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J Supercond Nov Magn (2015) 28:331­337 DOI 10.1007/s10948-014-2700-3

ORIGINAL PA P E R

Microwave Conductivity in Two-Band Superconductors V3+x Si 1-x
Oleg V. Dolgov · Alexander A. Golubov · Yuri A. Nefyodov · Alexei M. Shuvaev · Mikhail R. Trunin

Received: 9 June 2014 / Accepted: 11 August 2014 / Published online: 26 August 2014 © Springer Science+Business Media New York 2014

Abstract We present the results of investigations of the temperature dependences of complex conductivity (T ) = (T ) - i (T ) at frequency 9.4 GHz in series of single crystals V3+x Si1-x with different Si content. The data exhibit peculiarities typical for multiband superconductors, namely a nonlinear temperature dependence of resistivity above superconducting transition temperature Tc , suppression of superconducting transition temperature Tc by nonmagnetic impurities, a positive curvature of (T ) curves close to Tc , and a coherence peak in (T ) at T Tc /2. Using a two-band model in the weak-coupling regime, we demonstrate that the behavior of Tc and the evolution of (T ) with Si-content variation are consistently described by changing of the interband scattering rate.

Keywords Microwave surface impedance · Complex conductivity of superconductors · BCS theory · Two-band model

1 Introduction Possible existence of superconductors with different order parameters at the Fermi surface [1­3] was predicted soon after the microscopic theory of Bardeen-Cooper-Schrieffer (BCS) [4]. Recently, to explain the observed deviations from the predictions of the BCS theory, the realization of multiband superconductivity was suggested in a number of transition metals [5­7], Nb-doped SrTiO3 [8], rare earth Niborocarbides [9], the layered superconductor NbSe2 [10], as well as MgNi3 C [11], heavy-fermion compounds CeCoIn5 [12] and CePt3 Si [13] and in Nb3 Sn [14]. Recently, manifestations of two-band superconductivity are discovered in superconductors with higher transition temperature Tc : MgB2 (Tc = 39 K) where two superconducting gaps, , are resolved in some experiments and iron-pnictides (Tc 50 K) where the description of electronic transport and thermodynamical characteristics requires contribution of multiple bands which strongly differ by their kinetic properties (see recent reviews in [15, 16]). At the same time, the situation with A15 compounds is still controversial. Studies of the temperature variation of a superfluid density [17, 18] and an optical conductivity [19] in A15-structure superconductor V3 Si have revealed well-resolved multiband features. On the other hand, the measurements of the thermal condictivity [10], of the field dependence of the magnetic field penetration depth [20], and of the reversible magnetization [21] were interpreted within the single-band scenario. In this contradictory situation, further studies of V3 Si are very important. In this work,

O. V. Dolgov Max Planck Institute for Solid State Research, Stuttgart, Germany O. V. Dolgov P.N. Lebedev Physical Institute RAS ,Moscow, Russia A. A. Golubov Faculty of Science and Technology and MESA+ Institute of Nanotechnology, Enschede, Netherlands A. A. Golubov · M.R. Trunin Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Russia Yu.A. Nefyodov · M.R. Trunin ( ) Institute of Solid State Physics RAS, 142432 Chernogolovka, Moscow Russia e-mail: trunin@issp.ac.ru A.M. Shuvaev Institute of Solid State Physics, Vienna University of Technology, 1040 Vienna, Austria


332

J Supercond Nov Magn (2015) 28:331­337

we present the results of experimental and theoretical study of magnetic penetration depth and microwave conductivity in V3+x Si1-x single crystals with different Si content at frequency 9.4 GHz and demonstrate clear two-band features in the microwave response.

