# Superconducting properties of tin-copper amorphous thin films

###### Abstract

The thermodynamic properties of the superconducting state in the amorphous thin films have been characterized. The concentration of copper changes in the range from to . The calculations have been conducted in the framework of the strong-coupling formalism, wherein the Eliashberg functions determined in the tunnel experiment (Phys. Rev. B 51, 685 (1995)) have been used. The value of the Coulomb pseudopotential equal to has been adopted. It has been found that the critical temperature () decreases from K to K. The ratio of the energy gap to the critical temperature: , differs significantly from the BCS value: . Similarly behaves the ratio of the specific heat jump to the specific heat of the normal state: , and the parameter: , where is the thermodynamic critical field. In particular, and .

###### pacs:

74.20.Fg, 74.25.BtKeywords: Superconductivity, Amorphous systems, Thermodynamic properties.

The research on the superconducting state has lasted more than hundred years Onnes1911A (). One of the first superconductors discovered was tin, wherein the superconducting phase is characterized by the low value of the critical temperature: K Eisenstein1954A (), Matthias1963A (). It turns out that the disappearance of the crystal structure and the formation of the amorphous system can be the reason for the significant increase in the transition temperature. In particular, the amorphous alloys containing copper deserve particular attention Schmidt1999A ().

This paper presents the results of the theoretical analysis obtained for the superconducting state of the amorphous thin films, for the concentration of copper in the range from to .

Due to the high values of the electron-phonon coupling constant (), the numerical calculations have been performed in the framework of the strong-coupling theory, wherein: . The spectral functions () have been determined on the basis of the tunnel experiment Watson1995A (). The maximum phonon frequency () is equal to meV. A detailed dependence of the electron-phonon coupling constant on the concentration of copper is presented in Fig. 1.

The strong-coupling theory is based on the Eliashberg equations Eliashberg1960A ():

(1) |

(2) |

It should be noted that the Eliashberg set can be obtained from the Fröhlich Hamiltonian, which explicitly models the electron-phonon interaction Frohlich1952A (). The solutions of the Eliashberg equations are: the order parameter function (), and the wave function renormalization factor (). The order parameter for the superconducting state is defined as the ratio: . The quantity is the fermion Matsubara frequency: , whereas: , and denotes the Boltzmann constant. The symbol represents the pairing kernel for the electron-phonon interaction: .

The classical Eliashberg formalism is the one-parameter theory, in which the depairing electron correlations are modeled with the help of the Coulomb pseudopotential () Morel1962A (). In the study we have assumed: . The quantity represents the Heaviside function, is the cut-off frequency: .

The finite number of equations has been taken into account performing the numerical analysis (). The functions and are stable for . The Eliashberg equations have been solved using the numerical procedures tested and discussed in the papers: Ania2013A (), Ania2014A (), Ania2014B (), Ania2014C (), Ania2014D (), Ania2015A (), Ania2015B ().

In order to calculate the exact value of the order parameter, the solutions of the Eliashberg equations should be analytically continued on the real axis: and . Most effectively it can be done with the help of the Eliashberg equations in the mixed representation Marsiglio1988A ():

and

The symbols and represent the Bose-Einstein and the Fermi-Dirac functions.

Fig. 2 shows the plot of the temperature course of the order parameter and the wave function renormalization factor. It can be noted that the increase in the concentration of copper very significantly weakens the superconducting state, while contributing to the less pronounced fall in the value of the electron effective mass: , where the symbol represents the electron band mass.

Then, the overt dependence of the critical temperature on the copper concentration has been determined. The results are presented in Fig. 3. The critical temperature changes over the fairly wide range from K to K. Nevertheless, the full Eliashberg formalism allows us to reproduce the experimental results with a very good approximation Schmidt1999A (), Knorr1971A (), Leitz1980A (), Dutzi1984A (), Naugle1995A (). Additionally, let us point out that the values of determined with the use of the approximated McMillan or Allen-Dynes formulas are significantly understated McMillan1968A (), Allen1975A ().

The other quantities characterizing the superconducting phase (the thermodynamic critical field and the specific heat of the superconducting state ) have been determined on the basis of the free energy difference between the superconducting and the normal state Bardeen1964A ():

The symbol represents the value of the electron density of states at the Fermi level. The markings and refer to the superconducting state and the normal state, respectively.

The obtained curves are plotted in the lower panel in Fig. 4. Note that an increase in the value of x causes a strong decrease of the free energy difference, which is in a direct way reflected in the thermodynamic critical field (upper panel in Fig. 4):

(6) |

The specific heat of the superconducting state should be calculated on the basis of the formula: , where represents the specific heat of the normal state: , is the Sommerfeld constant. The specific heat difference between the superconducting and the normal state () is determined on the basis of:

(7) |

Fig. 5 presents the dependence of the specific heat for the superconducting state and the normal state on the temperature. The characteristic jump that exists at the critical temperature is marked with the vertical line.

The analyzed thermodynamic properties allow to determine the dimensionless parameters:

(8) |

The numerical calculations prove that for , the parameter increases from to , and the parameter decreases from to . Note that the BCS theory predicts: and Bardeen1957A (), Bardeen1957B (). Thus, even for the high concentration of copper, the thermodynamic critical field and the specific heat of the superconducting state is not correctly described by the BCS theory.

The physical values of the order parameter have been obtained by solving the Eliashberg equations in the mixed representation.

Fig. 6 shows the plot of the values of the function . It can be seen that the order parameter forms the characteristic deformed spirals of the size that is decreasing together with the temperature’s increase. Let us notice that the imaginary part of describes the damping effects Varelogiannis1997A (). On the other hand, the real part of the function is used to determine the exact values of the energy gap at the Fermi level:

(9) |

From the physical point of view, the most interesting is its value for the lowest temperature considered in the paper: . From here, meV, when .

Based on the results achieved, the values of the dimensionless parameter have been calculated:

(10) |

As a result, it has been obtained: . Note that in the BCS theory the ratio is the universal constant of the value equal to Bardeen1957A (), Bardeen1957B ().

In conclusion, within the full Eliashberg formalism we have determined the thermodynamic parameters of the superconducting state in the thin films, for the concentration of copper in the range from to .

The results show that the increase in the concentration of copper lowers the value of the critical temperature from K to K. The values of the remaining thermodynamic quantities, like the order parameter, the thermodynamic critical field and the specific heat of the superconducting state, differ materially from the expectations of the BCS theory.

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