Muon-spin rotation measurements of the penetration depth of the Mo{}_{3}Sb{}_{7} superconductor

Muon-spin rotation measurements of the penetration depth of the MoSb superconductor

R. Khasanov rustem.khasanov@psi.ch Laboratory for Muon Spin Spectroscopy, Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland    P.W. Klamut Institute of Low Temperature and Structure Research of the Polish Academy of Sciences, Okólna 2, 50-422 Wrocław, Poland    A. Shengelaya Physics Institute of Tbilisi State University, Chavchavadze 3, GE-0128 Tbilisi, Georgia    Z. Bukowski Institute of Low Temperature and Structure Research of the Polish Academy of Sciences, Okólna 2, 50-422 Wrocław, Poland Laboratory for Solid State Physics, ETH Zürich, CH-8093 Zürich, Switzerland    I.M. Savić Faculty of Physics, University of Belgrade, 11001 Belgrade, Serbia and Montenegro    C. Baines Laboratory for Muon Spin Spectroscopy, Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland    H. Keller Physik-Institut der Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland
Abstract

Measurements of the magnetic field penetration depth in superconductor MoSb ( K) were carried out by means of muon-spin-rotation. The absolute values of , the Ginzburg-Landau parameter , the first and the second critical fields at are  nm, ,  mT, and  T. The zero temperature value of the superconducting energy gap was found to be 0.35(1) meV corresponding to the ratio . At low temperatures saturates and becomes constant below , in agreement with what is expected for wave BCS superconductors. Our results suggest that MoSb is a BCS superconductor with the isotropic energy gap.

pacs:
74.70.Ad, 74.25.Op, 74.25.Ha, 76.75.+i
preprint: PREPRINT (August 1, 2019)

Recently, the attention was devoted to MoSb. This compound was originally discovered more than forty years ago Jensen66 (); Hulliger66 () and only recently was found to become a type-II superconductor with the transition temperature  K.Bukowski02 () The properties of MoSb in a superconducting state are rather unusual. Specific heat, resistivity and magnetic susceptibility experiments of Candolfi et al.Candolfi07 () suggest that MoSb can be classified as a coexistent superconductor – spin-fluctuating system. As discussed in Ref. Candolfi07, , factoring in the effect of spin fluctuations leads to renormalized values of the electron-phonon coupling constant and the Coulomb pseudopotential, which, being substituted to the McMillan expression, lead to between 1.4 K and 2.0 K. This is substantially closer to the experimentally observed  K than  K, which would be obtained without taking into account the effect of spin fluctuations.Candolfi07 ()

There is currently no agreement on the symmetry of the order parameter of MoSb. The recent specific heat experiments of Candolfi et al.Candolfi08 () suggest that the order parameter in MoSb is of conventional wave symmetry. Tran et al.Tran08 (), based again on the results of specific heat measurements, have reported the presence of two isotropic wave like gaps with and [ is the zero-temperature value of the superconducting energy gap]. In contrast, Andreev reflection measurements of Dmitriev et al.Dmitriev06 (); Dmitriev07a (); Dmitriev07b () reveal that the superconducting gap is highly anisotropic. The maximum to the minimum gap ratio was estimated to be and wave symmetry of the order parameter was proposed in a qualitative analysis.Dmitriev07a (); Dmitriev07b ()

The symmetry of the superconducting order parameter can be probed by measurements of the magnetic penetration depth . A fully gaped, isotropic pairing state produces a thermally activated behavior leading to an almost constant value of the superfluid density for .Khasanov05 (); Prozorov06 () Presence of line nodes in the gap leads to a continuum of low laying excitations, which result in a linear at low temperatures.Hardy93 (); Kadono04 () In two-gap superconductors with highly different gap to ratios the inflection point in is generally present.Carrington03 (); Khasanov07_La214 (); Khasanov07_Y123 ()

In this paper, we report the study of the magnetic field penetration depth in superconductor MoSb by means of muon-spin rotation. Measurements were performed down to 20 mK in a series of fields ranging from 0.02 T to 0.2 T. Our results are well explained assuming conventional superconductivity with the isotropic energy gap in agreement with the recent specific heat experiments of Candolfi et al.Candolfi08 () The zero-temperature value of the gap was found to be  meV corresponding to the ratio .

