Evidence of multiband superconductivity in the β -phase Mo 1−x Re x alloys
Shyam Sundar, L S Sharath Chandra, M K Chattopadhyay, S B Roy
EEvidence of multiband superconductivity in the β -phaseMo -x Re x alloys Shyam Sundar , L S Sharath Chandra , M K Chattopadhyay , and S B Roy , Homi Bhabha National Institute at RRCAT, Indore, Madhya Pradesh 452 013, India Magnetic and Superconducting Materials Section, Raja Ramanna Center for AdvancedTechnology, Indore, Madhya Pradesh 452013, India
Abstract
We present a detailed study of the superconducting properties in the β -phase Mo x Re x ( x =0 .
25 and 0.4) solid solution alloys pursued through magnetization and heat capacitymeasurements. The temperature dependence of the upper critical field H C (T) in these binaryalloys shows a deviation from the prediction of the Werthamer–Helfand–Hohenberg (WHH)theory. The temperature dependence of superfluid density estimated from the variation oflower critical field H C with temperature, cannot be explained within the framework of asingle superconducting energy gap. The heat capacity also shows an anomalous feature in itstemperature dependence. All these results can be reasonably explained by considering theexistence of two superconducting energy gaps in these Mo x Re x alloys. Initial results ofelectronic structure calculations and resonant photoelectron spectroscopy measurementssupport this possibility and suggest that the Re-5d like states at the Fermi level may notintermix with the Mo-5p and 5s like states in the β -phase Mo x Re x alloys and contribute quitedistinctly to the superconductivity of these alloys. Introduction
There has been a widespread interest recently in the multiband effect insuperconductors, especially after the discovery of superconductivity in iron pnictidecompounds [1]. Several new materials have been identified where the multibandeffect is considered to be important [2–10]. Even before the recent interest, multibandeffects were also known to govern the superconductivity in some well knownmaterials like MgB2, NbSe2, borocarbides, Nb3Sn and MgCNi3 [11–15]. All thesesuperconductors have complex crystallographic structures and complex Fermisurfaces. In this context it is interesting to note here that a two-band model was onceconsidered for understanding the superconductivity in elemental body centered cubic(bcc) Niobium [16]. However, to the best of our knowledge the same framework hasnot been used so far to investigate the superconducting properties in other cubicmetals and solid solution alloys. This is due to the fact that impurities, disorder andinter-band scattering can suppress the multiband effect in potential superconductors[2]. To this end, we re-investigate the interesting but not so well exploredsuperconducting properties of β -phase Mo x Re x binary solid solution alloys and showthat various interesting superconducting properties of these alloys with bcc crystalstructure can be explained within the framework of two superconducting gaps.Mo1- x Re x alloys possess excellent mechanical properties at elevated temperaturesand find widespread applications in aerospace and defense industries, medical fieldsand welding production [17–20]. Superconductivity has been observed in Mo1- x Re x alloys across a wide solid-solution range of its phase diagram. The superconductingtransition temperature T C in some of the alloy compositions is about an order ofagnitude higher than the T C = 0 . T C = 1 K in Re [21]. The T C ofMo1- x Re x alloys varies non linearly with x [22]. The Mo0 . .
40 alloy wasidentified as a strong coupling superconductor with a normalized energy gap 2 /k B T C= 5 .
0. This was well above the value of 3.52 predicted by theBardeen–Cooper–Schrieffer (BCS) theory of a weakly coupled superconductor [23].Shum et al provided an explanation for the enhancement of T C in Mo0 . . M Re/ M Mo = 1 .
