Multiband superconductivity and possible nodal gap in RbCr$_{3}$As$_{3}$ revealed by Andreev reflection and single-particle tunneling measurements

TL;DR

This study reveals multiband superconductivity with one potentially nodal or highly anisotropic gap in RbCr₃As₃, using Andreev reflection and tunneling measurements, highlighting directional dependence of observed gaps.

Contribution

First combined use of PCAR and tunneling spectroscopy to identify multiband superconductivity and gap anisotropy in RbCr₃As₃, revealing a possible nodal gap.

Findings

Evidence of two superconducting gaps at 1.8 meV and 5 meV.

Single-particle tunneling shows only the smaller gap, suggesting anisotropy or nodes.

The larger gap may have an s-wave symmetry, but its absence in tunneling indicates directional effects.

Abstract

By measuring point-contact Andreev reflection (PCAR) spectra in the newly discovered chromium-based quasi-one-dimensional superconductor RbCr$_{3}$As$_{3}$, we find clear evidence of two superconducting components, i.e., one having a gap value of about 1.8 meV and another with a gap value of about 5 meV. Since the current injection may have components in both the direction parallel and perpendicular to the [(Cr$_{3}$As$_{3}$)$^{-}$]$_\infty$ chains in the PCAR measurements, it naturally explains the two-component feature observed in this multi-band superconductor. Detailed analysis shows that the larger gap may have an $s$-wave nature. We then carry out the single-particle tunneling measurements based on a scanning tunneling spectroscope by using the needle-like sample as the tip, and in this case the measured current is mainly parallel to the [(Cr$_{3}$As$_{3}$)$^{-}$]$_\infty$ chains. The single-particle tunneling spectra show only one gap feature with a gap value of about 1.8 meV. Fitting to the single particle tunneling spectra indicates that the gap should have a large anisotropy or even node(s). We argue that the absence of the larger gap may be related to the direction of the injecting current. Therefore, our combined experiments show the multiband superconductivity with one gap being nodal or highly anisotropic.

Equations

$$G(V)={\int }_{-\infty }^{+\infty }\frac{df(E-V,T)}{dV}\left[1+A(E)-B(E)\right]dE,$$

$$G=w_1{G}_1+(1-w_1)G_2.$$

Full text

I Introduction

In materials containing 3d transition-metal elements, many new superconductors have been discovered and some of them show the unconventional superconductivity. The latter is argued to have a close relationship with the strong correlation effect of electrons in the 3d orbitals. For the compounds containing 3d transition metal chromium, superconductivity was first observed in the binary CrAs under pressure with a transition temperature $T_{c}\approx 2$ K CrAs . After that some new quasi-one-dimensional (quasi-1D) Cr-based superconductors were discovered at ambient pressure, and they are mainly classified into two families including 233 and 133 with the chemical formulas of $A_{2}$Cr3As3 and $A$Cr3As3 ($A=$ K, Rb, Cs), respectively K233 ; Rb233 ; Cs233 ; K133 ; Rb133 . Recently, the superconductor Na2Cr3As3 was also successfully synthesized Na233 , and it has the highest $T_{c}$ = 8.6 K among these Cr-based superconductors. This discovery has been extended to the Mo-based system, superconductivity with $T_{c}$ of about 10.3 K has been reported in K2Mo3As3 Mo233Ren . It has been claimed that the 133 phase may be more stable against water moisture in air than the 233 phase, thus it is easier to be handled for many elegant measurements. The structural characteristic of the Cr-based materials is that it contains the 1D [Cr3As3]∞ chains separated by alkali metal atoms K233 , which makes the physical properties and superconductivity of these materials very interesting.

