- Email: [email protected]
Description of the thermodynamic properties of BiH5 and BiH6 superconductors beyond the mean-field approximation
Solid State Communications 279 (2018) 27–29
Contents lists available at ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Communication
Description of the thermodynamic properties of BiH5 and BiH6 superconductors beyond the mean-field approximation M.W. Jarosik a , E.A. Drzazga a,* , I.A. Domagalska a , K.M. Szcze¸s´niak b , U. Ste¸pien´ a a b
Institute of Physics, Czestochowa University of Technology, Ave. Armii Krajowej 19, 42-200 Czestochowa, Poland Ul. Pomorska 37/55, Zawiercie 42-400, Poland
A R T I C L E
I N F O
Communicated by Y.E. Lozovik Keywords: Hydrogen-rich materials High-pressure effect Electron-phonon superconductivity Eliashberg theory Thermodynamic properties
A B S T R A C T
The ab initio calculations suggest (Y. Ma et al.), that the high-pressure (p = 200 GPa) superconducting state in BiH5 and BiH6 compounds characterizes with a high value of the critical temperature (TC ∼ 100 K). Due to the large value of the electron-phonon coupling constant (𝜆 ∼ 1.2), the thermodynamic parameters of the superconducting phase in BiH5 and BiH6 have been determined beyond the mean-field approximation - in the framework of the Eliashberg equations formalism. We have calculated the dependence of the order parameter (Δ) and the wave function renormalization factor on the temperature. Then we have estimated the free energy difference between the superconducting state and the normal state, the thermodynamic critical field (HC ) and the specific heat of the superconducting state (CS ) and the normal state (CN ). The values of the dimensionless ratios RΔ = 2Δ (0) ∕kB TC , RC = ΔC (TC ) ∕CN (TC ) and RH = TC CN (TC ) ∕HC2 (0) are equal RΔBiH = 4.17 and RΔBiH = 4.20, 5 6 RCBiH = 2.54 and RCBiH = 2.58, RHBiH = 0.146 and RHBiH = 0.146 respectively. 5
6
In 1935 Wigner and Huntington noted that hydrogen affected by the high pressure (∼25 GPa) should transit into a metallic state [1]. So far, it has been possible to experimentally demonstrate the metallization of liquid hydrogen for the pressure at 140 GPa, but the temperature was up to 3000 K [2]. In 2017 the information was spread that hydrogen metallization was observed at room temperature under the pressure at 495 GPa [3]. Unfortunately, this result has not been confirmed again. In 1968 Ashcroft noted that the metallic hydrogen should be a hightemperature superconductor [4]. He gave the following arguments: first of all, hydrogen has the lowest atomic mass of the nucleus, which implies the Debye frequency (𝜔D ) of the crystal lattice of solidified hydrogen to be high; secondly, there are no electrons on the internal shells in the hydrogen atom, which favors a strong electron-phonon coupling. In the following years, thanks to the rapid development of computer technology, more detailed calculations were carried out. The hydrogen metallization pressure was estimated to be around 400 GPa [4,5]. It has been suggested that in the molecular hydrogen phase, within the range of pressures from 400 to 500 GPa there should be a sharp rise in the critical temperature from approximately 80 K to about 350 K [6–9]. Above the pressure at 500 GPa the numerical calculations suggest that the metallic molecular hydrogen should dissociate into the atomic phase [6], whereas the critical temperature value
* Corresponding author. E-mail address: [email protected] (E.A. Drzazga). https://doi.org/10.1016/j.ssc.2018.05.009 Received 1 March 2018; Accepted 21 May 2018 Available online 25 May 2018 0038-1098/© 2018 Elsevier Ltd. All rights reserved.
5
6
should be within the limit from 300 K to 470 K [10]. In the literature on the subject there are also works analyzing the superconducting state of hydrogen for extremely high pressures (up to 3.5 TPa) [10–14]. The highest value of TC was foreseen at 630 K for p = 2 TPa. Due to the very high value of the hydrogen’s metallization pressure, in 2004 Ashcroft suggested to try to look for a high temperature superconducting state not in pure hydrogen but in hydrogenated compounds [15]. Induction of high temperature superconducting state in hydrogenated compounds should be supported by chemical precompression caused by heavier elements. The real breakthrough occurred only in December 2014, when Drozdov, Eremets, and Troyan have experimentally demonstrated the existence of a high-temperature superconducting state in H3 S (TC = 203 K for p = 155 GPa) [16,17]. In the presented work, we have determined the thermodynamic properties of the superconducting state in hydrogenated compounds BiH5 and BiH6 . Due to the high predicted value of the electronphonon coupling constant (𝜆BiH5 = 1.236 and 𝜆BiH6 = 1.259 [18]) we performed the calculations for both cases as a part of Eliashberg formalism [19]. A detailed description of Eliashberg equations and the computational methods used has been presented in the works [20–26]. The spectral function (𝛼 2 F (Ω)) used in numerical calculations has been determined by Y. Ma et al. with an use of the density functional the-
M.W. Jarosik et al.