2 BCS Theory and Two-Band Model Functional relationship between the measured surface impedance Z and conductivity in superconductors depends significantly on the dimensionless parameter S = (E - i /2)/(q vF ) [22], where the parameters with dimension of energy E , , and qvF have the following characteristic values: E is a gap value at T = 0 in the superconducting state and E 0 in the normal state, = 1/ is the inverse relaxation time of quasiparticles, vF is the Fermi velocity, and q is the wave number that characterizes the range of electromagnetic field variation: q 1/ in the superconducting state ( is the magnetic field penetration depth) and q 1/ in the normal state ( is skin depth). If S 1, then the relation between the field and the current in a superconductor is local, while for S 1, it is essentially nonlocal. In some cases, it is more convenient to compare these energy parameters with parameters that have the dimension of length, namely the mean free path l = vF / , the coherence length 0 = vF /( ), and the magnetic field penetration depth . V3 Si crystals are in the London limit since the coherence length (0) 10 nm is much less than the magnetic field penetration depth (0) 100 nm at T = 0, and therefore, the relationship between the current and field is local. In this case, the surface impedance Z = R + iX defines the complex conductivity = - i = i0/Z 2 , where = 2f is the measurement frequency and 0 = 4 · 10-7 H/m. Figure 1 shows the calculated temperature dependences of the conductivity components (T ) and (T ) according to the formulas of the BCS theory as a function of the parameter at fixed frequency f = 10 GHz. The conductivity values (0) decrease with increasing relaxation rate = 1/ , reflecting the increase of the penetration depth in a dirty London superconductor compared with a pure one (the left figure). The existence of a nonzero imaginary part of the conductivity in the normal state of pure superconductors associated with temporal dispersion of Drude conductivity 0 = ne2 / m: n = n - in = 0 /(1 + i ). On the right panel of Fig. 1, the BCS coherent peak near Tc in the real part (T ) of conductivity is visible. In a London (S 1) pure ( < ) superconductor, the coherent peak is absent. As the relaxation rate increases, the peak appears at and increases in amplitude as long as the transition

from pure ( < ) to dirty ( > ) limit occurs. Thereafter, coherent peak ceases to grow, reaching the value of order max / (Tc ) ln( /) [22]. Two-band model of superconductivity assumes the presence of two coupled gap functions 1 and 2 , which, in the weak-coupling regime, are described by the system of equations given in [23]. Let subscripts and take the values 1 or 2, corresponding to a number of an electronic band. The matrix of electron­boson interaction constants which determines the relation between replaces transition temperature and characteristic cutoff frequency of a phonon spectrum D in the single-band BCS theory: Tc = (2 D / ) exp(-1/ ) ( 1.78 is Euler's constant). Electronic scattering by nonmagnetic impurities is taken into account by scattering rates = 1/ , which in the considered weak-coupling regime do not depend on energy. Nondiagonal elements of the scattering and interaction matrices are related like 12 /21 = 12 / 21 . Each of gap functions found from the system of equations [23]is used to calculate microwave conductivity = - i for each band using the expressions for local (London) regime [22] derived in the single-band case. Total conductivity is determined by a sum of conductivities in each band taking into account partial plasma frequencies, pl 1 and pl 2 , without cross-terms, which is justified in the local limit [24]. In isotropic single-band superconductor, the presence of not too strong impurity concentration does not lead to changes of thermodynamic properties of a superconductor in particular its transition temperature. In multiband superconductors, the picture is quite different [23, 25]. In addition to intraband scattering processes of electrons (characterized by coefficients 11 and 22 ), when electron remains in the same energy band after scattering process, there are scattering processes with interband transitions (characterized by coefficients 12 and 21 ). Assume that the superconducting gap in band 1 is larger than in band 2. The physical meaning of the parameters and are as follows: 11 specifies the maximum value of Tc ; 22 characterizes the low-temperature region T < Tc /2, in particular, the temperature dependences of the conductivity curves (T ) and (T ) at T 0; 12 and 21 should be much smaller than 11 or 22 to display the twoband superconductivity effects were pronounced; increasing of intraband scattering rates 11 or 22 reduce the effectiveness of electromagnetic field screening that appears to reduce the contribution of the imaginary part of conductivity (T ) from the corresponding subband; increasing of 21 tends to equalize the superconducting gaps in different bands, thus smoothing the temperature dependence of (T ); and growth of 12 leads to decrease Tc of superconductor almost without changing its other parameters. The latter effect, which occurs in multiband superconductors in contrast to single-band ones, is illustrated in


J Supercond Nov Magn (2015) 28:331­337 Fig. 1 Evolution of the temperature dependences of the microwave conductivity components within the framework of BCS theory and local electrodynamics. Left imaginary part of the conductivity. Right the real part. Both components are normalized to the value of the imaginary part of the conductivity at 0 temperature in pure limit L (0)

333

1

0.5
=1 =0.5 =0.2 =0.1 =0.05 =0.002

0.8

0.4

''(T)/L''(0)

'(T)/L''(0)

0.6

0.3

0.4

0.2



=1 =0.5 =0.2 =0.01 =0.005 =0.002

0.2

0.1

0

0.25

0.5

0.75

1

0

0.25

0.5

0.75

1

T/TC

T/TC

Fig. 2. With increasing 12 , strongest suppression of superconductivity occurs in the band with larger gap as long as the gaps in the different bands are not equalized (Fig. 3). After that, the suppression of the critical temperature by interband scattering occurs less efficiently.

