Two-gap superconductivity in Mo$_{8}$Ga$_{41}$ and its evolution upon the V substitution

TL;DR

This study investigates the microscopic superconducting properties of Mo8Ga41 using muon spin rotation experiments, revealing two-gap superconductivity that is suppressed upon V substitution, leading to a single-gap state and potential competing phases.

Contribution

The paper provides the first microscopic evidence of two-gap superconductivity in Mo8Ga41 and shows how V substitution suppresses one gap, altering the superconducting state.

Findings

Two-gap superconductivity in Mo8Ga41 confirmed by μSR.

V substitution suppresses one superconducting gap.

Emergence of a competing state upon V doping.

Abstract

Zero-field and transverse-field muon spin rotation/relaxation ($\mu$SR) experiments were undertaken in order to elucidate microscopic properties of a strongly-coupled superconductor Mo$_{8}$Ga$_{41}$ with $T_{\text{c}}=9.8$ K. The upper critical field extracted from the transverse-field $\mu$SR data exhibits significant reduction with respect to the data from thermodynamic measurements indicating the coexistence of two independent length scales in the superconducting state. Accordingly, the temperature-dependent magnetic penetration depth of Mo$_{8}$Ga$_{41}$ is described using the model, in which two s-wave superconducting gaps are assumed. The V for Mo substitution in the parent compound leads to the complete suppression of one superconducting gap, and Mo$_{7}$VGa$_{41}$ is well described within the single s-wave gap scenario. The reduction in the superfluid density and the evolution of the low-temperature resistivity upon the V substitution indicate the emergence of a competing state in Mo$_{7}$VGa$_{41}$ that may be responsible for the closure of one of the superconducting gaps.

Equations

$$A(t)=A_{\text{s}}(0)\left[\frac{1}{3}+\frac{2}{3}\left(1-{\sigma }_{\text{KT}}^2{t}^2\right)\exp \left(-\frac{{\sigma }_{\text{KT}}^2{t}^2}{2}\right)\right]+A_{\text{bgd}},$$

$$A^{\text{TF}}(t)=$$

$$A_{\text{bgd}}^{\text{TF}}(0)\times \cos ({\gamma }_{\mu }{\mu }_0{H}_{\text{bgd}}t+\varphi ).$$

$${\sigma }_{\text{sc}}\approx 0.172\frac{1-b}{{\kappa }^2}\left[1+1.21(1-\sqrt{b}{)}^3\right],$$

$$\frac{{\lambda }^{-2}(T)}{{\lambda }^{-2}(0)}=\frac{\Delta (T)}{\Delta (0)}\text{tanh}\left[\frac{\Delta (T)}{2k_{\text{B}}T}\right],$$

$$\frac{{\lambda }^{-2}(T)}{{\lambda }^{-2}(0)}=\omega \frac{{\Delta }_{\text{s1}}(T)}{{\Delta }_{\text{s1}}(0)}\text{tanh}\left[\frac{{\Delta }_{\text{s1}}(T)}{2k_{\text{B}}T}\right]+$$

$$(1-\omega )\frac{{\Delta }_{\text{s2}}(T)}{{\Delta }_{\text{s2}}(0)}\text{tanh}\left[\frac{{\Delta }_{\text{s2}}(T)}{2k_{\text{B}}T}\right],$$