Clearly, the newly discovered Cr-based superconductors provide a new platform to research on the superconductivity related to the pairing of 3d electrons. The band structure calculations of $A_{2}$Cr3As3 reveal three sets of Fermi surfaces (FSs) which are mainly contributed by 3d orbitals of chromium. The FSs consist of two flat ones from quasi-1D $\alpha$ and $\beta$-bands as well as a three-dimensional (3D) one from 3D $\gamma$-band CaoCCal ; DaiJHPRL ; HuJPTripletPRB ; ZhouYCal . The quasi-1D FSs were then confirmed by angle-resolved photoemission spectroscopy measurements in K2Cr3As3 ARPES . Because of the complex band structure, different kinds of unconventional pairing symmetries were predicted for $A_{2}$Cr3As3 superconductors theoretically. One interesting proposal is that a nodal triplet $p_{z}$-wave pairing driven by the ferromagnetic fluctuations within the Cr sublattice occurs on the quasi-1D $\beta$ band HuJPTripletPRB . Another prediction is that the pairing can change from a spin-triplet state $f_{y(3x^{2}-y^{2})}$ with line nodes on the 3D $\gamma$-band to a spin-triplet fully gapped state $p_{z}\hat{z}$ at the quasi-1D $\alpha$-band when the strength ratio between Hund’s coupling and inter-orbital repulsion decreases ZhouYCal . Meanwhile from some other theoretical works, the quasi-1D and 3D FSs may couple with each other, which results in a strongly anisotropic singlet pairing and gap nodal rings on the 3D Fermi sphere when the local repulsion is strong enough KimCal .

Up to now many experimental results suggest unconventional superconductivity in $A_{2}$Cr3As3. The very large upper critical field $H_{c2}$ exceeding the Pauli limit may suggest a strong and unconventional pairing mechanism CaoZhuReview ; K233 ; Rb233 ; BalakirevHc2 . Furthermore, in K2Cr3As3, it was found that $T_{c}$ decreases significantly with increase of the density of non-magnetic impurities, which may suggest the sign-reversal gaps of superconductivity CaoGHImpurity . The similar conclusions have also been obtained from the absence of the Hebel-Slichter coherence peak of $1/T_{1}$ just below $T_{c}$ in NMR studies ImaiNMR ; ZhengGQNMR . In addition, $1/T_{1}$ decreases rapidly below $T_{c}$ following a function of $T^{5}$ at low temperatures in Rb2Cr3As3, which may be a proof of the presence of point nodes in the gap function ZhengGQNMR . The possible line-nodal gap was proved by low-temperature electronic specific-heat data LuoJLSH and the muon-spin relaxation data uSR in K2Cr3As3, as well as the penetration depth measurements in K2Cr3As3 and Rb2Cr3As3 YuanHQPD1 . The anisotropic superconductivity was observed in $A_{2}$Cr3As3 not only when the magnetic field is applied along or perpendicular to the $c$-axis, but also with the in-plane rotation of field CaoZhuReview , and the threefold in-plane $H_{c2}$ modulation was observed and argued to be related to the spin-triplet pairing ZhuZWInplanAnis .

The $A$Cr3As3 family has half amount of the alkali metal atoms comparing to the $A_{2}$Cr3As3 family, and the space groups for these two families are different K133 ; Rb133 . However, they both contain the 1D [Cr3As3]∞ chains but with different ionic valences of the Cr3As3 units. The normal-state electrical resistivity also has different temperature dependent behaviors in these two families of the materials K133 ; Rb133 . Theoretical calculation predicted that FSs are constructed by two or three quasi-1D and two 3D bands DaiJHCal ; YangFCal . The superconducting pairing symmetry can be triplet $f$ or $p_{z}$ wave or singlet $s^{\pm}$ wave with different interaction parameters YangFCal . The nodal superconducting gap structure in RbCr3As3 was inferred from the thermal conductivity measurements which show a sizable value of the thermal conductivity coefficient in the zero temperature limit at ambient field, and the magnetic field enhancement is clearly faster than that expected for a full-gap superconductor LiSYTC . However, more experiments are required to derive the information of the superconducting gap in the Cr-based family.