Solid State Communications 279 (2018) 27–29
can determine the renormalized electronic density of states in the superconducting phase: ] [ NS (𝜔) (𝜔 − iΓ) , (2) = Re √ NN (𝜔) (𝜔 − iΓ)2 − (Δ(𝜔))2 where we have assumed that the pair breaking parameter equals
Γ = 0.15 meV. We have plotted the obtained results in the drawings Fig. 1(b) and (c). Note that the characteristic maxima of the determined curves are formed in points 𝜔 = ± Δ. On their basis, one can experimentally analyze the evolution of the energy gap depending on the temperature. The exact form of the order parameter on the real axis is presented in the figures Fig. 2(a)–(f). When analyzing the case T = T0 , it can be seen that in the range of the lower frequencies (𝜔 < 𝜔d = 85 meV) non-zero is only the real part of the order parameter. This means no damping processes that are modeled by a function Im [Δ (𝜔)]. It is also worth noting that as the temperature rises, 𝜔d decreases significantly. Additionally, for illustrative purposes in drawings (g) and (h) we have plotted the values of the order parameter on the complex plane. In both analyzed cases, the values of Δ(𝜔) on the complex plane form characteristic spirals, whose radii decrease with increasing temperature. The maximum value of the wave function renormalization factor on the imaginary axis relatively precisely determines the ratio of the effective mass of the electron (m⋆ e ) to the electron band mass (me ). However, the exact value of m⋆ e ∕me should be determined on the basis of the following formula:
Fig. 1. (a) The order parameter as a function of the temperature obtained in the framework of the Eliashberg equations on the imaginary axis (lines). Symbols represent the values of the order parameter obtained on the basis of Eq. (1). (b)–(c) Renormalized density of states for the selected values of temperature.
ory [18]. Additionally, 1100 Matsubara frequencies were taken into account, and the value of Coulomb pseudo-potential (𝜇 ⋆ ) equals 0.1. was adopted. The stability of the Eliashberg equations’ solutions was obtained for a temperature higher or equal to T0 = 20 K. In Fig. 1 (a) we have plotted a temperature dependence of the order parameter. It was obtained as part of the imaginary axis formalism and using the equation:
Δ(T ) = Re[Δ(𝜔 = Δ(T ))],
(1)
m⋆ e = Z (𝜔 = 0) . me
where function of the order parameter on the real axis (Δ (𝜔)) was obtained using the method of the analytical extension, which has been described in detail in the paper [27]. We have obtained the following critical temperature values: TCBiH = 103 K and TCBiH = 100 K. It 5
Graphs of the ratio of the electron effective mass to electron band mass in the temperature range from 0 to TC , were collected in the drawing Fig. 3. It can be seen that in both cases the effective mass takes values slightly higher than two. This result proves a strong renormalization of the electron band mass by the electron-phonon interaction. Additionally, the ratio m⋆ e ∕me is slightly dependent on the temperature, and its value at the critical temperature can be calculated with good [ ] accuracy using the formula: m⋆ e ∕me T =TC = 1 + 𝜆 [30]. Let us note [ ] that for T = TC the numerical calculations give m⋆ e ∕me BiH5 = 2.3 and ] [ ⋆ me ∕me BiH = 2.33, which is in a good agreement with the analytical 6 result. Basing on the solutions of the Eliashberg equations on the imaginary axis (Δn and Zn ) we have calculated the free energy difference between the superconducting state and the normal state:
6
means that BiH5 and BiH6 compounds should belong to the family of high-temperature superconductors. Let us consider the order parameter Δ(T0 ) = Δ (0) estimated on the basis of Eq. (1). Hence the dimensionless ratio RΔ = 2Δ (0) ∕kB TC equals 4.17 and 4.2, for BiH5 and BiH6 . respectively. Compared with the predictions of the BCS theory, these are very high values, because RΔBCS = 3.53 [28,29]. Please note that the above result comes from the existence of the significant retardation and strong-coupling effects. They can be characterized by the parameter ]r = kB TC ∕𝜔ln , where [
the quantity 𝜔ln = exp 2𝜆 ∫0 max dΩ 𝛼 FΩ(Ω) ln (Ω) denotes the logarithmic phonon frequency. For BiH5 and BiH6 compounds it was obtained r = 0.11. In the BCS limit we have r = 0 [30]. Basing on the function of the order parameter on the real axis, one Ω
(3)
2
Fig. 2. (a)–(f) The real and imaginary parts of the order parameter on the real axis for the selected temperature values. (g) and (h) The values of the order parameter on the complex plane. The curves correspond to the frequency range from 0 to 𝜔D . 28
M.W. Jarosik et al.