Full text

I Introduction

Design of new superconducting materials following empirical rules based on chemical and structural considerations or rigorous analysis of the electronic structure is a challenging task. Recently, systematic work was performed on superconducting intermetallics with endohedral gallium clusters TGa${}_{\text{n}}$ (T is a transition metal)Xie et al. (2015). In this context, Mo8Ga41Bezinge et al. (1984) (the superconducting transition temperature $T_{\text{c}}\simeq 9.7$ K), Mo6Ga31Poole (2000) ($T_{\text{c}}\simeq 8$ K), ReGa5Xie et al. (2015) ($T_{\text{c}}\simeq 2.3$ K), Rh2Ga9Shibayama et al. (2007) ($T_{\text{c}}\simeq 1.9$ K) and Ir2Ga9Shibayama et al. (2007) ($T_{\text{c}}\simeq 2.2$ K) superconductors as well as non-superconducting V8Ga41Girgis et al. (1975) and PdGa5Grin et al. (1997) are considered. By analyzing their electronic structures, Xie et al.Xie et al. (2015) established the interrelations between the critical temperature $T_{\text{c}}$ and valence electron count, thus placing these structurally different intermetallic compounds into an individual class of superconductors.

Mo8Ga41 was firstly synthesized by Yvon et al.Yvon (1975) during their systematic investigation of Ga-rich intermetallic compounds with unusual crystal structures. View of the Mo8Ga41 crystal structure is shown in Fig. 1. The crystal structure is built by MoGa10 polyhedra, which are condensed on the triangular faces of GaGa12 cuboctahedron. A detailed description of the crystal structure can be found elsewhereVerchenko et al. (2016). From the first sight, Mo8Ga41 follows all of the Matthias’ empirical rules for superconductors: it contains a 4d transition metal (Mo), the compound may possess high density of electronic states, and its complex crystal structure is closely related to the cubic symmetry. Despite the fact that nowadays these rules seem to be obsolete, Yvon and co-authorsBezinge et al. (1984) found that Mo8Ga41 superconducts below $T_{\text{c}}=9.7$ K with the upper critical field of $\mu_{0}H_{\text{c2}}=8.6$ T. Results obtained by Yvon et al. were further pointed out on the unusual behavior of Mo8Ga41 in its superconducting state. Indeed, values of $T_{\text{c}}$ and $dH_{\text{c2}}/dT$ lead to the Werthamer-Helfand-Honenberg (WHH) prediction of the upper critical field $\mu_{0}H_{\text{c2}}\simeq 6.7$ T, which is significantly smaller than the experimental valueBezinge et al. (1984). Together with a relatively high critical temperature, this may indicate strong electron-phonon coupling in the superconducting regime.

In our recent studyVerchenko et al. (2016), we carried out comprehensive investigation of thermodynamic and transport properties of Mo8Ga41 in both, the normal and superconducting, states. Magnetization, heat capacity, and resistivity measurements confirmed bulk superconductivity at $T_{\text{c}}=9.8$ K in zero magnetic field. The superconducting transition was found to be characterized by a large value of the normalized specific heat jump $\Delta{}c_{\text{p}}/\gamma_{\text{N}}T_{\text{c}}=2.84$, which is twice as high as the weak-coupling BCS prediction. At the same time, the electronic contribution to the specific heat below $T_{\text{c}}$ obeyed a power-law rather than exponential decay behavior as is expected within the framework of BCS theory. Such disagreements have motivated us for further investigation of microscopic properties of Mo8Ga41.

Here, we present the results of muon spin rotation/relaxation ($\mu$SR) experiments on bulk Mo8Ga41 and its V-substituted derivative, Mo7VGa41. Transverse-field $\mu$SR (TF-$\mu$SR) experiments were undertaken in order to elucidate superconducting-state properties of both materials. The temperature-dependent magnetic penetration depth $\lambda$ and the upper critical field $\mu_{0}H_{\text{c2}}$ were extracted from the TF-$\mu$SR data and were used to investigate the order parameter in Mo8Ga41 and its evolution upon the V for Mo substitution.