Andreev reflection and single-particle tunneling measurements are very useful and combinatorial to detect the superconducting gap. In this work, we will present a detailed study of the superconducting gap structure on the RbCr3As3 single crystals by using the point-contact Andreev reflection (PCAR) and scanning tunneling microscopy/spectroscopy (STM/STS) measurements. The results show the existence of two superconducting components. The large gap has the value of about 5 meV, and shows only on the Andreev reflection spectra; while the smaller one with the value of about 1.8 meV can be observed both on the Andreev reflection and the STS measurements. We suggest that the smaller gap is highly anisotropic or even nodal based on the fittings to the STS data.

II Experimental methods and superconductivity characterization

The RbCr3As3 single crystals were grown by the deintercalation process of Rb+ ions from Rb2Cr3As3 precursors Rb133 . The samples are loosely assembled by many needle-like single crystals with diameter of sub- to several micrometers, and the $c$-axis is defined as the direction along the needle main axis or the [(Cr3As3)-]∞-chains of the sample. The RbCr3As3 samples are stable in the air. The X-ray diffraction data measured on a bundle of crystals can be found in the previous work Rb133 measured by a Bruker single-crystal X-ray diffractometer, and all the diffraction peaks can be well indexed with the hexagonal space group P63/m.

The needle-like single crystals are loosely assembled to form a bundle, and there is large separation between these needles; hence a direct measurement of the cross-sectional size of a bundle-like sample induces a very large error for determining the resistivity. In order to make a more elegant resistivity measurement, we separate a needle-like single crystal from a bundle of samples. The electrical resistivity was measured in a physical property measurement system (PPMS-9, Quantum Design) by using the standard four-probe method, and the current is applied along the $c$-axis of the sample. The widths and the thicknesses of the samples are detected by scanning electron microscope (SEM, Phenom ProX), and the typical values are from 3 to 7 $\mu$m for these samples. Figure 1 shows temperature dependent resistivity for a separate RbCr3As3 single crystal at 0 T, and the inset shows the enlarged view near the superconducting transition.The onset transition temperature $T_{c}$ is 7.17 K determined from the 90% of the normal-state resistivity, and the transition width is about 0.64 K determined by using criterion of 10% and 90% of the normal-state resistivity. The residual resistance ratio $RRR=\rho(300\ \mathrm{K})/\rho(8\ \mathrm{K}>T_{c})=2.17$ is consistent with previous reports Rb133 ; LiSYTC , but the measured residual resistivity $\rho(8\ \mathrm{K})=249$ $\mu\Omega\cdot$cm here is much smaller than the values obtained previously Rb133 . The resistivity in our measurements are reproducible, and the difference is about 20% for different samples. The obtained resistivity value is even smaller than that obtained by weighing the mass to calculate the cross-section LiSYTC . This may be caused by the disconnections located on some needle-like crystals, which increases the total resistivity in that work. We note that the resistivity here is also much larger than the values of $A_{2}$Cr3As3, and the possible explanation for this may be the crystal defects and lattice deformation induced by the deintercalation process Rb133 ; K133 . In addition, temperature dependent behaviors of resistivity are very different between these two systems Rb133 ; K133 , and the resistivity increases almost linearly with increase of temperature in RbCr3As3. This feature is similar to the one observed in cuprates which have a typical non-Fermi-liquid behavior resistivity , and it may be related to the spin scattering due to the magnetic frustration DaiJHCal in the material. The much larger resistivity in $A$Cr3As3 than that in $A_{2}$Cr3As3 also needs further investigation.