The MoSb single-crystal samples were grown through peritectic reaction between Mo metal and liquid Sb.Bukowski02 () The transverse field muon-spin rotation (TF-SR) experiments were performed at the M3 beam line at Paul Scherrer Institute (Villigen, Switzerland). For our experiments the ensemble of some sub- and millimeter size single crystals were mounted onto the silver plate to cover the area of approximately 50 mm. The silver sample holder was used because it gives a nonrelaxing muon signal and, hence, only contributes as temperature independent constant background. The crystals were oriented so that the magnetic field was preferably applied along the 001 crystallographic direction. The MoSb samples were field cooled from above down to  mK in magnetic fields ranging from 20 mT to 0.2 T. A full temperature scan (from 20 mK up to 2.5 K) was performed in a field of  T.

Figure 1 shows the muon-spin precession signals in  T above ( K) and below ( K) the superconducting transition temperature of MoSb. The difference in the relaxation rate above and below is due to the well-known fact that type-II superconductors exhibit a flux-line lattice leading to spatial inhomogeneity of the magnetic induction. As is shown by Brandt,Brandt88 (); Brandt03 () the second moment of this inhomogeneous field distribution is related to the magnetic field penetration depth in terms of .

Figure 1: (Color online) Transverse-field muon-spin precession signals from MoSb obtained in  T above the transition temperature ( K – black squares) and after field-cooling the sample below ( K – red circles). The solid lines correspond to the fit by means of Eq. 1. For visualization purpose the apparent precession frequencies are modified from the actual precession frequencies by the use of a rotating reference frame.

The analysis of TF-SR data was carried out in the time-domain by using the following functional form:Adroja05 ()

(1)

where the first term denotes the contribution from the MoSb sample and the second term is the background contribution from the Ag sample holder. and are the initial asymmetries arising from the sample and the background, and are the internal filed inside the sample and the applied external field seen in the Ag backing plate,  MHz/T is the muon gyromagnetic ratio, and is the initial phase. and are the muon-spin relaxation rates caused by the nuclear moments and the additional component appearing below due to nonuniform field distribution in the superconductor in the mixed state.

The analysis was carried out as follows. First, the muon-time spectra were fitted by means of Eq. (1) with all the parameters free. Then, the ratio was obtained as the mean value in the temperature interval of  K K. After that, by keeping the ratio fixed, the nuclear moment contribution was estimated by analyzing the experimental data above 2.1 K, (i.e. at , where ). The final analysis was made with , and as fixed, and and as free parameters. This allows to reduce the number of the fitting parameters and, as a consequence, to increase the accuracy in determination of .

Within BCS theory the zero-temperature magnetic penetration depth [] of isotropic superconductor does not depend on the magnetic field. Contrary, the nonlocal and the nonlinear response of the superconductor with nodes in the gap to the magnetic field, as well as, the faster suppression of the contribution of the smaller gap to the total superfluid density in a case of two-gap superconductor leads to the fact that , evaluated from SR experiments, is magnetic field dependent and it increases with increasing field.Amin00 (); Landau07 () Such behavior was observed in various hole-doped cuprates,Kadono04 (); Sonier00 (); Khasanov07_field-effect () and the double-gap NbSeSonier00 (); Fletcher07 () and MgB superconductors.Serventi04 (); Angst04 ()

Figure 2: (Color online) Field dependence of the SR depolarization rate measured at  mK in MoSb sample. The solid line corresponds to a fit of Eq. (2) to the experimental data with the parameters shown in the figure. The inset shows the dependence of normalized to its value at on the reduced magnetic field for MoSb, the two-gap superconductor MgB from Ref. Angst04, , and the hole-doped cuprate superconductor YBaCuO from Ref. Sonier00, .

Consideration of the ideal vortex lattice (VL) of an isotropic wave superconductor within Ginsburg-Landau approach leads to the following expression for the magnetic field dependence of the second moment of the magnetic field distribution:Brandt03 ()

(2)

Here is the magnetic induction, which for applied fields in the region is , is the coefficient depending on the symmetry of the VL ( for triangular VLKhasanov05 (); Brandt03 () and, as is shown below, for rectangular VL), is the first critical field, and is the upper critical field. Equation (2) accounts for reduction of due to stronger overlapping of vortices by their cores with increasing field. According to calculations of Brandt,Brandt03 () Eq. (2) describes with less than 5% error the field variation of for an ideal triangular vortex lattice and it holds for type-II superconductors with the value of the Ginzburg-Landau parameter ( is the coherence length) in the range of fields .

