Thermodynamics of the superconducting state in Calcium at 200 GPa
Abstract
The thermodynamic parameters of the superconducting state in Calcium under the pressure at 200 GPa were calculated. The Coulomb pseudopotential values () from to were taken into consideration. It has been shown, that the specific heat’s jump at the critical temperature and the thermodynamic critical field near zero Kelvin strongly decrease with . The dimensionless ratios and significantly differ from the predictions based on the BCS model. In particular, decreases from to with the Coulomb pseudopotential; whereas increases from to . The numerical results have been supplemented by the analytical approach.
pacs:
74.20.Fg, 74.25.Bt, 74.62.FjI Introduction
By using the advanced technique, it is experimentally possible to explore the properties of the superconducting state under the high pressure (). In particular, the above researches enable: (i) the test of the theories for the superconducting state, (ii) as well as to improve the properties of the superconductors, and (iii) to create the new superconductors. At present, the elemental superconductors are known, however of them superconduct if the pressure is applied Schilling (). The most interesting elements are: Lithium, where the critical temperature () rises rapidly to K at GPa Deemyad (), Yttrium with the maximum value of the critical temperature of K at GPa Hamlin (), and Calcium which has the highest observed value of ( K at GPa) Yabuuchi (), Okada (), Yin ().
The thermodynamic properties of the superconductors under the high pressure can be analyzed in the framework of the Eliashberg approach Eliashberg (), Marsiglio (), Carbotte (). In this formalism the complicated form of the electronphonon interaction is modeled by Eliashberg function (). We notice that the firstprinciple calculations of require the knowledge of the electronic wave functions, the phonon spectrum, and the electronphonon matrix elements between two singleelectron Bloch states. Experimentally, the form of the Eliashberg function can be directly obtained from the second derivative of  curve for the tunnel junction () McMillan ().
From the physical point of view, the Eliashberg approach represents particularly important method of the analysis, since it enables the calculation of the thermodynamic parameters on the quantitative level. In particular, the exact form of the free energy difference between the superconducting and normal state should be calculated on the basis of the socalled Eliashberg equations Bardeen (). In the considered case, the input parameters are the Eliashberg function and the Coulomb pseudopotential (), where models the Coulomb repulsion between electrons. We notice that its value is selected in such way that determined on the basis of the Eliashberg equations equals the experimental value of the critical temperature.
In the presented paper, we have analysed the thermodynamic properties of Calcium under the pressure at GPa by using the Eliashberg approach. In particular, the following parameters were taken into account: the specific heat in the superconducting state (), the specific heat in the normal state (), and the thermodynamic critical field (). Additionally, the dimensionless ratios and have been determined BCS ().
For Calcium, the dependence of the critical temperature on the pressure has been obtained experimentally by Okada et al. in 1996 Okada () and then by Yabuuchi et al. in 2006 Yabuuchi (). The results prove that the critical temperature grows with the pressure from the value of about K ( GPa) to K ( GPa). On the basis of the Yabuuchi’s results it is easy to show that the values of the Coulomb pseudopotential can be large. For example, and SzczesniakDurajski (), SzczesniakSzczesniak (). In the case of the pressure GPa, the experimental value of is still unknown, therefore the wide range of the Coulomb pseudopotential’s values have been considered in the paper; .
Ii The Eliashberg Equations
The Eliashberg functions for Calcium were determined by Yin et al. Yin (). At the pressure GPa, the structure is clearly favored, and the linearresponse calculations indicate that it is also dynamically stable. Additionally, the Yin’s results have shown that the strong electronphonon coupling persists and can be high ( K). In the paper, the thermodynamic parameters for Calcium have been calculated by using the Eliashberg equations on the imaginary axis, the Yin’s Eliashberg function has been taken into account.
The Eliashberg set has the following form:
(1) 
and
(2) 
The wave function renormalization factor is denoted by , and the order parameter function by . The th Matsubara frequency has the form: , where and is the Boltzmann constant. We notice that the value of the order parameter is given by the ratio: . In Eqs. (1) and (2) the symbol denotes the electronphonon pairing kernel:
(3) 
where the maximum phonon frequency is equal to meV. Finally, denotes the Heaviside unit function and is the cutoff frequency ().