94, disturbs the phonon spectrum and leads to the quasi local vibrationor Brout–Visscher mode [25]. This mode contributes appreciably to theelectron-phonon coupling function α F (ω) and to 2 /k B T C . However, subsequent pointcontact spectroscopy studies by Tulina and Zaitsev pointed out that the enhancementof the electron-phonon coupling ( λ ep ) from the lattice softening alone could notexplain the enhancement of T C in Mo1- x Re x alloys [26]. They argued that there mustbe a significant contribution from the electronic factor N(0) (N(0) is the electrondensity of states and is the matrix element of the electron phonon coupling)towards the enhancement of T C in these alloys [26].Molybdenum has an unoccupied d band just above the Fermi level and it is quite clearthat the addition of Re fills those unoccupied states and enhances the density of states(DOS) of the alloy system [21]. According to Matthias’ empirical rule [21], the T C forsolid solutions of transition metals shows its maxima at valence electrons per atom( e/a ratio) around 4.7 and 6.4. The maximum in the T C also corresponds to themaximum in the Sommerfeld coefficient of electronic heat capacity γ . This indicatesthat the maximum in the T C is observed for the maximum electron density of states atthe Fermi level [21]. However, the electron density of states for the solid solutionscorresponding to e/a = 6 . e/a = 4 . x Re x solid solutions belong to the former regime [27].Hence, the exact reason for the enhancement of T C in Mo1- x Re x solid solutions stillremains a matter of debate [24, 26].The electronic properties of the Mo1- x Re x alloys are also quite interesting and theexistence of Fermi pockets and associated electronic topological transition (ETT)have been established in the Mo1- x Re x alloys above the critical concentration x C =0 .
11 through various experimental and theoretical studies [28–32]. The directevidence of this ETT has been obtained recently with the help of angle resolvedphotoemission spectroscopy measurements along the H-N direction of the Brillouinzone [33]. However, any correlation between the ETT and the superconductingproperties of the Mo1- x Re x alloys is yet to be established. Apart from all theseinteresting microscopic properties, the Mo1- x Re x alloys may also be useful forsuperconducting radio-frequency cavity applications [34].In this paper we present a detailed study of temperature( T ) and magnetic field ( H )dependence of magnetization ( M ) and heat capacity ( C ) in the Mo1- x Re x ( x = 0 . C ( T ) in these alloys canbe explained by considering the existence of two superconducting gaps. A positivecurvature is observed in the temperature dependence of the upper critical field HC HC
1) can also beunderstood by considering the existence of two superconducting gaps. We note thatthis multiband effect in the Mo1- x Re x alloys is observed in that alloy compositionange where the appearance of the Fermi pockets above a critical value x > xC hasearlier been reported in the literature [33]. Experimental details
Polycrystalline samples of Mo x Re x , where ( x = 0 .
25, 0.40) were prepared by meltingconstituent elements with purity better than 99.95% in an arc furnace under 99.999%argon atmosphere. The samples were flipped and remelted six times to improve thehomogeneity. Figure 1 shows the x-ray diffraction patterns of these alloys obtainedwith a Geigerflex diffractometer (Rigaku, Japan) which indicate that these sampleshave formed in the bcc phase (space group: Im3m). The lattice parameters obtainedare about 3 .
135 ± 0 .
001 A0 and 3 .
126 ± 0 .
001 A0, respectively, for x = 0 .
25 and x =0 .
40. The heat capacity measurements were performed in the temperature range 2–15K in various applied magnetic fields up to 3 T using a Physical Property MeasurementSystem (PPMS; Quantum Design, USA). The magnetization measurements wereperformed using a Superconducting Quantum Interference Device (SQUID) basedVibrating Sample Magnetometer (SQUID-VSM; Quantum Design, USA).
Results and discussion
Figures 2( a ) and ( b ) show the temperature dependence of heat capacity C ( T ) for theMo0 . .
25 and Mo0 . .
40 alloys, respectively, at various applied magneticfields. The superconducting transition temperature T C is estimated as that temperaturewhere the temperature derivative of the heat capacity is minimum. The estimatedvalue of T C is 9 . . . .
25 alloy and 12 . . . .
40 alloy. The application of a magnetic field shifts the T C to lowertemperatures. A field of 2 T suppresses the superconductivity to below 2 K in theMo0 . .
25 alloy, whereas about 3 T is needed to achieve the same suppression inthe Mo0 . .