The samples for PCAR or STS measurements are also divided from a bundle of single crystals, and the typical sizes of selected samples are 300-500 $\mu$m in length and about several micrometers in width. For the PCAR measurements, the point contact was constructed by sticking a thin gold wire with the diameter of 16 $\mu$m to the sample by using a tiny bit of silver paste (DuPont, 4929N), and a typical configuration is shown in the inset of Fig. 2(a). The length of the sample in the point-contact region was about 20 $\mu$m, and two ends of the sample were buried in silver paste more than 100 $\mu$m in length to reduce the contact resistance of these electrodes. Since the contact constructed by silver paste has a finite size, the current injection may have components both parallel and perpendicular to the [(Cr3As3)-]∞-chains. The PCAR measurements were carried out in PPMS-9 by using a home-made setup. The ac modulation for the differential conductance measurement is several microamperes with the frequency of 985 Hz in the lock-in measurements.

The STS measurements were carried out in an ultra-high vacuum, low-temperature and high-magnetic field scanning tunneling microscope (USM-1300, Unisoku Co., Ltd.). We made the STS tip by a separate RbCr3As3 single crystal in the following steps. We first found a very thin wire of the sample by tweezers, and then cut off one end to get a relatively fresh terminal for the STS tip. Afterwards, we used the silver paste to glue most part of the sample on a Pt/Ir tip, and made the newly-cut prominent end face towards a flat Au(111) single-crystal flake sample (purity 99.999%). The single-particle tunneling spectra were recorded between the needle-like sample tip and the Au single crystal in an ultra-high vacuum chamber with a base pressure of about $1\times 10^{-10}$ torr. The direction of injecting current is mainly along the [(Cr3As3)-]∞-chains or $c$-axis of the sample for the STS measurements. The setpoint condition for the spectrum measurements is $I_{set}=50$ pA and $V_{set}=10$ mV, and the ac modulation is 0.5 mV with the frequency of 871.773 Hz.

III Point-contact Andreev reflection results

Figure 2(a) shows the measured PCAR spectra at different temperatures and 0 T. Since the upper critical field is very high, e.g., $\mu_{0}H_{c2}=72.4$ T in the zero-temperature limit and $\mu_{0}H_{c2}\approx 52$ T at 3 K for RbCr3As3 samples Rb133 , the highest magnetic field of 9 T provided by our PPMS gives only negligible influence on the spectrum at 3 K. For this reason, we only show the temperature dependence of PCAR spectra to investigate the superconducting gap of this material. One can see that there are always zero-bias peaks on the spectra taken below $T_{c}$. This kind of peak often appears on the spectra taken by the PCAR measurements on an unconventional superconductor, especially when the injecting current is along the nodal direction. In cuprate superconductors, this has been well investigated both in theory PCAR cuprate theory1 ; PCAR cuprate theory2 ; PCAR cuprate theory3 and by experiments LauraGreene ; WeiJ ; PCAR cuprate experiment . It should be noted that there are a pair of hump features at about $\pm 5$ meV on the spectrum measured at 3 K, and they are marked by black arrows in Fig. 2(a). We will show that this corresponds to the second superconducting gap in RbCr3As3 through the fitting. Beside the superconducting gap features, some unexpected small dips of $dI/dV$ can be observed at low temperatures and energies just above the superconducting gap energy, e.g., from 10 to 15 meV for the spectrum measured at 3 K. The bottom energies of these dips decrease with increase of temperature. This kind of dip feature seems to be very common in point-contact measurements and was argued to be induced by the critical current effect or the heating effect when the voltage was high and the point contact was not in the pure ballistic limit JcPC . From temperature dependence of the spectra, one can see that the height of the zero-bias peak decreases with increase of temperature, and the spectrum shape changes to a V-shaped normal-state background when the temperature reaches 7 K $\approx T_{c}$. To get rid of the normal-state background signal, we normalized the spectrum measured at 3 K by dividing the one measured at 7 K, and the normalized curve is shown by the open circles in Fig. 2(b-e). The differential conductance at zero-bias is about 1.45 times larger than the value of the high-energy normal-state background for the normalized spectrum at 3 K, which suggests that the barrier height of the junction is small and the contact is closer to the Andreev reflection (normal metal/superconductor, NS) limit instead of the tunneling (normal metal/insulator/superconductor, NIS) limit.

