 K. This T_c is lower than that of CsV₃Sb₅ (T_c ∼ 2.5 K) whereas very close to that of KV₃Sb₅ (T_c ∼ 0.93 K).^[21,22]

Zoom In Zoom Out Reset image size

The charge ordering transition also has a remarkable influence on the magnetic property of RbV₃Sb₅. As shown in Fig. 2(c), the magnetization M(T) curve exhibits a relatively weak temperature dependence with a small absolute value above T^*, reflecting the Pauli paramagnetism of RbV₃Sb₅. In contrast, when T < T^*, there is a sharp drop in the M(T) curve because of the decreased carrier density originating from the partially gapped FS by the charge ordering transition. In addition, the nearly overlapped zero-field-cooling (ZFC) and field-cooling (FC) M(T) curves also suggest that this anomaly should be due to certain density wave transition not an antiferromagnetic one. Figure 2(d) shows the temperature dependence of heat capacity C_p(T) of RbV₃Sb₅ single crystals measured between T = 2 and 117 K at zero field. It can be seen that there is a jump at ∼102 K, in agreement with the T^* obtained from resistivity and magnetization measurements. The jump in the C_p(T) curve of RbV₃Sb₅ is similar to those of KV₃Sb₅ and CsV₃Sb₅,^[20–22] suggesting the same origin of this anomaly of heat capacity from the charge ordering transition. The electronic specific heat coefficient γ and phonon specific heat coefficient β can be obtained from the linear fit of low-temperature heat capacity using the formula C_p/T = γ + β T² [inset of Fig. 2(d)]. The fitted γ and β is 17(1) mJ·mol⁻¹·K⁻² and 3.63(2) mJ·mol⁻¹·K⁻⁴, respectively. The latter gives the Debye temperature Θ_D = 168.9(3) K using the formula Θ_D = (12π⁴ NR/5β)^1/3, where N is the number of atoms per formula unit and R is the gas constant. The electron-phonon coupling λ_e-ph can be estimated with the values of Θ_D and T_c using McMillan's formula^[29]

$$\begin{eqnarray}&&{\lambda }_{{\rm{e-ph}}}=\displaystyle \frac{1.04+{\mu }^{\ast }\mathrm{ln}({\Theta }_{{\rm{D}}}/1.45{T}_{{\rm{c}}})}{(1-0.62{\mu }^{\ast })\mathrm{ln}({\Theta }_{{\rm{D}}}/1.45{T}_{{\rm{c}}})-1.04},\end{eqnarray} \tag{ 1 }$$

where μ^* is the repulsive screened Coulomb potential and is usually between 0.1 and 0.15. Assuming μ^\ast = 0.13, the calculated λ_e-ph is about 0.489, implying that RbV₃Sb₅ is a weakly coupled BCS superconductor.^[30]