II Experimental details

Mo8Ga41 and Mo7VGa41 specimens were synthesized as described earlierVerchenko et al. (2016) in the form of small submillimeter-size crystals. Elemental composition of the obtained crystals was investigated using a JSM JEOL 6490-LV scanning electron microscope operated at 30 kV and equipped with an energy-dispersive x-ray detection system INCA x-Sight. For quantitative analysis, molybdenum and vanadium elements, and gallium phosphide provided by MAC Analytical Standards were used as external standards. According to the EDX spectroscopy results, crystals of Mo8Ga41 and Mo7VGa41 contain 16.3(5) at. $\%$ Mo, 83.7(8) at. $\%$ Ga, and 14.4(5) at. $\%$ Mo, 2.0(2) at. $\%$ V, and 83.6(9) at. $\%$ Ga, respectively, leading to the Mo8.0(2)Ga41.0(4) and Mo7.1(2)V1.0(1)Ga41.0(4) formula units. In the case of Mo7VGa41sm , elemental mapping was used to check the distribution of Mo and V species across the surface, which was found to be uniform confirming local homogeneity of crystals. For measurements of thermodynamic properties, several crystals of Mo7VGa41 were glued together and measured as a polycrystalline sample. Magnetization curves were registered in magnetic fields between 0 T and 14 T at temperatures between 1.8 K and 10 K using the VSM setup of a Physical Property Measurement System (PPMS, Quantum Design). Heat capacity was measured in magnetic fields up to 12 T by the relaxation method using the Heat Capacity option of PPMS. Further, crystals of Mo8Ga41 and Mo7VGa41 were crushed by grinding in an agate mortar and analyzed by powder x-ray diffraction technique. The experiments were performed on a Huber Guinier Camera G670 [Image plate detector, Cu x-ray source, Ge (111) monochromator, $\lambda=1.540598$ Å]. PXRD patterns of Mo8Ga41 and Mo7VGa41 agree with the V8Ga41 structure type and show no impurity phases in the specimens. Finally, the obtained powders of Mo8Ga41 and Mo7VGa41 were pressed into cylindrical pellets with the diameter of 6 mm and the height of 1 mm at room temperature at the external pressure of 100 bar. These pellets were used for the $\mu$SR experiments.

$\mu$SR measurements were performed at the Paul Scherrer Institute (PSI), Villigen, Switzerland. Zero-field (ZF) and transverse-field (TF) $\mu$SR experiments were carried out on the GPS and DOLLY spectrometers located at the $\pi$M3 and $\pi$E1 beamlines, respectively. Experiments were performed in the temperature range from 1.6 to 15 K in GPS and from 0.29 to 15 K in DOLLY instruments. ZF-$\mu$SR experiments were performed in zero applied field. In TF-$\mu$SR experiments the sample was field-cooled from above $T_{c}$ in series of fields ranging from 30 mT to 0.59 T.

III Results and Discussion

III.1 Mo8Ga41

III.1.1 Zero-field $\mu$SR experiments

Figure 2 shows ZF-$\mu$SR time-spectra of Mo8Ga41 measured at $T=0.29$ K and 12.5 K. The obtained spectra below and above $T_{\text{c}}$ reveal no significant difference indicating the absence of internal coherent magnetic fields in Mo8Ga41 that may appear, for instance, as a result of long-range magnetic ordering. On the contrary, our ZF-$\mu$SR data show that the only nuclear component is present, which can be analyzed by using the static Kubo-Toyabe depolarization functionKubo (1981):

[TABLE]

where $A_{\text{s}}(0)$ is the initial asymmetry for muons stopped in the sample, $\sigma_{\text{KT}}$ is the muon depolarization rate, and $A_{\text{bgd}}$ is the background contribution from muons that missed the sample. The muon depolarization rate $\sigma_{\text{KT}}$ does not change with temperature within its standard deviation (see the inset in Fig. 2). This clearly indicates the absence of any spontaneous coherent magnetic fields in the superconducting state of Mo8Ga41.