Satisfactory fit of Eq. (2) to the experimental data would suggest that there is no significant change of the penetration depth in the range of magnetic fields of the experiment. We performed an analysis of the magnetic field dependence of the muon depolarization rate measured at  mK (see Fig. 2), which is a good approximation of . The solid line in Fig. 2 corresponds to the fit of experimental by means of Eq. (2). The fit yields  nm and  T. A good agreement between the experiment and the theory (see Fig. 2) allows to conclude that can be consistently fitted within the behavior expected for conventional isotropic superconductor. For comparison in the inset of Fig. 2 we plot the dependence of the inverse squared penetration depth normalized to its value at on the reduced field for MoSb studied here, the two-gap superconductor MgB from Ref. Angst04, , and the hole-doped cuprate superconductor YBaCuO from Ref. Sonier00, . It is seen that in a case of MgB and YBaCuO , evaluated from muon experiments, is field dependent and decreases rather strongly with increasing field.

Figure 3: (Color online) (a) Theoretical profiles ( nm,  nm, and  T) calculated for , 0.05, 0.1, and 0.15 s. (b) Dependence of the product on the reduced field for the triangular (circles) and the rectangular (stars) VL’s. Calculations were made by using the numerical approach of Brandt Brandt03 () for . The solid lines are obtained by means of Eq. (2) with for triangular and for rectangular VL’s, respectively. The insets are the contour plots of field variation within triangular and rectangular VL’s.

The absolute error in , estimated from the fit of by means of Eq (2), is %. There are, however, other sources of uncertainties which could also affect and, consequently, increase the error: (i) The use of Eq. (2) to extract from requires that the VL is well ordered and it is triangular, as opposed to disordered or rectangular VL. (ii) The fitting function Eq. (1) assumes that the contribution from the sample is described by a single line of Gaussian shape as opposed to asymmetric magnetic field distribution generally observed in good quality single crystals.Sonier00 () (iii) The background signal due to the Ag backing plate was assumed to have a temperature-independent frequency and is non-relaxing at all . However, when the sample goes superconducting it excludes some field, even in the mixed state. These small fields are very inhomogeneous and could cause as a shift of the background signal to higher fields and an increase of a slow relaxation. In the following we are going to discuss in detail the sources (i) and (ii). We believe that the source (iii) does not play a substantial role here since even at a  mK is only 0.16 mT smaller than the applied field and is constant in the whole temperature range (from 20 mK up to ). It should be mentioned, however, that experience with more robust superconductors has shown that even though the background field does not usually change perceptibly, the excluded flux can affect the background relaxation rate.

To account for possible random deviations of the flux core positions from their ideal ones (VL disorder) and for broadening of SR spectra due to nuclear depolarization, the field distribution of an ideal VL [] is, generally, convoluted with a Gaussian distribution in terms of:Brandt88a (); Sonier94 (); Maisuradze08 ()

(3)

Here and is the contribution to the Gaussian broadening of due to VL disorder. The theoretical profiles for various ’s are shown in Fig. 3 (a). It is obvious that both, the nuclear moment contribution and the VL disorder broaden the profiles, thus requiring that the total second moment of SR line needs to be the sum of three components:Maisuradze08 () . This implies that neglecting the VL disorder leads to overestimation of and, as a consequence, to underestimation of . On the other hand, , obtained from the fit of asymmetric line by using symmetric Gaussian function [see Eq. (1)], becomes underestimated.Sonier00 () In a case of MoSb the main contribution to comes, most probably, from the nuclear moment term  s, which is comparable with in the whole field range, rather than from , which for good quality single crystals corresponds, typically, to %, of (see, e.g., Ref. Sonier00, and references therein). Such big also leads to the fact that the shape of profile is close to the Gaussian one as is demonstrated by the solid blue line in Fig. 3.

Dependences of on the reduced field for triangular and rectangular VL’s, obtained by using numerical calculations of Brandt, Brandt03 () are shown in Fig.3 (b). Solid lines correspond to Eq. (2) with (red line) and (blue line). As is seen, in a case of rectangular VL can still be satisfactory described by means of Eq. (2) with the only 5% bigger value of the coefficient . This implies that the uncertainty related to different VL symmetries would result in % additional error in . To summarize, by taking into account the above presented arguments we believe that the true overall uncertainty in the absolute value of is about 10-15 %, and  nm.

The zero–temperature value of the superconducting coherence length may be estimated from as , which results in  nm ( is the magnetic flux quantum). Using the values of and , the zero-temperature value of the Ginzburg-Landau parameter is . The value of the first critical field can also be calculated by means of Eq. (4) from Ref. Brandt03, as  mT. Note that the zero-temperature values of , , , and the first and the second critical fields are in agreement with those reported in the literature.Bukowski02 (); Dmitriev06 (); Dmitriev07b (); Candolfi08 (); Tran08 ()

Figure 4: (Color online) Temperature dependence of of MoSb reconstructed from measured in  T. The black solid and the dash-dotted blue lines are the fits by means of clean and dirty weak-coupling BCS models. The blue dashed line represents calculated by using two-gap alpha modelCarrington03 (); Khasanov07_La214 () with the parameters from Ref. Tran08, . See text for details.