3 Comparison with Experiment Crystals V3+x Si1-x were grown by the method of zone melting in Ar atmosphere from polycrystalline blanks with pre-prepared composition which were also exposed to one
Fig. 2 Critical temperature Tc of two-band superconductor as a function of interband scattering rate 12 . 11 = 0.289, 12 = 0.006, 21 = 0.002, 11 = 22 = 5 K, pl 1 = pl 2 , = 0.5 K. The solid line shows the calculation for the case of 22 = 0.205 and dashed line for 22 = 0.235. Squares depict the experimental points (see the next section). The insets show the evolution of the curves (T ) with changing 12 (bottom left curves correspond to 22 = 0.235 and the curves on the right at the top to 22 = 0.205)

pass zone melting in vacuum. For this study, selected five single-phase (i.e., consisting only the phase V3 Si) samples have been selected with Si-content 25, 24, 23, 22, and 21 atomic percent respectively. The samples have the form of thin plates with typical dimensions 2 â 1 â 0.5mm3 . For all crystals, the width of the superconducting transition did not exceed 0.5 K, as is seen by measuring temperature dependences of their dynamic susceptibility. For measurements of surface impedance components, we used the "hot finger" method. We placed the sample into the center of the cylindrical superconducting niobium cavity resonating at the frequency f = /2 =

18 1
12 12 12 12 12 = 16 K 12 = 20 K = = = = 0K 4K 8K 12 K

''(T)/''(0)

16

0.75 0.5 0.25 0

14

TC , K

12

5

T, K 10

5

15
12 12 12 12 = = = = 0K 4K 8K 12 K

10 T, K

15

0.8 0.6 0.4

10
12 = 16 K 12 = 20 K

''(T)/''(0)

8 0

0.2 0

22 = 0.235 22 = 0.205

5

10

15

20

25

30

35

12, K


334 Fig. 3 Temperature dependences of the superconducting gaps on the interband scattering rates 12 . Calculation parameters are the same as in Fig. 2 with 22 = 0.205

J Supercond Nov Magn (2015) 28:331­337

40
1 2 1 2 1 2 , , , , , ,
12 12 12 12 12 12

30

= = = = = =

0K 0K 10 K 10 K 35 K 35 K

1,2, K

20

10

0

5

10 T, K

15

20

9.4 GHz in TE011 mode and having high unloaded quality factor Q0 2 â 107 . Since the sample is located at the antinode of the microwave magnetic field, the resonance frequency f(T ) of the system and its quality factor Q(T ) can be easily related to the surface impedance Z(T ) = R(T ) + iX (T ) of the sample using simple relations [26]. The real part of Z(T ), the surface resistance R(T ), is proportional to the microwave power absorbed by the sample, whereas the imaginary part, the surface reactance X(T ), is the reactive component which defines the

electromagnetic field screening. In the superconducting state, X(T ) directly provides the magnetic field penetration depth (T ) = X(T )/0 . In the temperature range 2 T 20 K, the results of measurements of the surface impedance components are showninFig. 4. For all samples, except for the stoichiometric one (25% of silicon), the normal skin effect condition, R(T ) = X(T ), is satisfied at T > Tc . The reason for the discrepancy of R(T ) and X(T ) curves in the sample with the stoichiometric composition at Tc < T < 30 K is the

Fig. 4 Surface resistance R(T ) (circles) and reactance X(T ) (squares) of V3+x Si1-x crystals with various Si content

0.2
R, X, R, X, R, X, R, X, R, X, 21% 21% 22% 22% 23% 23% 24% 24% 25% 25% Si Si Si Si Si Si Si Si Si Si

0.15

Z,

0.1

0.05

0

5

10

15

20

T, K


J Supercond Nov Magn (2015) 28:331­337 Table 1 The parameters of the investigated crystals V % Si Tc , K ° (0), A 12 , K
22 3 +x

335 Si
1 -x

21 8.9 3,260 33 0.205

22 9.85 2,160 26 0.205

23 10.5 1,860 23 0.205

24 12.5 1,640 15 0.205

25 16.6 1,520 4 0.235

temporal dispersion of the Drude conductivity (estimated, 0.3 for this sample). At higher temperatures T > 30 K, when the processes of electron-phonon scattering become significant and the relaxation time is reduced, the equality R(T ) = X(T ) is also achieved in stoichiometric crystals [17]. The validity of the normal skin effect makes it possible to obtain the resistivities (T ) = 2R 2 (T )/0 , as shown in Fig. 5. The data demonstrate strong deviations from the Bloch-Gruneisen behavior, namely the tendency to satura¨ tion of the resistivity at high temperatures. This behavior is very similar to that of the resistivity (T ) in iron pnictide superconductors and can be explained within the two band scenario [27]. Table 1 shows the measured (critical temperature Tc and penetration depth (0) = X(0)/0 ) and the fitting parameters 12 and 22 of the investigated samples V3+x Si1-x . The constants of electron­phonon interaction 11 , 12 , 21 and the relative distribution of the electron within the subbands pl 1 and pl 2 are kept the same for all samples in the series, and the parameters 22 , 11 = 22 = 12 were adjusted for each sample. The last unknown