III.1.2 Transverse-field $\mu$SR experiments

In order to elucidate microscopic properties of the superconducting state in Mo8Ga41, including the structure of superconducting gap, transverse-field $\mu$SR experiments were carried out in magnetic fields of 30, 70, 200, and 490 mT. Figure 3 shows TF-$\mu$SR time-spectra of Mo8Ga41 measured at $T=0.29$ K and 10 K in $\mu_{0}H=30$ mT. The spectrum below $T_{\text{c}}$ clearly exhibits faster relaxation with respect to the spectrum at 10 K due to the formation of the flux line lattice in the superconducting state. The following oscillatory decaying Gaussian function was used to fit the experimental data (the fitting results are shown as solid lines in Fig. 3):

[TABLE]

Here, $A^{\text{TF}}_{\text{s}}(0)$ and $A^{\text{TF}}_{\text{bgd}}(0)$ are initial asymmetries belonging to the sample and the background contributions, respectively. $\gamma_{\mu}/2\pi=135.5$ MHz/T is the muon gyromagnetic ratio, $\mu_{0}H_{\text{int}}$ and $\mu_{0}H_{\text{bgd}}$ are the internal and background magnetic fields, respectively, $\phi$ is the initial phase, and $\sigma$ is the Gaussian muon spin relaxation rate. Above $T_{\text{c}}$, the muon spin relaxation rate is referred to the temperature independent nuclear magnetic dipolar contribution $\sigma_{\text{nm}}$, while below $T_{\text{c}}$, the total relaxation rate consists of the nuclear and superconducting contributions, $\sigma=\sqrt{\sigma^{2}_{\text{sc}}+\sigma^{2}_{\text{nm}}}$. By fitting the data at $T=10$ K, we obtained $\sigma_{\text{nm}}=0.16$ $\mu{}\text{s}^{-1}$. This value was further used to extract $\sigma_{\text{sc}}(T)$ and $\mu_{0}H_{\text{int}}(T)$ at temperatures below $T_{\text{c}}$ according to Eq. 2. In the superconducting state, Fourier transforms of the asymmetry spectra are well described by the simple Gaussian approximation what is normally the case for powder samples of anisotropic superconductors. Fourier analysis of the asymmetry spectra shows no indications of disorder of the vortex lattice.

From the data taken at each temperature below the transition temperature $T_{\text{c}}$, we reconstructed $\sigma_{\text{sc}}$ as a function of transverse field for $\mu_{0}H=30$, 70, 200 and 490 mT. The resulting $\sigma_{\text{sc}}(T)$ dependencies for various transverse fields are shown in Fig. 4. The temperature-dependent variation of $\sigma_{\text{sc}}$ exhibits a clear enhancement below $T_{\text{c}}$. Figure 4 also implies that $\sigma_{\text{sc}}$ measured at similar temperatures decreases with increasing measuring field [see Fig. 5(a) where $\sigma_{\text{sc}}(\mu_{0}H)$ at $T=3$ K is depicted]. The more precise scan performed at $T=1.7$ K shows that before coming down, $\sigma_{\text{sc}}$ first increases by going through a maximum at around 20 mT [Fig. 5(b)]. Note that this is the typical behavior observed in various type-II superconductors experimentallyCardwell and Ginley (2002) and as well as predicted theoreticallyBrandt (2003).

In terms of Ginzburg-Landau treatment of the vortex state, BrandtBrandt (2003) has shown that $\sigma_{\text{sc}}$ can be calculated as a function of the reduced field $b$ and the Ginzburg-Landau coefficient $\kappa$ for the case of single-gap s-wave superconductivity:

[TABLE]

where $b=\frac{\mu_{0}H}{\mu_{0}H_{\text{c2}}}$, $\mu_{0}H_{\text{c2}}$ is the upper critical field, $\kappa=\lambda/\xi$, $\lambda$ is the magnetic penetration depth, and $\xi$ is the Ginsburg-Landau coherence length. Eq. (3) is valid for $b$ exceeding $0.25/\kappa^{1.3}$ (where the maximum of $\sigma_{\text{sc}}$ occurs) and $\kappa\gtrsim 5$. For Mo8Ga41, the value of $\xi=15.8$ nm was estimated from the upper critical field at zero temperatureVerchenko et al. (2016), thus, Eq. (3) can be used for $\lambda>79$ nm.

