Non-BCS thermodynamic properties of {\rm H_{2}S} superconductor

Non-BCS thermodynamic properties of superconductor

Artur P. Durajski adurajski@wip.pcz.pl    Radosław Szczȩśniak szczesni@wip.pcz.pl    Yinwei Li yinwei˙li@jsnu.edu.cn 1. Institute of Physics, Czȩstochowa University of Technology, Ave. Armii Krajowej 19, 42-200 Czȩstochowa, Poland 2. Institute of Physics, Jan Długosz University, Ave. Armii Krajowej 13/15, 42-200 Czȩstochowa, Poland 3. School of Physics and Electronic Engineering, Jiangsu Normal University, Xuzhou 221116, People’s Republic of China
July 16, 2019
Abstract

The present paper determines the thermodynamic properties of the superconducting state in the compound. The values of the pressure from GPa to GPa were taken into consideration. The calculations were performed in the framework of the Eliashberg formalism. In the first step, the experimental course of the dependence of the critical temperature on the pressure was reproduced: K, whereas the Coulomb pseudopotential equal to was adopted. Next, the following quantities were calculated: the order parameter at the temperature of zero Kelvin (), the specific heat jump at the critical temperature (), and the thermodynamic critical field (). It was found that the values of the dimensionless ratios: , , and deviate from the predictions of the BCS theory: , , and . Generalizing the results on the whole family of the -type compounds, it was shown that the maximum value of the critical temperature can be equal to K, while , and adopt the following values: , , and , respectively.

Keywords: Superconductors; Hydrogen sulfide; Thermodynamic properties.

pacs:
74.20.Fg, 74.25.Bt, 74.62.Fj

The experimental results, which prove that the compound under the influence of the high pressure () has the extremely high values of the critical temperature (), were presented in December, 2014 Drozdov et al. (2014). In particular, it was shown that in the range of the pressures from GPa to GPa, the critical temperature increases from K to K. Additionally, we should underline the fact that the strong isotope effect () was observed, which clearly suggests the electron-phonon origin of the superconducting state Hirsch and Marsiglio (2015); Bernstein et al. (2015); Papaconstantopoulos et al. (2015); Flores-Livas et al. (2015); Akashi et al. (2015); Zhang et al. (2015). Interestingly, as the result of the dissociation of the starting compound, most likely of the scheme: Duan et al. (2014a), Duan et al. (2014b), the superconducting state with the critical temperature of up to K ( GPa) was induced Drozdov et al. (2014). From the physical point of view, the obtained result indicates that the superconductor of the highest known value of was just discovered.

Even before the release of the experimental results, the extensive theoretical studies of the superconducting condensate in the compound were performed in the paper Li et al. (2014). In the framework of the ab initio calculations, it was found that the analyzed system enters the metallic state above the pressure of GPa. Next, the existence of the superconducting state was proven in the pressures range from GPa to GPa, wherein the highest value of equal to K was obtained for GPa (the Cmca structure). It should be noted that the predictions included in the publication Li et al. (2014) agree with the experimental results Drozdov et al. (2014). However, there was no structural transition observed between the phases P-1 and Cmca.

The results presented in the paper Drozdov et al. (2014) prove that depending on the method to handle the expected final values of the temperature and the pressure one can prepare the compound or the system . Note that the above results are in the agreement with the theoretical predictions that suggest the stability of the system below GPa Duan et al. (2014b). On the other hand, the compound seems to be stable for GPa, wherein , , and are unstable in the considered range of the pressures.

The superconductor can be further enriched with hydrogen Strobel et al. (2011). The case was very carefully analyzed in the work Duan et al. (2014a). On the basis of the ab initio calculations, it was shown that the metallization of this system takes place for GPa, while for GPa the record value of equal to K was obtained (the - structure). Note that the similarly high values of the critical temperature can be obtained in the hydrogen-rich compounds of the type: ( K for GPa) Wang et al. (2012), Szczȩśniak and Durajski (2013a), ( K for GPa) Jin et al. (2010), Flores-Livas et al. (2012), Szczȩśniak and Durajski (2013b), ( K for GPa) Abe and Ashcroft (2011), Szczȩśniak et al. (2013), and ( K for GPa) Li et al. (2010).

Historically, the physical properties of the compound have been studied for many years. On the molecular level is formally the analogue of . However, in the solid phase its properties are significantly different due to the fact that hydrogen sulfide is very weak hydrogen-bonded Ikeda (2001). It should be noted that has the complicated pressure-temperature phase diagram. In the area: GPa and K, as many as seven crystal structures are distinct Cockcroft and Fitch (1990), Shimizu et al. (1991), Shimizu et al. (1992), Endo et al. (1994), Endo et al. (1996), Fujihisa et al. (1998). On the other hand, the recently published theoretical results call into question the original findings on the number and the type of the existing crystal structures Li et al. (2014). It is also worth noting that to the present date there is no final consensus on the pressure at which the molecular dissociation occurs at the room temperature. In literature one can find the two characteristic values of the pressure: GPa and GPa Sakashita et al. (1997), Fujihisa et al. (2004), Shimizu et al. (1997).