parameter of interband scattering 21 was calculated from the ratio of 21 /12 = 21 / 12 [23]. The best agreement with the experiment gives the values 11 = 0.289, 2 2 -3 -3 12 = 6 · 10 , 21 = 2 · 10 , pl 1 /pl 2 = 0.07/0.93, D = 500 K. In Fig. 6, the symbols show the temperature dependences of the imaginary part of the microwave conductivity (T ) in V3+x Si1-x crystals normalized to its value at 0 temperature (0). These curves were obtained by the formulas of local electrodynamics (T ) = (T ) - i (T ) = i0 /Z 2 (T ) using the experimental dependences of the surface impedance shown in Fig. 4. All of them have a distinctive feature, namely the presence of near Tc region with positive curvature which is not explained in the framework of the single-band BCS model (dashed line in Fig. 6). Solid lines show the calculations within the two-band model of superconductivity. As seen from Fig. 6, the twoband model describes well the evolution of the experimental curves. The temperature dependences of the real part of conductivity (T ) are shown in Fig. 7. In all samples, except the stoichiometric (25% Si) one, the BCS coherent peak is observed. Unlike the single-band BCS theory in which the coherent peak occurs at T 0.85Tc , in Fig. 7, it is shifted to lower temperatures that is explained in the framework of two-band theory, namely the peak is formed by electrons from the subband with a smaller superconducting gap. The main difference between experimental and theoretical curves of (T )/ (Tc ) consists in the amplitude of the coherent peak. The calculated peak amplitudes in Fig. 7 for all samples, except

Fig. 5 Resistivities of V3+x Si1-x crystals with various Si content

150
21% 22% 23% 24% 25% Si Si Si Si Si

100

, cm
50 0 0

100

200

300

T, K


336 Fig. 6 The imaginary part of microwave conductivity in V3+x Si1-x crystals in superconducting state. The symbols correspond to experimental data and the solid lines to calculations within the two-band model. For the sake of presentation clarity, each of the curves corresponding to 22% of Si and higher are shifted in the vertical direction by 0.1. Dashed line is calculation in the single-band BCS model

J Supercond Nov Magn (2015) 28:331­337

1.2

''(T)/''(0)

1 0.8 0.6 0.4 0.2 0 5 10 T, K

25% 25% 24% 24% 23% 23% 22% 22% 21% 21%

Si, Si, Si, Si, Si, Si, Si, Si, Si, Si,

exp. theor. exp. theor. exp. theor. exp. theor. exp. theor.

15

the stoichiometric one, reached their maximum values in the dirty limit. At the same time, the experimental data show approximately two times larger peak than theoretical calculation.

4 Conclusions In conclusion, in this work, we have presented the results of the study of the temperature dependences of microwave conductivity in V3+x Si1-x crystals with various Si content at frequency 9.4 GHz. The data exhibit peculiarities

typical for multiband superconductors, namely nonlinear temperature dependence of resistivity above superconducting transition temperature Tc , suppression of Tc by nonmagnetic impurities, an upward curvature of (T ) dependences near Tc , and a coherent peak of (T ) at T Tc /2. The data are interpreted within a two-band model in the weakcoupling regime. We have demonstrated that the behavior of Tc and the evolution of (T ) with Si-content variation are consistently described by the variation of the interband impurity scattering rate. The results strongly support the viewpoint that V3 Si belongs to the class of multiband superconductors.

'(T)

Fig. 7 The real part (T )/ (Tc )) of microwave conductivity in superconducting state of V3+x Si1-x crystals. The solid lines show the results of calculations within a two-band model of superconductivity. The open circles are the experimental data normalized in order to match the peak values of the calculated curves

2
21% Si 24% Si 25% Si

1.5

1

0.5

0

0

5

10 T, K

15

20


J Supercond Nov Magn (2015) 28:331­337 Acknowledgments We would like to thank V.A. Marchenko for producing V3+x Si1-x crystals. Helpful discussions with I.I. Mazin are gratefully acknowledged. This work was supported by megagrant no.14Y26.31.0007 funded by Russian Ministry of Education and Science and in part by RFBR grant no.12-02-01009 and Dutch FOM.

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