The $\sigma_{\text{sc}}(\mu_{0}H)$ data obtained from $T$-scans [Fig. 5(a)] as well as from the $\mu_{0}H$-scan [Fig. 5(b)] were fitted using Eq. (3). Firstly, the fitting with $\mu_{0}H_{\text{c2}}$ and $\lambda$ treated as variable parameters (solid red lines in Fig. 5) was employed yeilding the values of upper critical field $\mu_{0}H_{\text{c2}}$ that are significantly smaller than that obtained from thermodynamic measurements previouslyVerchenko et al. (2016). Indeed, as shown in Fig. 5(a), by fitting the $\sigma_{\text{sc}}(\mu_{0}H)$ data we obtained $\mu_{0}H_{\text{c2}}=3.77(4)$ T and $\lambda=216.1(2)$ nm at $T=3$ K, compared to 6.7 T from magnetization and heat-capacity measurementsVerchenko et al. (2016). On the other hand, if such a high value of $\mu_{0}H_{\text{c2}}$ is adopted, $\sigma_{\text{sc}}$ has to be significantly enhanced, in poor agreement with the $\mu$SR data. The same situation is observed in the case of $\mu_{0}H$-scan [Fig. 5(b)], where the values of $\mu_{0}H_{\text{c2}}=4.9(3)$ T and $\lambda=215(1)$ nm were obtained at $T=1.7$ K.

Values of the upper critical field of Mo8Ga41 are summarized in Fig. 6. Obviously, the entire plot of $\mu_{0}H_{\text{c2}}(T)$ calculated from the TF-$\mu$SR data exhibits large reduction in the upper critical field compared to thermodynamic measurements. This situation is a characteristic feature of multigap superconductivity that leads to two or more distinct length scales in the superconducting stateLee et al. (1999). Therefore, we further studied magnetic penetration depth in order to clarify this multigap behavior.

III.1.3 Magnetic penetration depth

The temperature-dependent inverse squared magnetic penetration depth $\lambda^{-2}$ of Mo8Ga41 obtained from the TF-$\mu$SR data using Eq. (3) is shown in Fig. 7. Upon increasing temperature, $\lambda$ increases monotonically and tends to infinity at $T=T_{\text{c}}$. Concurrently, $\lambda^{-2}$ decreases with increasing temperature and vanishes at the transition temperature. The temperature dependence of $\lambda^{-2}$ is indicative of the order parameter, i.e., absolute value of the gap energy, as well as symmetry of the superconducting gap. Assuming weak-coupling BCS-type s-wave superconductivity with a single gap $\Delta$ revealsTinkham (2004)

[TABLE]

in the dirty limit. At non-zero temperatures, the gap $\Delta(T)$ is assumed to follow the function $\Delta(T)=\Delta(0)\,{\rm tanh}[1.82(1.018(T_{\text{c}}/T-1))^{0.51}]$. This function has been found to well represent the temperature dependence at any coupling strengthPadamsee et al. (1973).

The fit using Eq. (4) is shown in Fig. 7 as a dashed line. The single-gap BCS-type model satisfactorily describes the experimental data with $\Delta(0)=1.80(7)$ meV, $\Delta(0)/k_{B}T_{\text{c}}=2.1$, and $\lambda(0)=192(2)$ nm. The reduced value of $\chi^{2}=0.66$ has been obtained with $n=13$ degrees of freedom. The results of fitting, from the first sight, hint at the single-gap BCS-type superconductivity in Mo8Ga41. However, previously we established two non-BCS-type featuresVerchenko et al. (2016): (i) the superconducting state is characterized by the strong electron-phonon coupling with $\lambda_{\text{ep}}=0.9$, (ii) the normalized specific heat jump $\Delta{}c_{\text{p}}/\gamma_{\text{N}}T_{\text{c}}=2.84$ is much larger than in the weak-coupling BCS limit. Also, in the current study we found that (iii) the upper critical field from the TF-$\mu$SR data shows the significant reduction with respect to the thermodynamic data. These features suggest that the single-gap BCS-type model should be avoided, and the multigap model should be used instead.