The present paper determines the thermodynamic parameters of the superconducting state in the compound in the range of the pressures from GPa to GPa. Then, the study generalizes the results on the entire family of the compounds of the -type (also additionally hydrogenated). The numerical calculations were performed in the framework of the Eliashberg formalism due to the significant strong-coupling and retardation effects.

The Eliashberg equations for the order parameter function and for the wave function renormalization factor take the form Marsiglio et al. (1988):

and

where:

(3)

The order parameter is defined as: .

The imaginary axis functions ( and ) should be calculated from Eliashberg (1960):

(4)

and

(5)

where: is the Matsubara frequency and . The symbol represents the Boltzmann constant. The pairing kernel is given by:

(6)

The Eliashberg function () models the structure of the electron-phonon interaction and is the maximum phonon frequency. For the superconductor , in the range of the pressure from GPa to GPa, the Eliashberg functions were calculated in the paper Li et al. (2014). The maximum phonon frequency is in the order of meV. The function describes the depairing Coulomb interaction, where represents the Coulomb pseudopotential Morel and Anderson (1962). The symbol is the Heaviside unit function and denotes the cut-off energy (). The Bose function and the Fermi function is given by the symbol and , respectively.

The Eliashberg equations have been solved for . The functions and are stable for K. Note that the above was based on the numerical methods used in the publications: Szczȩśniak and Durajski (2013a), Szczȩśniak and Durajski (2014a), Durajski and Szczȩśniak (2014), Szczȩśniak and Durajski (2014b).

Fig. 1 (A)-(B) shows the plots of the exemplary dependence of the order parameter and the wave function renormalization factor on the temperature ( GPa). The wide range of the values of the Coulomb pseudopotential was taken into account. It turns out that the resulting curves can be reproduced with the very good approximation by using the functions: and , where: , , , and (the electron-phonon coupling constant () was defined in Tab. 1).

In the first case, it is clear that the high values of the order parameter correspond to a high values of the critical temperature (Fig. 1 (A)). It should be noted that the full shape of the function cannot be properly identified within the framework of the classical BCS theory, as: Eschrig (2001).

Figure 1: The dependence of (A) the order parameter and (B) the wave function renormalization factor on the temperature for the selected values of the Coulomb pseudopotential. The figures (C)-(D) present the overt form of the functions and for the selected temperatures. The shaded area is the rescaled Eliashberg function.

On the other hand, the wave function renormalization factor determines the value of the electron effective mass: , where denotes the electron band mass. The results plotted in Fig. 1 (B) prove that the effective mass of the electrons is large in the superconducting state. From the physical point of view the obtained result comes from the existence of the significant strong-coupling effects in the compound, which are characterized by the electron-phonon coupling constant. Of course, these effects cannot be ignored in the quantitative analysis. Let us notice that the BCS model predicts: .

The curves in Fig. 1 (A) and (B) were obtained on the basis of the following expressions: and , while the example solutions of the Eliashberg equations are presented in Fig. 1 (C) and (D). It can be seen that the order parameter has the zero imaginary part at the low frequencies, due to the absence of the damping effects Varelogiannis (1997). At the higher frequencies both and are characterized by the very complicated shape clearly correlated with the shape of the Eliashberg function.

Fig. 2 (A) presents the influence of the Coulomb pseudopotential on the value of the critical temperature in the compound. It can be seen that in the case of the weak electron depairing correlations (), can reach the high value of the order of K.

The numerical results obtained with the help of the Eliashberg equations can be reproduced with the very good accuracy using the modified Allen-Dynes formula:

(7)

whereas the symbols appearing in Eq. (7) were defined in Tab. 1. In contrast, the numerical parameters were selected using the method of the least squares on the basis of numerical values of the function .

Figure 2: The critical temperature as a function of (A) the Coulomb pseudopotential for GPa and (B) the pressure for . The experimental results are taken from Drozdov et al. (2014).
Figure 3: The critical temperature as a function of the coupling constants and for the superconductors of the -type.
Quantity Value ( GPa)
1.28
82.70 meV
115.05 meV
-
-
-
-
Table 1: The quantities , , and represent the electron-phonon coupling constant, the logarithmic phonon frequency, and the second moment of the normalized weight function. The parameters and are the strong-coupling correction function and the shape correction function, respectively Allen and Dynes (1975).

Additionally, Fig. 2 (A) shows the plot of the values of the critical temperature obtained with the help of the classical Allen-Dynes and McMillan formulas Allen and Dynes (1975), McMillan (1968). It was found that the classical formulas significantly understate for the higher values of the Coulomb pseudopotential.

Fig. 2 (B) presents the experimental dependence of the critical temperature on the pressure for the compound Drozdov et al. (2014). Note that the obtained results can be reproduced using the Eliashberg equations or Eq. (7) adopting: . In particular, for the pressure values from GPa to GPa, it was obtained: K.