The MR [=[ρ_ab(μ₀ H)−ρ_ab (0)]/ρ_ab (0)] of RbV₃Sb₅ is negligible above T^* and increases gradually below T^* [Fig. 3(a)], consistent with the ρ_ab (T) data [Fig. 2(a)]. At low temperature, the MR does not saturate up to 14 T and the Shubnikov-de Haas (SdH) QOs can be clearly observed at low-temperature and high-field region [inset of Fig. 3(a)]. The MR at 2 K can be fitted using the formula MR = A(μ₀ H)^α with α = 1.001(5) [inset of Fig. 3(a)], such linear behavior of MR extends to T^*, especially at μ₀ H > 3 T. Figure 3(b) shows the field dependence of Hall resistivity ρ_yx (T,μ₀ H) at several typical temperatures. At high temperature, the values of ρ_yx (T,μ₀ H) are negative with nearly linear dependence on field. When decreasing temperature below 50 K, the ρ_yx (T,μ₀ H) becomes positive but the linear field dependence is almost unchanged in high-field region. Similar to the MR curves, the SdH QOs appear at low temperatures. The Hall coefficient R_H obtained from the linear fits of ρ_yx (T,μ₀ H) curves are shown in Fig. 3(c). The strong temperature dependence of R_H implies that RbV₃Sb₅ is a multi-band metal, consistent with theoretical calculations and ARPES measurements of KV₃Sb₅ and CsV₃Sb₅.^[20,21,24] At high temperature, the negative R_H suggests that the electron-type carriers are dominant, which could originate from the electron pockets around Γ and K points of BZ.^[20,21,24] The most remarkable feature is that the weakly temperature-dependent R_H starts to decrease rapidly below T^* and changes its sign to positive at about 40 K. Such behavior is very similar to the typical CDW materials NbSe₂ and TaSe₂,^[31] and SDW system GdSi.^[28] Both theory and STM results indicate that the q _CDW connects the M point when the Fermi level is close to the van Hove filling as in the case of AV₃Sb₅.^[18–21,25] Moreover, there are a band with van Hove singularity and a pair of Dirac-cone like bands near the M point,^[25] which can form hole pockets especially when the E_F shifts downward slightly due to the slight Rb deficiency.^[20,21] Therefore, the charge order may lead to the gap opening of hole bands not electron ones. It seems very peculiar that the ρ_ab (T) becomes smaller with positive R_H in the charge ordering state even though the portions of hole-type FSs are gapped. Here, we explain this phenomenon in the framework of two-band model. According to the two-band model in low-field region,^[32]

$$\begin{eqnarray}&&{R}_{{\rm{H}}}=\displaystyle \frac{{\rho }_{yx}}{{\mu }_{0}H}=\displaystyle \frac{{n}_{{\rm{h}}}{\mu }_{{\rm{h}}}^{2}-{n}_{{\rm{e}}}{\mu }_{{\rm{e}}}^{2}}{e{({n}_{{\rm{h}}}{\mu }_{{\rm{h}}}+{n}_{{\rm{e}}}{\mu }_{{\rm{e}}})}^{2}},\end{eqnarray} \tag{ 2 }$$

where μ_e,h and n_e,h are the mobilities and densities of electron- and hole-type carriers, respectively. Because of zero-field ρ_ab (0) = 1/σ_ab (0) = 1/(n_h eμ_h + n_e eμ_e),

$$\begin{eqnarray}&&{R}_{{\rm{H}}}/{\rho }_{ab}(0)=\displaystyle \frac{{n}_{{\rm{h}}}{\mu }_{{\rm{h}}}^{2}-{n}_{{\rm{e}}}{\mu }_{{\rm{e}}}^{2}}{{n}_{{\rm{h}}}{\mu }_{{\rm{h}}}+{n}_{{\rm{e}}}{\mu }_{{\rm{e}}}}.\end{eqnarray} \tag{ 3 }$$

The derived R_H/ρ_ab (0) with the dimension of mobility is shown in Fig. 3(d). According to Eq. (3), if the ${n}_{{\rm{e}}}{\mu }_{{\rm{e}}}^{2}$ is much larger than the ${n}_{{\rm{h}}}{\mu }_{{\rm{h}}}^{2}$ which should be the case of T > T^*, the R_H/ρ_ab (0) is negative and the 1/(e|R_H|) will be close to n_e, which is about 1.6 × 10²² cm⁻³ at 300 K. On the other hand, when the T is just below T^*, the μ_h may still not increase remarkably and the n_h decreases continuously because the FS reconstruction has not finished yet, manifesting from the drops of M(T) curves shown in Fig. 2(c). This would result in an even negative value of R_H/ρ_ab (0), which can be clearly seen in the inset of Fig. 3(d). In contrast, when the T is far below T^* (<70 K), the n_h becomes insensitive to temperature and the μ_h may be much larger than the μ_e because both of electron and hole mobilities have the temperature dependence BT⁻ⁿ with different B and n values. It will lead to a sign reversal of R_H/ρ_ab (0) to positive even the n_h is smaller than the value above T^*. This also explains the even smaller ρ_ab (T) below T^*. Since the strongly CDW-like coupled portions of FSs near M point may play a negative role in conductivity above T^*, the carrier scattering around this area can be effectively reduced when entering charge ordering state, and thus the μ_h can enhance significantly.^[33] Similar discussion about the sign change of R_H has been developed by Ong for 2D multiband system and applied to Sr₂RuO₄ and CDW material 2H-NbSe₂.^[34–36]