Solid State Communications 279 (2018) 27–29
wherein the specific heat of the normal state can be determined on the basis of the formula: CN (T )∕kB 𝜌(0) = 𝛾 kB T, where 𝛾 = 23 𝜋 2 (1 + 𝜆) denotes the Sommerfeld constant. Obtained results were collected in the drawing Fig. 4. It can be seen that the calculated thermodynamic functions assume similar values for BiH5 and BiH6 , compounds, which results from similar values of the electron-phonon coupling constants. Additionally, it should be emphasized that the values of the thermodynamic critical field and specific heat cannot be correctly calculated within the framework of the meanfield BCS theory. To find out about this, let us estimate the dimensionless parameters RC = ΔC (TC ) ∕CN (TC ) and RH = TC CN (TC ) ∕HC2 (0). For BiH5 compound we have received RC = 2.54 i RH = 0.146, while for BiH6 compound we have RC = 2.58 and RH = 0.146. Recall that the BCS theory gives accordingly 1.43 and 0.168 [28,29]. In summary, in the presented work we have determined the values of the thermodynamic parameters of the superconducting state in BiH5 and BiH6 compounds. The systems were located under the pressure at 200 GPa. We have shown that for the Coulomb pseudopotential’s value of 0.1 the superconducting state with a very high critical temperature of the order of 100 K is induced in both studied compounds. Additionally, within the framework of Eliashberg formalism, we have determined the values of: the energy gap, the electron effective mass, the thermodynamic critical field and the specific heat of the superconducting state. Performed calculations prove that the thermodynamic parameters of the superconducting state in BiH5 and BiH6 compounds cannot be calculated correctly in the framework of the mean-field BCS theory.
Fig. 3. The ratio of the electron effective mass to the electron band mass determined for BiH5 and BiH6 compounds as a part of the Eliashberg formalism of the imaginary axis (lines) and with the aid of formula (3) (stars).
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
Fig. 4. (a) The free energy as a function of the temperature (lower panel) and the thermodynamic critical field (upper panel). (b) The specific heat of the superconducting state and the normal state as a function of the temperature.
∑ ΔF = −2𝜋 kB T 𝜌 (0) n=1 M
(√
)
𝜔2n + Δ2n − ||𝜔n ||
|𝜔 | × (ZnS − ZnN √ | n | ), 𝜔2n + Δ2n
[18] [19] [20]
(4)
[21] [22] [23]
where 𝜌 (0) denotes the value of the electron density of states at the Fermi level, 𝜔n is the fermion Matsubara frequency, and symbols S and N refer respectively to the superconducting and normal (metallic) state. On this basis, the thermodynamic critical field can be estimated: H √ C = 𝜌 (0 )
√
[
−8𝜋 ΔF ∕𝜌 (0)
]
[24] [25] [26]
(5)
[27] [28] [29] [30]
and the difference in specific heat between the superconducting state and the normal state (ΔC = CS − CN ): [ ] 2 ΔC (T ) 1 d ΔF ∕𝜌 (0) , (6) =− 2 kB 𝜌 (0) 𝛽 d( k B T )
29
E. Wigner, H.B. Huntington, J. Chem. Phys. 3 (1935) 764. S.T. Weir, A.C. Mitchell, W.J. Nellis, Phys. Rev. Lett. 76 (1996) 1860. R.P. Dias, I.F. Silvera, Science 355 (2017) 715. N.W. Ashcroft, Phys. Rev. Lett. 21 (1968) 1748. M. Stadele, R.M. Martin, Phys. Rev. Lett. 84 (2000) 6070. P. Cudazzo, G. Profeta, A. Sanna, A. Floris, A. Continenza, S. Massidda, E.K.U. Gross, Phys. Rev. Lett. 100 (2008) 257001. R. Szcze¸s´ niak, E.A. Drzazga, Solid State Sci. 19 (2013) 167. Y. Yan, J. Gong, Y. Liu, Phys. Lett. A 375 (2011) 1264. R. Szcze¸s´niak, D. Szcze¸s´niak, E.A. Drzazga, Solid State Commun. 152 (2012) 2023. E.G. Maksimov, D.Y. Savrasov, Solid State Commun. 119 (2001) 569. R. Szcze¸s´niak, M.W. Jarosik, Solid State Commun. 