40 alloy. In such a case the
C(T ) in the normal state can be expressedby the functional form
C(T ) = Ce + CL where Ce = γ T is the electronic contributionto heat capacity and C L = βT + δT is the lattice contribution to heat capacity[35].Figures 2( c ) and ( d ) show the plots of C/T versus T of these alloys in the normal stateobtained by applying high magnetic fields, which suppressed the superconductingtransition temperature below 2 K. The C/T is linear in T C/T from linearity appears at temperatures well below T C ( H = 0) (bluedashed line in figures 2( c ) and ( d )). The temperature dependence of heat capacity C ( T )can be fitted with the functional form γ T + βT + δT (red solid line in figures 2( c )and ( d )) over an extended temperature range well above T C ( H = 0). The Debyetemperature θ D can be estimated from the coefficient β as θ . /β . Theestimated Debye temperature θ D is 440 ± 4 K for the Mo0 . .
25 alloy and θ D is373 ± 2 K for the Mo0 . .
40 alloy. Morin and Maita [27] reported a θ D value of340 K for Mo0 . .
40 alloy, while θ D value reported by Stewart and Giorgi [23]for the same alloy was 325 K. The value of Sommerfeld coefficient of electronic heatcapacity γ is estimated to be about 3 .
83 ± 0 .
02 mJ mol-1 K2 and 4 .
48 ± 0 .
02 mJ mol-1K fortheMo0 . .
25 andMo0 . .
40 alloys, respectively. The γ value reportedearlier for Mo0 . .
40 alloy agrees well with the present results [23]. igure 1:
X-ray diffraction pattern for Mo0 . .
25 alloy and Mo0 . .
40 in the range 30–900obtained using Cu-K α radiation. The most intense peak of each of the patterns is scaled to 1000 inorder to present the patterns in the same scale. The intensity of x-ray diffraction pattern ofMo0 . .
40 alloy is shifted upwards by 1000 (for the clarity). The samples are found to have a bccstructure and space group: Im3m.
Figures 3( a ) and ( b ) show the magnetization ( M ) as a function of magnetic field ( H ) atvarious temperatures below the T C of the Mo0 . .
25 and Mo0 . .
40 alloys,respectively. The insets show the expanded view of the field dependence ofmagnetization near H C . The upper critical field HC T C ( H ) of the jump in the C ( T ) at various applied magnetic fields. Figures 3( c ) and ( d )show the magnetic field dependence of magnetization M ( H ) below T C of theMo0 . .
25 and Mo0 . .
40 alloys, respectively, with an enlarged low H region.The measurements were performed after cooling the sample in the zero magnetic fieldto the desired temperature T < T
C from well above T C. The data have been correctedfor the demagnetization effects. At low fields, the magnetization M ( H ) is linear in - H indicating that the sample is in the Meissner state. A procedure of a linear fit of anumber of data points near H appl = 0 after equating M to H , is used to estimate thedemagnetization factor in these alloys. Then the effective magnetic field isestimated as H eff = H appl - αM . The lower critical field H C , below which a type-IIsuperconductor remains in the Meissner state, is in principle estimated from thedeviation from the linearity in the low field M versus H plot. However, suchestimation of H C may be impaired by the Bean–Livingston surface barrier and /orgeometrical barrier effects [36, 37]. In order to estimate the H C , a straight line isfitted to the M-H curve and the difference M between the measured magnetization andthe fitted curve is estimated for a wide magnetic field region [37–39]. The ( M ) / isthen plotted as a function of H and the value of H C is estimated as the field at which afitted straight line to this curve crosses the H axis [37–39]. We have observed thatwhile this procedure is applicable for determining H C in the Mo0 . .
40 alloy,( M ) / is not linear in H for the Mo0 . .
25 alloy. Hence, for this latter alloy HC M-H curve deviates from linearity. Since H C will be different for different criteria, we have estimated HC M/ d H at high fields extrapolates to zero. We have also crosscheckedsome of the estimated HC M/ d H from the measured isothermal M ( H ) curves bothn increasing and decreasing cycle [40]. The temperature dependence of H C is shownfor the Mo0 . .
25 and Mo0 . .
40 alloys in figure 4. The derivative(d HC / d T )T = T C estimated by fitting a straight line to the data points just below T C turns out to be about -0 .
159 ± 0 .
005 T K-1 for the Mo0 . .
25 alloy and -0 .
29 ±0 .