The two-gap model can be introduced using the linear combination of two s-wave superconducting gaps:

[TABLE]

where $\omega$ is the linear combination coefficient describing the contribution of each gap, and $\Delta_{\text{s}i}(T)=\Delta_{\text{s}i}(0)\,{\rm tanh}[1.82(1.018(T_{\text{c}}/T-1))^{0.51}]$ is the temperature dependence of each gap. Fit of the $\lambda^{-2}(T)$ data using Eq. 5 is shown as a solid red line in Fig. 7 yielding $\Delta_{\text{s1}}(0)=4.3(6)$ meV, $\Delta_{\text{s2}}(0)=1.76(6)$ meV, and $\lambda(0)=216.2(4)$ nm. This fitting is characterized by the reduced value of $\chi^{2}=0.50$ with $n=12$ degrees of freedom for $\omega=0.7$. The value of $\omega$ was found during the least-squares fit and then was fixed in order to reduce possible correlations between parameters. The use of the two-gap model in comparison to the single-gap model slightly reduces the value of $\chi^{2}$ with almost the same number of degrees of freedom. In the case of Mo8Ga41, the two-gap model implies the existence of two s-wave superconducting gaps with similar energies and almost equal contributions ($\omega=0.7$). Certainly, this model replicates the single-gap BCS-type model when describing the $\lambda^{-2}(T)$ data, and the fitting results are very similar (compare the solid and dashed lines in Fig. 7). However, keeping in mind the non-BCS-type features listed above, we exclude the single-gap scenario from the consideration. At the same time, the $\lambda^{-2}(T)$ data corroborate the multigap behavior of Mo8Ga41 and reveal that the introduction of two superconducting gaps is sufficient to explain superconducting behavior of this compound. Note that similar behavior was reported for SrPt3P, which was suggested to be a two-band superconductor with equal gapsKhasanov et al. (2014). Future studies that will help in resolving the gap structure in Mo8Ga41 are highly desirable.

III.2 V for Mo substitution in Mo7VGa41

In our previous studyVerchenko et al. (2016), Mo${}_{\text{8-x}}$V${}_{\text{x}}$Ga41 was also found to be superconducting with the transition temperature $T_{\text{c}}$ being just slightly lower than that for the parent compound. Zero-field and transverse-field $\mu$SR experiments were conducted therefore on Mo7VGa41 having $T_{\text{c}}\simeq 9.2$ K at zero magnetic fieldVerchenko et al. (2016). The analysis of the experimental data was performed similarly to that described previously for Mo8Ga41 sample.

Figure 8(a) shows a summary of the temperature-dependent upper critical field of Mo7VGa41. The $\mu_{0}H_{\text{c2}}$ data from the magnetization and heat capacity experiments as well as that obtained in TF-$\mu$SR studies are presented. In contrast to the unsubstituted Mo8Ga41, the TF-$\mu$SR $\mu_{0}H_{\text{c2}}$ values agree well with the data from thermodynamic measurements for Mo7VGa41. This implies that only one characteristic length scale is needed to describe the upper critical field. In accordance with this observation, the temperature dependence of the magnetic penetration depth of Mo7VGa41 presented as the $\lambda^{-2}(T)$ plot in Fig. 8(b) was satisfactorily fitted within the single-gap BCS-type model using Eq. (4). The fit yields $\Delta(0)=1.87(9)$ meV and $\lambda(0)=292(4)$ nm with the value of $\Delta(0)/k_{B}T_{\text{c}}=2.36$ exceeding the weak-coupling BCS limit of 1.76. Thus, from the TF-$\mu$SR data we suggest a simple s-wave superconducting state in Mo7VGa41 accompanied by the strong electron-phonon coupling.