Let us notice that Eq. (7) allows to discuss possible to achieve values of the critical temperature for the whole family of the compounds of the -type. In the first step it should be noted that contributions to the Eliashberg function derived from sulfur and hydrogen (both for and ) are very clearly separated Duan et al. (2014a), Li et al. (2014). In particular, in the frequency range from to meV the crucial is the electron-phonon interaction derived from sulfur, while above meV significant is the contribution derived from hydrogen. Based on the above fact, the model Eliashberg function was factorized as follows:

where and are the contributions to the electron-phonon coupling constant derived respectively from sulfur and hydrogen. On the other hand, the symbols and represent the maximum phonon frequencies.

Using the equations included in Tab. 1, it can be shown that:

(9)

and

(11)
Figure 4: (A) The free energy difference between the superconducting and normal state, (B) the thermodynamic critical field, and (C) the specific heat of the superconducting state and the normal state as a function of the temperature for the selected values of the Coulomb pseudopotential. The vertical lines indicate the position of the characteristic specific heat jump at .

Fig. 3 presents the dependence of the critical temperature on and . It was adopted: , , meV, meV Durajski et al. (2012), Szczȩśniak and Jarosik (2012), and the selected values of .

The obtained results show that for the low values of the Coulomb pseudopotential, the maximum critical temperature can be equal even to K. From the physical point of view, the result shows the possibility of induction of the superconducting state with the critical temperature comparable to the room temperature in the compounds of the -type.

The thermodynamic critical field and the specific heat of the superconducting state can be calculated in the Eliashberg formalism using the formula for the free energy difference between the superconducting state () and the normal state ():

where denotes the value of the electron density of states on the Fermi level.

The formula for the thermodynamic critical field has the form: . On the other hand, the specific heat of the superconducting state can be expressed using the formula: , where and . The symbol denotes the Sommerfeld constant: .

Figure 5: The dimensionless thermodynamic ratios as a function of the Coulomb pseudopotential for GPa (left panel) and as a function of the pressure for (right panel).
Figure 6: The dimensionless thermodynamic ratios as a function of the coupling constants and for the -type superconductors.

The examples of the results obtained for GPa are plotted in Fig. 4. It can be seen that below the critical temperature, the difference in the free energy takes the negative values, which from the physical point of view shows the thermodynamic stability of the superconducting phase (Fig. 4 (A)). It should also be noted that the values of strongly depend on the Coulomb pseudopotential. The strong dependence of the free energy difference on the Coulomb pseudopotential transfers directly to the thermodynamic critical field and the specific heat of the superconducting state (see Fig. 4 (B) and (C)). In particular, it was obtained: and , while .

The designated thermodynamic functions allow to calculate the dimensionless ratios: , , and , where . It is worth noting that, in the BCS theory, the quantities: , , and adopt the universal values, which are equal respectively to: , , and Bardeen et al. (1957a), Bardeen et al. (1957b). In the case of ( GPa), the obtained results are presented in Fig. 5. It is easy to note that the values - differ significantly from the predictions of the classical BCS theory, with the largest derogations observed in the case of the weak electron depairing correlations.

Fig. 5 shows the plots of the dimensionless ratios as a function of the pressure ( GPa). The numerical results show that the thermodynamic parameters - cannot be correctly estimated in the framework of the BCS theory for the wide range of pressure: , , and . From the physical standpoint, the results presented above arise from the existence of the strong-coupling and retardation effects in . Note that in the simplest case, it can be characterized by the ratio: Carbotte (1990). Thus, the results included in Fig. 5 can be reproduced with the help of the following formulas:

(13)
(14)

and

(15)

It should be underlined that the number parameters appearing in Eq. (13) - Eq. (15) were selected on the basis of numerical values of - depending on the Coulomb pseudopotential. Eq. (13) - Eq. (15) can be used to estimate - for the whole family of the -type compounds. The obtained results are plotted in Fig. 6. It was found that the values, which maximally differ from the results of the BCS theory, are respectively: , , and .

In the presented paper, the thermodynamic parameters of the superconducting state inducing in the compound were determined ( GPa). The calculations were carried out in the framework of the Eliashberg formalism. It has been shown that the theoretical analysis is able to reproduce the experimental dependence of the critical temperature on the pressure, assuming the relatively low value of the electron depairing correlations ().

The other thermodynamic functions such as: the order parameter, the thermodynamic critical field or the specific heat of the superconducting state deviate from the expectations of the classical BCS theory (as evidenced by the values of the parameters - ). It turns out that this result is associated with the existence of the significant strong-coupling and retardation effects appearing in the compound.

Generalizing the obtained results, we showed that the maximum value of the critical temperature in the family of the compounds of the -type can be equal even to K, which from the physical point of view means the possibility of the existence of the superconducting phase at the room temperature. The values of the parameters - for the family of differ also significantly from the expectations of the BCS theory.

Acknowledgements.
Y. L. acknowledges funding from the National Natural Science Foundation of China under Grant Nos. 11204111, 11404148, the Natural Science Foundation of Jiangsu province under Grant No. BK20130223, and the PAPD of Jiangsu Higher Education Institutions.

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