01 T K-1 for the Mo0 . .
40 alloy. Within the framework of Werthamer,Helfand and Hohenberg (WHH) model [41], the temperature dependence of H C canbe expressed in the dirty limit as [1]where t = T /T C , ħ = 2 eH(ν f τ/ πT C ) = ( /π ) H C T C / (-d H C / d T ) T = T C with Fermivelocity ν f and the relaxation time of electrons τ , α M = 3 / mνf τ = H C (0)/1.84√2 T C and λ SO = 1 / πT C τ τ H C estimated using the WHH model (dashed lines infigure 4) by taking experimentally obtained (d HC / d T )T = T C matches with theexperimental observations only at temperatures close to the T C. This deviation fromthe WHH model indicates that the HC (T) line in these alloys has a positive curvature.We have also tried to fit the HC (T ) over a large temperature range by taking(d HC / d T )T = T C and T C as fitting parameters and the corresponding fit is shown assolid lines in figure 4. The fitted curve matches with the experimental data at lowtemperatures and deviates at temperatures close to T C for both the alloys. The valuesof (d H C / d T )T = T C obtained are -0 . .
002 T K-1 and -0 . .
005 T K-1 for theMo0 . .
25 and Mo0 . .
40 alloys, respectively. These values arecomparatively higher than those estimated experimentally, which leads to a deviationat temperature close to T C. The value of T C obtained as fitting parameter is 9 . .
06K for the Mo0 . .
25 alloy and 12 . . . .
40 alloy. Thesevalues are smaller than those observed experimentally. For both the alloys, the fittingparameter α M lies between 0.08 to 0.18 and λ SO is about zero. This indicates that theparamagnetic effects and spin orbit interaction are negligible in these alloys. Thevalues of the temperature dependent H C for the present alloys are comparable tothose reported earlier in the literature [42, 43]. Even in those earlier reports thetemperature dependence of H C for various Mo1- x Re x solid solutions showeddeviation from the predictions of theoretical model available at that time, namely theAbrikosov–Gorkov model [42]. igure 2. Temperature dependence of heat capacity of ( a ) Mo0 . .
25 and ( b ) Mo0 . .
40 alloysin different magnetic fields. The superconducting transition temperature T C is 9 . . . . . .
25 and Mo0 . .
40 alloys. The panels ( c ) and ( d ) present the C/T versus T of these alloys in the normal state. The open symbols are experimental points. The bluedashed line is the linear fit. The red solid line is the fit using C(T ) = γ T + βT + δT . The temperature dependence of HC . .
25 andMo0 . .
40 alloys shows the usual form HC (T ) = HC (T /T C )
2] down tothe lowest temperature [44]. The fit using the above equation at low temperaturesyields HC ( ) = 68 . . . .
25 and HC ( ) = 81 . . . .
40. The values of HC (T ) for the Mo0 . .
40 alloy are comparable tothose of the Mo0 . .
36 alloy reported earlier in the literature [43]. However, thetemperature dependence of HC . .
25 alloy is different from thatreported for the same composition [42].For a superconductor in the local limit with ξ( )<< λ (where ξ (0) and λ are coherencelength and penetration depth, respectively), the normalized super fluid density ρ s ( T )in the framework of local London model is given by [45, 46] [2]The Figure 6(a)and (b) show the temperature dependence of HC (T )/HC ( ) of theMo0 . .
25 and Mo0 . .
40 alloys, respectively, which represent the superfluiddensity in these alloys. The open symbols are the experimental data points. For asingle gap superconductor, the normalized superfluid density can be expressed as [47][3]where
F (E) is the Fermi function and Here, Δ ( T ) is thesuperconducting gap [45, 47]. For an isotropic superconductor, Δ ( T ) is given by Δ (T ) = Δ (0) tanh{1 . . (T C /T - 1 ) ] . } where Δ (0) is the superconducting gap atabsolute zero [48]. igure 3. Magnetic field dependence of magnetization of ( a ) Mo0 . .
25 and ( b ) Mo0 . . T C. The insets show the expanded view of the field dependence of magnetization near HC
2. Theupper critical field HC M-H curves reduces to zero. The panels ( c ) and ( d ) present the magnetic field dependence of magnetizationat various temperatures below T C of the Mo0 . .
25 and the Mo0 . .