In order to get in-depth understanding of superconducting gaps from the PCAR spectra, we fit the data with the Blonder-Tinkham-Klapwijk (BTK) theory BTK . The expression of differential conductance $G=dI/dV$ from BTK theory reads,

[TABLE]

where $f(E,T)$ is the Fermi distribution function, and $A(E)$ and $B(E)$ represent the contributions from the Andreev reflection and normal reflection, respectively. Then the PCAR spectra can be described by the superconducting gap $\Delta$, the junction barrier height $Z$, the broadening factor $\Gamma$, and temperature $T$ PCAR cuprate theory3 . Since there is an obvious two-gap feature on the PCAR spectrum measured at 3 K, we use a two-gap BTK model MgB2PCAR to fit the normalized differential conductance $G$ as

[TABLE]

Here $w_{1}$ is the weight for the smaller gap, and $G_{1}$ and $G_{2}$ are the normalized differential conductances originated from different bands with smaller and larger superconducting gaps, respectively. The BTK fitting results by different kinds of superconducting gaps are shown in Fig. 2(b-e). It is found that the PCAR spectra can be well fitted by the BTK model. For the $p$-wave gap, we use the gap function of $\Delta_{p}\sin\theta$. The gap minimum $\Delta_{\mathrm{min}}=-\Delta_{p}$ while the gap maximum $\Delta_{\mathrm{max}}=+\Delta_{p}$, and then the ratio $\Delta_{\mathrm{min}}/\Delta_{\mathrm{max}}=-1$. For the anisotropic-$s$-wave gap, we use the gap function of $\Delta_{\mathrm{ani-}s}(x\cos 2\theta+1-x)$, and $\Delta_{\mathrm{min}}/\Delta_{\mathrm{max}}=1-2x$ which reflects the gap anisotropy. One can see that the fitting results are very consistent with the measured spectra whether the smaller gap takes an $s$-wave [Fig. 2(b)] or a $p$-wave form [Fig. 2(c)]. Therefore, we can not make judgement whether the smaller gap has the gap nodes from the fitting results. However, for the larger gap, the $p$-wave gap function cannot fit the data well with any fitting parameters, and an example is shown in Fig. 2(d). The fitting curve deviates from the experimental data in the ranges marked by the orange arrows. We then use the anisotropic-$s$-wave gap function to fit the larger gap, and the fitting requires the gap anisotropy $x\leq 0.2$ or $\Delta_{\mathrm{min}}/\Delta_{\mathrm{max}}\geq 0.6$. The fitting result with $x=0.2$ is shown in Fig. 2(e). Hence for the larger gap, there may be a little gap anisotropy exists, but the possibility for the nodal gap is rare from the fitting results. In order to obtain the temperature dependent behavior of the superconducting gap, we use the two-gap BTK model with different gap functions to fit the normalized PCAR spectra at different temperatures, and the fitting results are shown in Fig. 3(a,d). Figure 3(c,d) show the obtained temperature dependence of the two superconducting gaps derived from the fittings, while the solid lines are the theoretical calculations carried out by using the Bardeen-Cooper-Schrieffer (BCS) gap model.

In order to verify our experimental results, we carried out the PCAR experiments on different samples, and the data for two other samples are shown in Fig. 4. The normalized spectra measured in Sample-2 and Sample-3 show the similar two-gap feature as in Sample-1, but with slightly different gap values obtained from the fitting procedures. The spectrum weight from the band(s) with smaller gap is much larger than that with the larger gap, and the weight ratios are all about 3:1.