III.3 Comparison of Mo8Ga41 with its V-substituted analog Mo7VGa41

The experimental data presented above reveal that Mo8Ga41 and Mo7VGa41 are different in two important aspects. First of all, application of the equation of Brandt leads to different results. The Brandt equation (Eq. 3) is formulated for the case of single-gap s-wave superconductivity. By this reason, the values of $\mu_{0}H_{\text{c2}}$ extracted for Mo7VGa41 using Eq. 3 are in good agreement with thermodynamic measurements, since Mo7VGa41 exhibits actually a single-gap s-wave superconductivity. The different situation is observed for Mo8Ga41. In this case, Eq. 3 yields the values of $\mu_{0}H_{\text{c2}}$, which are significantly lower than those from thermodynamic measurements, indicating that Mo8Ga41 is not a single-gap superconductor. Fourier analysis of the asymmetry spectra reveals no signs of disorder of the vortex lattice that may cause the Brandt equation being not applicable. The $\lambda(\mu_{0}H)$ dependence calculated using Eq. 3 also reveals the different behavior for Mo8Ga41 and Mo7VGa41 (Fig. 9). The field dependence of $\lambda$ observed for Mo8Ga41 is reminiscent of that of NbSe2Sonier et al. (1997), which is also a two-gap superconductorHuang et al. (2007). At the same time, there is no field dependence in the case of Mo7VGa41 in compliance with its conventional behavior. The substitution of just one out of eight Mo atoms by one V atom leads to the complete suppression of one of the superconducting energy gaps.

The second aspect is that the V substitution is accompanied by more than a factor of 2 reduction of the superfluid density $\rho_{\text{s}}\propto\lambda^{-2}$. Indeed, from the fits of the experimental data we get $\lambda^{-2}(0)\simeq 21$ and 12 $\mu\text{m}^{-2}$ for Mo8Ga41 and Mo7VGa41, respectively.

In order to explain such discrepancies, three different scenarios are going to be compared. According to the first one, the structural disorder caused by mixing Mo and V atoms in one crystallographic position may lead to the reduction in the superfluid density. From the single-crystal x-ray diffraction dataVerchenko et al. (2016), it is known that the substitution occurs in two crystallographic positions, and the average V content is less than 15 at. %. Such small amount of disorder in the crystal structure should not cause a drastic change of $\rho_{\text{s}}$. Moreover, as in the case of the K${}_{\text{1-x}}$Na${}_{\text{x}}$Fe2As2 solid solution Kim et al. (2014), the reduction in $\rho_{\text{s}}$ caused by structural disorder should be accompanied by the significant reduction in $T_{\text{c}}$, whereas the Mo${}_{\text{8-x}}$V${}_{\text{x}}$Ga41 solid solution demonstrates no significant change of $T_{\text{c}}$. By substituting V for Mo, the transition temperature just slightly reduces from $T_{\text{c}}=9.8$ K in Mo8Ga41 to 9.2 K in Mo7VGa41Verchenko et al. (2016).

In contrast to K${}_{\text{1-x}}$Na${}_{\text{x}}$Fe2As2, the V for Mo substitution is not isovalent. Rather, the change in the electron count is observed, which enables us to formulate the second scenario, with the heterovalent substitution being responsible for the reduction in $\rho_{\text{s}}$. The effect of heterovalent substitution on the electronic structure of Mo${}_{\text{8-x}}$V${}_{\text{x}}$Ga41 was analyzed in our previous studyVerchenko et al. (2016). By using the rigid-band shift approximation, it was shown that no significant change in the density of states at the Fermi level occurs with increasing V content in the solid solution. Thus, the heterovalent nature of the substitution also does not explain the observed reduction in $\rho_{\text{s}}$.