40 alloys, respectively, inlow H regime. Magnetization results presented here are in the form of closely spaced data points. The dotted lines in figures 6( a ) and ( b ) show the temperature dependence ofnormalized superfluid density estimated using the equation (3) for an isotropic singlegap superconductor with Δ (0) = 5 . . . .
25 alloy and Δ (0) =20 . . . .
40 alloy, . Figure 4.
Temperature dependence of the upper critical field H C for the Mo1- x Re x alloys. The dashedlines are the fit using Werthamer, Helfand and Hohenberg (WHH) model by taking experimentallyobtained (d HC / d T )T = T C. The fit matches with the experimental observations only at temperaturesclose to T C The solid lines represent the fits to the data by taking (d HC / d T )T = T C and T C as fittingparameters. In this case, the experimental data points deviate from WHH model at temperatures closeto T C. Figure 5.
Temperature dependence of the lower critical field HC x Re x binary alloys. Thesolid lines represent the fits to the data using the form HC (T ) = HC (T /T C ) ]. respectively. The goodness of fit is estimated from the Pearson’s χ O i is the experimental value, Ei is expected or thetheoretical value and n is the number of data points. The value of χ . .
25 alloy and 0.26 for the Mo0 . .
40 alloy. The estimatedtheoretical curve matches well with the experimental data at high temperatures.However, marked deviation observed at low temperatures indicates the possibility ofthe existence of two superconducting gaps [49] or the presence of a single anisotropicgap [50]. For a two gap superconductor, the normalized superfluid density can beexpressed as [47]. [4] where Δ S and Δ L are the small and large superconducting gap, respectively. Theparameter c is the fraction that the small gap contributes to the superconductivity. Atlow temperatures ( T /T C < .
5) where ( T ) varies within 10% of Δ (0), the equation(4) reduces to [5]The fit to the temperature dependence of superfluid density at low temperatures usingequation (5) can distinguish between the presence of a single anisotropic gap and twosuperconducting gaps. In case of the presence of a single anisotropic gap, theparameter c in equation (5) will approach unity or zero. Any one of the Δ S (0) and Δ L (0)should also approach zero and the other should have a value less than 1 . k B T C [51].If the system has two superconducting gaps, then the parameter c will have a valuesuch that 0 < c < Δ S (0) and Δ L (0) will have non zero values. The insetsto the figures 6( a ) and ( b ) show the fit to the superfluid density at low temperaturesusing equation (5) for the Mo0 . .
25 and Mo0 . .
40 alloys, respectively. Thefits indicate the existence of two superconducting gaps in these alloys and negate thepossibility of a single anisotropic gap. The value of Δ L (0) is very close to the BCStheoretical limit of 1 . k B T C . Hence, we have used equation (4) to fit the superfluiddensity in whole temperature range (red solid lines in the figures 6( a ) and ( b )) byconsidering two isotropic superconducting gaps. The χ is about 0.13 for theMo0 . .
25 alloy and 0.0074 for the Mo0 . .
40 alloy. Note that χ values aresmaller for two gap models as compared to that for single gap models. The values of Δ L (0) (ΔS(0)) = 18 . . . . . .
25 and Δ L (0) (ΔS(0) =22 .
5± 0 . .
0± 0 . . .
40 are slightly higher (quite lower) thanthe BCS limit of 1 . k B T C. The estimated value of c is about 25 ± 1% and 12 ± 1%for the Mo0 . .
25 and Mo0 . .
40 alloys, respectively. igure 6.
Temperature dependence of superfluid density of ( a ) Mo0 . .
25 and ( b ) Mo0 . . T < . T C using equation (5) to know whether the system has two superconducting gaps or ananisotropic gap. The analysis shows that the temperature dependence of superfluid density of thesealloys can be explained only after considering the existence of two superconducting gaps. Additional evidence for the existence of two superconducting gaps in the presentMo1- x Re x alloys can be obtained directly from the temperature dependence of heatcapacity in the superconducting state. The electronic heat capacity in thesuperconducting state CS in the zero magnetic field is estimated by subtracting thecontribution of the lattice heat capacity C L from the total heat capacity and is plottedas CS/γ T
C in figures 7( a ) and ( b ). The values of heat capacity jump at T C , Δ CS/γT
Care about 1.7 and 2 for the Mo0 . .
25 and Mo0 . .