BTK theory can only describe the spectra measured on the contacts in the ballistic regime, and we thus check the status of the contact junction. The residual residual resistivity $\rho(8\ \mathrm{K})\approx 259\ \mu\Omega\cdot$cm in RbCr3As3. If we take a simplified model of a spherical Fermi surface with average Fermi momentum $\overline{k_{F}}=0.5\pi/a_{0}$ DaiJHCal ; YangFCal and the lattice constant $a_{0}=9.37$ Å  Rb133 , we can estimate the mean free path $l\approx 17$ nm. The junction resistance varies from 159 to 4 $\Omega$ in our point-contact measurements, which corresponds to the size of the contact $a$ from 11 to 68 nm based on the Sharvin formula $R_{s}=4\rho l/(3\pi a^{2})$ PCreview . Here the mean free path approximately equals to the contact size, which means that the contact may be between the diffusive regime and the ballistic regime. Although the fitting formulas are different in these two regimes, the obtained superconducting gap values are very close to each other PCreview . In addition, the point contact is usually consisted by many individual contacts which have bigger contact resistances and smaller sizes PCreview . So the contact here may still locate in the ballistic regime.

We have two more arguments to address the reason that the point-contact junction is not in the thermal regime. Firstly, the differential conductivity increases with increasing of the bias voltage in high bias-voltage range as shown in Fig. 2(a), which means that the junction resistance decreases with increasing of the bias voltage. This contradicts what expected in the thermal regime of a point contact, since the contact resistance would increase due to stronger heating effect with larger bias voltage. In addition, the V-shaped spectra obtained above $T_{c}$ by PCAR measurements are similar to the ones measured by STS. Secondly, the barrier height $Z$ from the fitting procedure is always very small for all three point-contact junctions, which means that the contact is close to the ballistic regime or the intermediate regime. From the reasons mentioned above, we conclude that the contact is not in the thermal region, and we can get the correct gap information from the measured PCAR spectra.

IV Scanning tunneling spectroscopy results

The above-mentioned Andreev reflection results show two-superconducting-gap feature clearly. However, we can not judge whether the smaller gap is nodal or nodeless from the PCAR data and fitting results presented above. We then do the further measurements by STS at temperature as low as 400 mK. The needle-like sample, which was attached to the end of the Pt/Ir tip, was used as the tip for the STS measurements. The tunneling spectra were taken between the sample and a flat gold flake as illustrated by the schematic figure in the inset of Fig. 5(b). The tunneling spectra measured at different temperatures are displayed in Fig. 5(a) and (b) for Sample-4 and Sample-5, respectively. One can see that the normal state spectra taken above $T_{c}$ show the V-shape, which is somewhat similar to the normal-state background obtained by PCAR measurements. In addition, the superconducting spectra are of the typical single-particle tunneling type with only one pair of coherence peaks for both samples, which means that we detect mainly the contribution from one superconducting component. We use the spectra measured above $T_{c}$ as the backgrounds and show the normalized spectra at 400 mK in Fig. 6(a-f) for Sample-4 and 5. Then the Dynes model Dynes , which can be regarded as the situation in BTK model with $Z=\infty$, was used to fit the normalized spectra using different kinds of gap functions. The fitting results are also shown in Fig. 6(a-f) as solid lines. For Sample-4, the zero-bias differential conductance is as low as about 24% of the normal state background far above the superconducting gap. Obviously from Fig. 6(a), one can see that the fitting curves by using an isotropic $s$-wave gap with any fitting parameters cannot catch up the main features of the measured spectrum. This suggests that the gap anisotropy is necessary for the superconducting gap from the fitting procedure. We find that the minimum anisotropy for the fitting corresponds to the ratio of $\Delta_{\mathrm{min}}/\Delta_{\mathrm{max}}=0.4$, and the fitting result is shown in Fig. 6(b). In addition, the fitting by a $p$-wave gap can also catch up the main features of the experimental data as shown in Fig. 6(c) for Sample-4. For Sample-5, the zero-bias differential conductance is about 61% of the normal state value, and the in-gap suppression of density of states is much smaller than that in Sample-4. Nevertheless, the model by an isotropic $s$-wave gap cannot fit the experimental data well in the energy ranges near the arrow positions. The gap anisotropy is also necessary for the fitting procedure. The minimum gap anisotropy expressed by $\Delta_{\mathrm{min}}/\Delta_{\mathrm{max}}=0.9$ is required for the fitting, and the fitting result is shown in Fig. 6(e). Besides, the model with a $p$-wave gap can also fit the experimental data well as shown in Fig. 6(f).