Finally, the third scenario considers competing states that cause the reduction in the superfluid density $\rho_{\text{s}}$. The Ba(Fe${}_{\text{1-x}}$Co${}_{\text{x}}$)2As2 solid solutionGordon et al. (2010) is an example, in which the competition between superconductivity and magnetism is observed. On the phase diagram of Ba(Fe${}_{\text{1-x}}$Co${}_{\text{x}}$)2As2, the superconducting dome is intersected with the spin-density-wave field yielding the intermediate region of competing states. Measurements of the magnetic penetration depth $\lambda$ clearly show that entering the intersected region results in the drastic increase in $\lambda$, which is equivalent to the reduction in $\rho_{\text{s}}$Gordon et al. (2010).

At this point, we make an assumption that the emergence of some competing state in Mo7VGa41 may lead to the suppression of one of the superconducting gaps accompanied by the significant reduction in the superfluid density. The collected ZF-$\mu$SR data show that both Mo8Ga41 and Mo7VGa41sm lack spontaneous coherent magnetic fields in the superconducting state, thus excluding magnetic order as a possible competing state from the consideration. To obtain further information, we analyze the low-temperature resistivity dataVerchenko et al. (2016) for Mo8Ga41 and Mo7VGa41. The $\rho(T)$ data at low temperatures can be fitted using equation $\rho(T)=\rho_{0}+AT^{n}$ (Figure 10). The fitting yields the exponents $n=1.49(8)$ for Mo8Ga41 and $n=1.21(3)$ for Mo7VGa41. This reduction in the exponent upon increasing V content may indicate the onset of an incipient charge-density-wave state in Mo7VGa41. A similar situation has been observed in Ta4Pd3Te16Pan et al. (2015), where the suppression of $n$ under pressure signifies enhancement of CDW fluctuations. The presence of CDW state in the V-substituted Mo8Ga41 would make one band unavailable for superconductivity and reduce the density of the supercarriers due to the opening of the energy gap associated with the CDW regime.

IV Conclusions

A comprehensive study by means of muon spin rotation/relaxation technique was carried out on a strongly-coupled superconductor Mo8Ga41 and its V-substituted analog Mo7VGa41. ZF-$\mu$SR experiments show that both compounds lack any spontaneous coherent magnetic fields in the superconducting state. TF-$\mu$SR experiments elucidate microscopic properties of the superconducting state, which turns out to be substantially different in Mo8Ga41 and Mo7VGa41. The upper critical field of Mo8Ga41 extracted from the TF-$\mu$SR data is lower than in thermodynamic measurements indicating the multigap superconductivity that leads to two or more independent length scales in the superconducting state. Accordingly, the temperature-dependent magnetic penetration depth was approximated by the model, in which two s-wave superconducting gaps with similar energies and almost equal contributions are assumed. The V for Mo substitution does not affect $T_{\text{c}}$ significantly, but leads to the complete suppression of one superconducting gap. This follows from the substantial reduction in the superfluid density $\rho_{\text{s}}$ and from the single-gap behavior of both the upper critical field and magnetic penetration depth of Mo7VGa41. We speculate that the emergence of a competing state in Mo7VGa41 is responsible for the closure of one of the superconducting gaps.

Acknowledgements.

This work was performed at the Swiss Muon Source, Paul Scherrer Institute, Villigen, Switzerland. Part of the experimental work was done by Gustavo Prack, Judith Suter and Clemens Spinnler (Swiss Nanoscience Institute, University of Basel, Basel, Switzerland) within their lab course at PSI. R.K. acknowledges Stefan Holenstein and Jean-Christophe Orain for their help during the $\mu$SR experiments. The work has been supported by the Russian Science Foundation, grant #17-13-01033. V.Yu.V. appreciates the support from the European Regional Development Fund, project TK134. A.A.T. is grateful for the financial support by the Federal Ministry for Education and Research under the Sofja Kovalevskaya Award of the Alexander von Humboldt Foundation.