149 (2009) 2053. R. Szcze¸s´ niak, A.M. Duda, E.A. Drzazga, Physica C 501 (2014) 7. J.M. McMahon, D.M. Ceperley, Phys. Rev. B 84 (2011) 144515. J.M. McMahon, D.M. Ceperley, Phys. Rev. Lett. 106 (2011) 165302. N.W. Ashcroft, Phys. Rev. Lett. 92 (2004) 187002. A. P. Drozdov, M. I. Eremets, and I. A. Troyan, arXiv: 1412.0460 (2014). A.P. Drozdov, M.I. Eremets, I.A. Troyan, V. Ksenofontov, S.I. Shylin, Nature 525 (2015) 73. Y. Ma, D. Duan, D. Li, Y. Liu, F. Tian, H. Yu, C. Xu, Z. Shao, B. Liu, and T. Cui, arXiv:1511.05291 (2015). G.M. Eliashberg, Sov. Phys. - JETP 11 (1960) 696. D. Szcze¸s´niak, A.P. Durajski, R. Szcze¸s´niak, J. Phys. Condensed Mattter 26 (2014) 255701. D. Szcze¸s´ niak, T.P. Zemła, Supercond. Sci. Technol. 28 (2015) 085018. R. Szcze¸s´niak, A.P. Durajski, D. Szcze¸s´niak, Phys. Status Solidi B 253 (2016) 538. ´ A.M. Duda, K.A. Szewczyk, M.W. Jarosik, K.M. Szcze¸s´niak, M.A. Sowinska, D. Szcze¸s´niak, Characterization of the superconducting state in hafnium hydride under high pressure, Phys. B Condens. Matter (2017), https://doi.org/10.1016/j. physb.2017.10.107. A.P. Durajski, R. Szcze¸s´niak, A.M. Duda, Solid State Commun. 195 (2014) 55. M.W. Jarosik, I.A. Wrona, A.M. Duda, Solid State Commun. 219 (2015) 1. R. Szcze¸s´niak, M.W. Jarosik, A.M. Duda, Adv. Condens. Matter Phys. 2015 (2015) 969564. K.S.D. Beach, R.J. Gooding, F. Marsiglio, Phys. Rev. B 61 (2000) 5147. J. Bardeen, L.N. Cooper, J.R. Schrieffer, Phys. Rev. 106 (1957) 162. J. Bardeen, L.N. Cooper, J.R. Schrieffer, Phys. Rev. 108 (1957) 1175. J.P. Carbotte, F. Marsiglio, in: K.H. Bennemann, J.B. Ketterson (Eds.), The Physics of Superconductors, Springer Berlin Heidelberg, 2003.
Contents lists available at ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Communication
Description of the thermodynamic properties of BiH5 and BiH6 superconductors beyond the mean-field approximation M.W. Jarosik a , E.A. Drzazga a,* , I.A. Domagalska a , K.M. Szcze¸s´niak b , U. Ste¸pien´ a a b
Institute of Physics, Czestochowa University of Technology, Ave. Armii Krajowej 19, 42-200 Czestochowa, Poland Ul. Pomorska 37/55, Zawiercie 42-400, Poland
A R T I C L E
I N F O
Communicated by Y.E. Lozovik Keywords: Hydrogen-rich materials High-pressure effect Electron-phonon superconductivity Eliashberg theory Thermodynamic properties
A B S T R A C T
The ab initio calculations suggest (Y. Ma et al.), that the high-pressure (p = 200 GPa) superconducting state in BiH5 and BiH6 compounds characterizes with a high value of the critical temperature (TC ∼ 100 K). Due to the large value of the electron-phonon coupling constant (𝜆 ∼ 1.2), the thermodynamic parameters of the superconducting phase in BiH5 and BiH6 have been determined beyond the mean-field approximation - in the framework of the Eliashberg equations formalism. We have calculated the dependence of the order parameter (Δ) and the wave function renormalization factor on the temperature. Then we have estimated the free energy difference between the superconducting state and the normal state, the thermodynamic critical field (HC ) and the specific heat of the superconducting state (CS ) and the normal state (CN ). The values of the dimensionless ratios RΔ = 2Δ (0) ∕kB TC , RC = ΔC (TC ) ∕CN (TC ) and RH = TC CN (TC ) ∕HC2 (0) are equal RΔBiH = 4.17 and RΔBiH = 4.20, 5 6 RCBiH = 2.54 and RCBiH = 2.58, RHBiH = 0.146 and RHBiH = 0.146 respectively. 5
6
In 1935 Wigner and Huntington noted that hydrogen affected by the high pressure (∼25 GPa) should transit into a metallic state [1]. So far, it has been possible to experimentally demonstrate the metallization of liquid hydrogen for the pressure at 140 GPa, but the temperature was up to 3000 K [2]. In 2017 the information was spread that hydrogen metallization was observed at room temperature under the pressure at 495 GPa [3]. Unfortunately, this result has not been confirmed again. In 1968 Ashcroft noted that the metallic hydrogen should be a hightemperature superconductor [4]. He gave the following arguments: first of all, hydrogen has the lowest atomic mass of the nucleus, which implies the Debye frequency (𝜔D ) of the crystal lattice of solidified hydrogen to be