40 alloys, respectively andthese are substantially higher than the BCS value of 1.43 for the weak couplingsuperconductors. This again suggests that the superconductivity in the present binaryMo1- x Re x alloys is rather unconventional.The electronic heat capacity in the superconducting state for a superconductor withtwo superconducting gaps corresponding to two bands without interband scattering isgiven by [52] [6]where C Si (i = 1 , ) corresponds to heat capacity resulting from superconducting gap Δ i and c = γ /γ is the fraction that the small gap contributes to the superconductivityand γ = γ γ
2. Here γ γ
2) is the normal state γ for the band 1 (band 2) that issuperconducting.Here, the C Si /γ i T C is given by [53, 54]7]where E i = x + δ i , f i = ( (α i T C E i /T )) -1 and α i = Δ i ( )/k B T C . The Δ i ( ) is thesuperconducting gap at absolute zero. For an isotropics wave superconductor, δ i = Δ i (T )/ Δ i ( ) , where Δ i ( ) is a constant, δ i = ( Δ i (T )/ Δ i ( ) ) cos n φ for line nodes and δ i = ( Δ i (T )/ Δ i ( ) ) sin n θ for point nodes, where θ and φ are the polar and azimuthal anglesover the Fermi surface. The equation (6) reduces to a single gap model when c = 0.The dotted blue lines in figures 7( a ) and ( b ) represent the temperature dependence ofheat capacity in the superconducting state with a single isotropic superconducting gapΔ(0) = 19 . . . .
25 alloy and Δ(0) = 26 . . . .
40 alloy, respectively. The goodness of fit χ is about 0.1 for theMo0 . .
25 alloy and 0.114 for the Mo0 . .
40 alloy. However, at lowtemperatures, the value of C S /γT C obtained experimentally is higher than thatcorresponding to the model fitting using single isotropic gap. We have also observedthat the model fitting by considering a single anisotropic gap (not shown here for thesake of clarity) cannot explain the temperature dependence of the heat capacity inthese Mo1- x Re x alloys. Then we have fitted our experimental results using equations(6) and (7) (solid red line in figures 7( a ) and ( b )) and found that the two isotropicsuperconducting gaps can explain the temperature dependence of heat capacity in thesuperconducting state. The χ . .
25 alloy and0.013 for the Mo0 . .
40 alloy. Similar to the fitting of temperature dependence ofsuperfluid density, the χ L (0) (Δ S (0) ) = 20 . . . . . .
25 alloy andΔ L (0) (Δ S ( . . . . . .
40 alloy is higher(lower) than the BCS limit of 1 . k B T C . The contribution from the smaller gap isabout 10 ± 1% in the Mo0 . .
25 alloy whereas it is about 2 . .
2% in theMo0 . .
40 alloy. These values, however, are relatively less as compared to thatestimated from superfluid density. Such behavior has been observed earlier in anothersuperconductor PrPt4Ge12 [46, 55]. This is probably due to the fact that thesuperfluid density estimated from H C is a local property whereas heat capacity is abulk property. We have also not considered the effect of inter-band scattering inanalyzing the temperature dependence of heat capacity in the superconducting state. Itis to be noted here that in an earlier study of the temperature dependence of electronicheat capacity in Mo0 . .
40 alloy, a clear deviation from the exponential behavior(corresponding to single energy gap) was indeed observed [23] but not analyzedfurther. igure 7.
Temperature dependence of the electronic heat capacity in the superconducting state C S /γT C plotted as a function of T/T C for ( a ) Mo0 . .
25 and ( b ) Mo0 . .
40. The lines are fits to theexperimental data (open symbols). The analysis shows that the temperature dependence of the heatcapacity in these alloys can be explained by considering the existence of two superconducting gaps.
In the case of two-gap superconductors, the magnetic field dependence of theelectronic part of heat capacity C S /T at temperatures well below T C should show twodistinct linear regions with a change of slope at intermediate fields [2]. We haveshown in figure 8, the plot of C S/ T as a function of H/H C at 2 K for both theMo0 . .
25 and Mo0 . .
40 alloys. The C S /T corresponding to Mo0 . . . .