From the mathematical point of view the exact solution of the Eliashberg equations represents a complicated problem. We notice that formally the Eliashberg set contains the infinite number of the nonlinear algebraic equations; in addition every equation has the integral kernel, which is dependent on the form of the Eliashberg function. It is possible to prove, that if we limit the number of the Matsubara frequencies, the solutions of the Eliashberg equations lose the convergence only in the area of the very low temperatures. In the paper we assume . In this case the functions and are stable for K. The Eliashberg equations have been solved by using iterative method Szczesniak ().
Iii The Numerical Results
In Fig. 1 the order parameter as a function of the number has been presented. We have considered the selected values of the Coulomb pseudopotential and temperatures. It is easy to see that decreases with number’s growth. In particular, the strong fall of the order parameter appears for low values of ; for higher values the function saturates. Very similar dependence on the number possesses the wave function renormalization factor (see Fig. 2). However, the function saturates considerably slower than .
The growth of the Coulomb pseudopotential’s value differently influences on the order parameter and the wave function renormalization factor. In the first case the values of the order parameter strongly decrease; whereas the function is practically not changing. From the physical point of view the above facts mean, that together with the increasing of decreases only the value of the critical temperature; the electron effective mass remains fixed ().
The solutions of the Eliashberg equations also very unlikely evolve with the temperature. Most clearly this fact is possible to observe in Fig. 3, where the dependences of and on the temperature have been plotted. In particular, the obtained results show that the temperature dependence of the order parameter can be modeled by the function: , where the parameters , , and are collected in Tab. 1. On the other hand, the wave function renormalization factor is slightly depended on the temperature.
meV  K  

In order to obtain the specific heats and the thermodynamic critical field, we have to calculate the free energy difference between the superconducting and normal state () Bardeen ():
(4) 
The functions and represent the wave function renormalization factors for the superconducting (S) and normal (N) state respectively.
The specific heat difference between the superconducting and normal state can be obtained by using the expression:
(5) 
On the other hand, the specific heat in normal state is given as:
(6) 
where . In Fig. 4 the specific heat for the superconducting and normal state as a function of the temperature has been shown. It is easy to see that the specific heat’s jump at the critical temperature decreases with the growth of the Coulomb pseudopotential. In particular, .
The thermodynamic critical field has been calculated by using the formula:
(7) 
In Fig. 5 the dependence of on the temperature has been presented. We can see, that the value of the thermodynamic critical field near the temperature of zero Kelvin () also strongly decreases with ; .
The dimensionless ratios and on the basis of the calculated thermodynamic functions have been determined. We notice that in the framework of the BCS model these parameters have the universal values: and BCS (). For Calcium the theoretical data have been collected in Fig. 6. We see that even for large values of the ratios significantly diverge from the values predicted by the BCS model.
Below, we have given the formulas, which enable the simple calculations of and . In particular:
(8) 
and
(9) 
We notice that the numerical coefficients in Eqs. (8) and (9) by the leastsquares analysis of exact and values have been chosen. Finally, the critical temperature by using the modified AllenDynes formula should be calculated SzczesniakDurajski1 ():
(10) 
where the strongcoupling correction function () and the shape correction function () are given by:
(11) 
and
(12) 
The functions and have the form:
(13) 
and
(14) 
The parameter is the second moment of the normalized weight function, denotes the logarithmic phonon frequency and is called the electronphonon coupling constant. In the case of Calcium under the pressure at GPa the following results have been obtained: , meV and .
Iv Summary
The thermodynamic parameters of the superconducting state in Calcium under the pressure at 200 GPa have been analyzed in the paper. The numerical calculations in the framework of the Eliashberg approach have been made. On the basis of the exact Eliashberg solutions the specific heats and the thermodynamic critical field have been determined.
For the wide range of the Coulomb pseudopotential values (), it has been shown that the specific heat’s jump at the critical temperature and the thermodynamic critical field near zero Kelvin strongly decrease with .
The dimensionless ratios and significantly differ from the predictions based on the BCS model even for high values of the Coulomb pseudopotential. In particular, and .
Acknowledgements.
The authors wish to thank Prof. K. Dziliński for providing excellent working conditions and the financial support. We also thank Mr M.W. Jarosik and Mr D. Szczȩśniak for the productive scientific discussion, that improved the quality of the presented paper. All numerical calculations were based on the Eliashberg function sent to us by: Prof. W.E. Pickett and Dr Z.P. Yin for whom we are very thankful.References

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