The STS data show the clear existence of the superconducting gap with the gap value from 1.46 to 1.97 meV which is consistent with the smaller gap values from the PCAR measurements. However, the normalized zero-bias conductance ranging from about 0.24 to 0.6 makes it difficult to determine the precise gap function of this gap. The finite zero-bias conductance may be attributed to the impurity scattering effect for a nodal superconductor. Of course the finite zero-bias conductance can also be induced by the imperfection of the tip and a large scattering effect is inevitable. Although the origin for the zero-bias conductance is unclear yet, we confidently conclude that the gap anisotropy is necessary for the fitting procedure.

V Discussion and conclusion

In order to determine the coupling strength, we need to evaluate the gap ratio $2\Delta/k_{B}T_{c}$. For the anisotropic or nodal gap, it is better to use the average gap value to calculate the gap ratio. Thus we use the formula $\overline{\Delta}^{2}=\frac{1}{2\pi}\int_{0}^{2\pi}\Delta^{2}(\theta)d\theta$ to calculate the average gap $\overline{\Delta}$. All the obtained average gap values are plotted in Fig. 7(a) from different fittings to the spectra measured in different samples. The average gap values change with gap functions or gap anisotropy ratios $\Delta_{\mathrm{min}}/\Delta_{\mathrm{max}}$. In Fig. 7(a), the solid lines, which just connects two data points, are used for guidance; they cover the range of anisotropy $\Delta_{\mathrm{min}}/\Delta_{\mathrm{max}}$ in which the experimental data can be well fitted based on the corresponding gap function. The broadening factor $\Gamma$ used for the fitting in Sample-4 is smallest among all the values for the fittings in other samples, and the fitting provides a constrain that the smaller gap is highly anisotropic or even nodal. If the smaller gap takes a nodal $p$-wave form as predicted by the theory, the mean value of the average gap is $\langle\overline{\Delta}_{p1}\rangle=1.26\pm 0.23$ meV, and the calculated gap ratio is ${2\langle\overline{\Delta}_{p1}}\rangle/{k_{B}T_{c}}=4.1\pm 0.8$. Such gap ratio is larger than 3.03 from weak coupling BCS theory for a $p$-wave gap. If the gap is an anisotropic $s$-wave one with $\Delta_{\mathrm{min}}/\Delta_{\mathrm{max}}=0.4$, the average gap value is just a little bit larger than that from the $p$-wave gap fitting. For the larger gap obtained from the PCAR spectra, it should be nodeless from the fitting procedure and the largest gap anisotropy corresponds to a minimum $\Delta_{\mathrm{min}}/\Delta_{\mathrm{max}}$ value of about 0.6. If the larger gap is an isotropic $s$-wave gap, the mean value of the gap is $\langle{\Delta}_{s2}\rangle=5.0\pm 0.43$ meV which corresponds to a gap ratio of ${2\langle{\Delta}_{s2}\rangle}/{k_{B}T_{c}}\approx 16$. This gap ratio is much larger than 3.53 from weak coupling BCS theory for an $s$-wave gap. The large values of the gap ratio suggest the strong coupling and unconventional superconductivity in this material, and could suggest a pairing probably not mediated by phonons. We note that the recent specific heat measurements revealed a single nodal gap in single crystal of K2Cr3As3; the corresponding ${2\overline{\Delta}}/{k_{B}T_{c}}\approx 11.1$ LuoJLSH which is also a very large value and may be consistent with our observation of the larger gap in RbCr3As3. Another penetration depth measurement reported a possible $d$-wave gap with ${2\overline{\Delta}}/{k_{B}T_{c}}\approx 3.1$ in polycrystalline K2Cr3As3 YuanHQPD , and such ratio is similar to the smaller gap ratio from our measurements in RbCr3As3. In this point of view, there may also be two-gap feature in K2Cr3As3, while the two gaps were selectively detected by different measurement techniques. This is similar to our experimental result here.