Zoom In Zoom Out Reset image size

Analysis of SdH QOs provides insight on the features of FSs and carriers further. After subtracting the slowly changed part of ρ_ab (μ₀ H) (≡⟨ρ_ab ⟩), the oscillation parts of resistivity Δ ρ_ab = ρ_ab − ⟨ρ_ab ⟩ as a function of 1/(μ₀ H) for H || c at several representative temperatures are shown in Fig. 4(a). The amplitudes of QOs decrease with increasing temperature or decreasing field, but still observable at 30 K. The fast Fourier transform (FFT) spectra of the QOs reveal two principal frequencies F_α = 33.5 T and F_β = 117.2 T [Fig. 4(b)]. Both of them are slightly smaller than those in KV₃Sb₅,^[24] indicating that RbV₃Sb₅ has smaller extremal orbits of FSs than KV₃Sb₅. According to the Onsager relation F = (ħ/2π e)A_F, where A_F is the area of extremal orbit of FS, the determined A_F is 0.0032 and 0.011 Å⁻² for α and β extremal orbits, respectively. These A_F's are very small, taking only about 0.0934% and 0.321% of the whole area of BZ in the k_x –k_y plane when taking the lattice parameter a = 5.4715 Å.^[20] The effective mass m^* can be extracted from the temperature dependence of the amplitude of FFT peak using the Lifshitz–Kosevich (LK) formula

$$\begin{eqnarray}&&\Delta {\rho }_{ab}\propto \displaystyle \frac{X}{\sinh X},\end{eqnarray} \tag{ 4 }$$

where X = 2π² k_B T/ħω_c = 14.69m^* T/μ₀ H_avg with ħω_c being the cyclotron frequency and μ₀ H_avg (=9 T) being the average value of the field window used for the FFT of QOs.^[37,38] As shown in Fig. 4(c), the temperature dependence of FFT amplitude of F_α can be fitted very well using Eq. (4) and the obtained m^* is 0.091(2)m_e, where m_e is the bare electron mass. This value is even smaller than that in KV₃Sb₅ (0.125m_e for the α orbit).^[24] The small extremal cross sections of FSs accompanied with such light m^* could be related to the highly dispersive bands near either M point or along the Γ–K path of BZ.^{[20,21,24,25]}

Zoom In Zoom Out Reset image size

In summary, we have carried out the detailed study on physical properties of RbV₃Sb₅ single crystals grown by the self-flux method. RbV₃Sb₅ single crystals exhibit a superconducting transition at T_{c, onset} = 0.92 K with a weakly coupling strength, accompanied with anomalies of properties at T^* ∼ 102–103 K. The high-temperature anomaly could be related to the formation of charge ordering state and it results in the sign change of R_H, which can be partially ascribed to the enhancement of mobility for hole-type carriers due to the reduced carrier scattering by the gapping of strongly CDW-like coupled portions of FSs. Furthermore, there are some very small FSs with rather low m^*, indicating the existence of highly dispersive bands near E_F in RbV₃Sb₅. Moreover, due to the similar electronic structure of RbV₃Sb₅ to KV₃Sb₅ and CsV₃Sb₅,^[20] RbV₃Sb₅ should also be a candidate of Z₂ topological metal. Therefore, the V-based kagome metals AV₃Sb₅ provide a unique platform to explore the interplay between nontrivial band topology, electronic correlation and possible unconventional superconductivity.