As a next step we are going to analyze the temperature dependence of the magnetic penetration depth. , presented in Fig. 4, was reconstructed from measured in a field of 0.02 T by using Eq. (2). , needed for such reconstruction, was assumed to follow the Werthamer-Helfand-Hohenberg predictionWerthamer66 () in agreement with the results of Ref. Candolfi08, . It is worth to mention here that the temperature dependence of is much less affected by the above described uncertainties than the absolute value. Indeed, both, Sonier94 (); Maisuradze08 () and the relaxation related to the excluded flux increase with decreasing in much the same way as , which causes mainly a correction to the coefficient in Eq. (2), while is temperature independent.

Figure 4 implies that below 0.7 K is temperature independent as expected for the superconductor with fully gaped states. The experimental dependence was analyzed within the dirty- and the clean-limit approaches. In the dirty-limit theory has the form:Tinkham75 ()

(4)

while in the clean-limit:Tinkham75 ()

(5)

Here is the Fermi function. The temperature dependence of the gap was approximated by .Carrington03 () As is seen, both, the dirty- and the clean-limit approaches, describe the experimental data reasonably well. The fitted curves are almost undistinguishable from the each other. The results of the fits are 2.11(2) K, 0.35(1) meV and 2.12(2) K, 0.39(1) meV for the dirty- and the clean-limit BCS model, respectively. The corresponding gap to ratios were found to be and . There are few reasons why, we believe, MoSb studied here is in the dirty limit: (i) The maximum gap value obtained in Andreev reflection experiments was found to be  meV which is close to 0.35(1) meV observed within our dirty limit calculations. (ii) The residual resistivity ratio measured on similar single crystals was found to be rather low .Bukowski02 () Note that the isostructural compound RuSn has ,Chakoumakos98 () which is bigger by more than 2 orders of magnitude. (iii) A reasonable assumption about Fermi velocity  m/s can be made within the free electron approximation by taking  eV.Dashjav02 () This allows us to readily estimate the BCS-Pippard coherence length  nm which is approximately 50 times bigger than  nm obtained experimentally. Considering that ( is the mean free path) we obtain that the coherence length in MoSb is limited by and, correspondingly,  nm.

Now we are going to comment shortly the results of the specific heat experiments of Tran et al.Tran08 () It was shown that the analysis of the electronic specific heat data within the framework of the phenomenological alpha model suggests the presence of two gaps with and and the relative weight of the bigger gap of 0.7.Tran08 () The curve simulated by using these parameters and  nm, is shown in Fig. 4 by blue dashed line. Both contributions were assumed to be in the dirty limit [see Eq. (4)] and the similar alpha model, but adapted for calculation of the superfluid density, was used.Carrington03 (); Khasanov07_La214 () The agreement between the simulated curve and the experimental data is rather good. We should emphasize, however, that in two-gap superconductor the contribution of the smaller gap to the total superfluid density decreases very fast with increasing field, thus leading to strong suppression of .Serventi04 (); Angst04 (); Khasanov07_La214 () This is inconsistent with the data presented in the inset of Fig. 2 revealing that in MoSb .

To conclude, the superconductor MoSb ( K) was studied by means of muon-spin rotation. The main results are: (i) The absolute values of the magnetic field penetration depth , the Ginzburg-Landau parameter , and the first and the second critical fields at were found to be  nm, ,  mT, and  T. (ii) Over the whole temperature range (from down to 20 mK) the temperature dependence of is consistent with what is expected for a single-gap wave BCS superconductor. (iii) The ratio was found, suggesting that MoSb superconductor is in the intermediate-coupling regime. (iv) The magnetic penetration depth is field independent, in agreement with what is expected for a superconductor with an isotropic energy gap. (v) The value of the electronic mean-free path was estimated to be  nm. This relatively short value suggests that strong scattering processes play an important role in the electronic properties of MoSb.

This work was performed at the Swiss Muon Source (SS), Paul Scherrer Institute (PSI, Switzerland). The authors are grateful to A. Amato and D. Herlach for providing the instrumental support during the SR experiments, and A. Maisuradze for calculating dependences. The work was supported by the K. Alex Müller Foundation and in part by the SCOPES grant No. IB7420-110784.

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