From the PCAR data, the weight of the larger gap $w_{2}$ is about $(25\pm 5)\%$ from the fitting procedure, but this weight seems to be negligible for the STS spectra. The possible reason for this result may be related to the different configurations for two kinds of measurements. As shown in Fig. 7(b), the current injection takes some stochastic directions in the PCAR measurements, while the tunneling current is mainly along the $k_{z}$ direction for the STS measurements. $A$Cr3As3 is a typical multi-band system, and a simplified schematic image of the Fermi surface is shown in Fig. 7(c). The Fermi surface consists of three flat sheets from quasi-1D bands and two 3D sheets from 3D bands DaiJHCal ; YangFCal . The density of states contributed by the 3D sheets is much larger than those from the flat sheets DaiJHCal ; YangFCal . In the STS measurements, we see only one component with the gap value comparable to the smaller one derived in the PCAR experiments. In the STM measurements the tunneling matrix element problem may weaken the detection of the 3D band(s) STMreview1 ; STMreview2 . Here in RbCr3As3, the 3D Fermi surfaces are composed by the Cr-$3d_{x^{2}-y^{2}}$ and $3d_{xy}$ orbitals, and the 1D Fermi surfaces have the contribution from Cr-$3d_{z^{2}}$ orbital according to the theoretical calculations DaiJHCal ; YangFCal . Therefore it is possible that the Au single crystal with an approximate $s$-wave electronic structure have much weaker wave function overlapping with the planar Cr-$3d_{x^{2}-y^{2}/xy}$ orbital bands DavisMatrix ; Zhenyu , which leads to a negligible tunneling matrix element along $k_{z}$ direction from the 3D Fermi surface of RbCr3As3. Consequently, the major contribution of the tunneling spectra may come from the 1D Cr-$3d_{z^{2}}$ orbits in STS measurements. However, the reason why we cannot detect the larger superconducting gap by STS measurements needs further theoretical explanations. Since our measurements do not have the capability to assign superconducting gaps to FSs in momentum space, we can only conclude that there is a multi-gap feature with strong-coupling superconductivity in RbCr3As3. In addition, the smaller gap, which was observed by both PCAR and STS measurements with different configurations, is anisotropic or even nodal; while the larger gap is nodelss. According to the theoretical predictions, the superconducting pairing symmetry can be triplet $f$-, $p_{z}$- or singlet $s^{\pm}$-wave with different interaction parameters YangFCal . Our data and fitting results provide useful information for future theoretical work.

In summary, we observe clear two-gap feature from PCAR measurements in RbCr3As3. The gap values yielded from the fitting to the PCAR spectra are about 1.8 and 5 meV. The larger gap may be nodeless. In the STS measurement by using the needle-like sample RbCr3As3 as the tip, we see however only one gap with the value comparable with that of the smaller gap yielded from the PCAR measurements. The smaller gap may be highly anisotropic or even with gap node(s) inferred from STS measurements and related fitting. The absence of the larger gap in the STS measurements may be explained as that the injecting current is mainly along the $k_{z}$ direction. The large ratio of $2\overline{\Delta}/k_{B}T_{c}$ suggests the unconventional superconductivity in this material.

Acknowledgements.

We appreciate the kind help in the PCAR measurements by Cong Ren, Lei Shan and Xingyuan Hou, and useful discussions with Yi Zhou, Xin Lu and Zhenyu Wang. This work was supported by the National Key R&D Program of China (Grant No. 2016YFA0300401), the National Natural Science Foundation of China (Grant No. 11534005, and No. 11774402), and the Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDB25000000).

∗ huanyang@nju.edu.cn

† hhwen@nju.edu.cn