25 alloy, the slope ofthe low field linear portion is slightly higher as compared to that at high fields, whichis similar to other two gap superconductors such as MgB2 and NbSe2 [56]. Suchbehaviour is observed when the smaller of the two gaps vanishes at low fields and thecorresponding normal electron contribution increases. However, in the case of theMo0 . .
40 alloy, the change in the slope is quite subtle and is also reversed ascompared to that of the Mo0 . .
25 alloy. This may be due to the enhancedinter-band scattering [2] in the Mo0 . .
40 alloy. The present β -phase Mo1- x Re x binary alloys have the bcc crystal structure, which is analogous to the elementalmolybdenum. In this structure, the Mo and Re atoms randomly occupy the corner ofthe cube (0, 0, 0) and the body center (0.5, 0.5, 0.5). Thus, the presence of twosuperconducting gaps in these alloys at the first sight is quite surprising. However, theconcentration of Re in the present alloys is higher than the critical concentration x C =0 .
11 at which the existence of electronic topological transition in β -phase Mo1- x Re x binary alloys has been reported [28–33]. For the Mo1- x Re x alloys above x C, a bandcrosses the Fermi level along the H-N direction of the Brillouin zone [33]. Initialresults of our band structure calculations and resonant photoelectron spectroscopyexperiments reveal that the there is a charge transfer from Re to Mo when Re islloyed with the Mo [57]. Our study also reveals that there is a substantial change inthe structure of density of states in Mo1- x Re x alloys just below the Fermi level [57].The density of states at the Fermi level of the Mo1- x Re x alloys are mainly derivedfrom the narrow Re 5 d like states and the broad Mo5 p as well as Mo5 s like states. TheRe 5 d like states are not intermixed with the Mo 5 p like and Mo 5 s like states. Theseinitial results [57] when compared with the results of angle resolved photoemissionstudies reported in literatures [33], indicate that the Re 5 d like states can be linked tothe band that crosses the Fermi level along the H-N direction of the Brillouin zonewhen Re is alloyed with Mo. It is natural to expect that the Fermi velocity in thesenarrow Re 5 d like states is distinctly different from that in the broad Mo 5 p like andMo 5 s like states. Therefore, we conjecture that these narrow Re 5 d like statescontribute to the superconductivity separately from the broad Mo 5 p like and Mo 5 s like states. It is also to be noted that the multiband superconductivity at the electronictopological transition has been observed in systems such as URhGe [58] and the hightemperature superconducting pnictides [59]. Figure 8.
Magnetic field dependence of C S /T at 2 K as a function of H/H C for the Mo . Re . andMo . Re . alloys. The C S /T corresponding to Mo . Re . alloy is shifted upwards for clarity. Themagnetic field dependence of the heat capacity shows two linear regions and a change in slope atintermediate magnetic fields in these alloys. Conclusion
In summary, we have observed various anomalous features in the superconductingproperties, namely the upper critical field and superfluid density of the β -phaseMo x Re x ( x = 0 .
25 and 0.4) alloys. These anomalous features are suggestive of theexistence of two superconducting gaps in these binary alloy superconductors. Furthersupport for the presence of a multiband effect is obtained from the temperaturedependence of the heat capacity in the superconducting state. At first sight, thepossibility of such multiband effects in these Mo x Re x alloys with relatively simplecrystal structures is quite surprising. However, there are reports in the literature[28–33] which suggest the existence of an ETT in the Mo1- x Re x alloys with thecritical concentration x C = 0 .
11. In this direction, preliminary results [57] of ourelectronic structure calculation and resonance photoelectron spectroscopyexperiments in the present Mo x Re x ( x = 0 .
25 and 0.4) alloys reveal the existence ofnarrow Re 5 d like states and the broad Mo 5 p as well as Mo 5 s like states at the Fermilevel, which contribute to a large enhancement in the density of states at the Fermievel. These narrow Re 5 d like states along with the broad Mo 5 p and Mo 5 s likestates are possibly the source of the multiband effect in the present Mo x Re x alloys. Acknowledgments
We would like to thank R K Meena for help in sample preparation, V S Tiwari andGurvinderjit Singh for the x-ray diffraction measurements and V Ganesan and DVenkateshwaralu for some of the heat capacity measurements.
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