high; secondly, there are no electrons on the internal shells in the hydrogen atom, which favors a strong electron-phonon coupling. In the following years, thanks to the rapid development of computer technology, more detailed calculations were carried out. The hydrogen metallization pressure was estimated to be around 400 GPa [4,5]. It has been suggested that in the molecular hydrogen phase, within the range of pressures from 400 to 500 GPa there should be a sharp rise in the critical temperature from approximately 80 K to about 350 K [6–9]. Above the pressure at 500 GPa the numerical calculations suggest that the metallic molecular hydrogen should dissociate into the atomic phase [6], whereas the critical temperature value
* Corresponding author. E-mail address: [email protected] (E.A. Drzazga). https://doi.org/10.1016/j.ssc.2018.05.009 Received 1 March 2018; Accepted 21 May 2018 Available online 25 May 2018 0038-1098/© 2018 Elsevier Ltd. All rights reserved.
5
6
should be within the limit from 300 K to 470 K [10]. In the literature on the subject there are also works analyzing the superconducting state of hydrogen for extremely high pressures (up to 3.5 TPa) [10–14]. The highest value of TC was foreseen at 630 K for p = 2 TPa. Due to the very high value of the hydrogen’s metallization pressure, in 2004 Ashcroft suggested to try to look for a high temperature superconducting state not in pure hydrogen but in hydrogenated compounds [15]. Induction of high temperature superconducting state in hydrogenated compounds should be supported by chemical precompression caused by heavier elements. The real breakthrough occurred only in December 2014, when Drozdov, Eremets, and Troyan have experimentally demonstrated the existence of a high-temperature superconducting state in H3 S (TC = 203 K for p = 155 GPa) [16,17]. In the presented work, we have determined the thermodynamic properties of the superconducting state in hydrogenated compounds BiH5 and BiH6 . Due to the high predicted value of the electronphonon coupling constant (𝜆BiH5 = 1.236 and 𝜆BiH6 = 1.259 [18]) we performed the calculations for both cases as a part of Eliashberg formalism [19]. A detailed description of Eliashberg equations and the computational methods used has been presented in the works [20–26]. The spectral function (𝛼 2 F (Ω)) used in numerical calculations has been determined by Y. Ma et al. with an use of the density functional the-
M.W. Jarosik et al.
Solid State Communications 279 (2018) 27–29
can determine the renormalized electronic density of states in the superconducting phase: ] [ NS (𝜔) (𝜔 − iΓ) , (2) = Re √ NN (𝜔) (𝜔 − iΓ)2 − (Δ(𝜔))2 where we have assumed that the pair breaking parameter equals
Γ = 0.15 meV. We have plotted the obtained results in the drawings Fig. 1(b) and (c). Note that the characteristic maxima of the determined curves are formed in points 𝜔 = ± Δ. On their basis, one can experimentally analyze the evolution of the energy gap depending on the temperature. The exact form of the order parameter on the real axis is presented in the figures Fig. 2(a)–(f). When analyzing the case T = T0 , it can be seen that in the range of the lower frequencies (𝜔 < 𝜔d = 85 meV) non-zero is only the real part of the order parameter. This means no damping processes that are modeled by a function Im [Δ (𝜔)]. It is also worth noting that as the temperature rises, 𝜔d decreases significantly. Additionally, for illustrative purposes in drawings (g) and (h) we have plotted the values of the order parameter on the complex plane. In both analyzed cases, the values of Δ(𝜔) on the complex plane form characteristic spirals, whose radii decrease with increasing temperature. The maximum value of the wave function renormalization factor on the imaginary axis relatively precisely determines the ratio of the effective mass of the electron (m⋆ e ) to the electron band mass (me ). However, the exact value of m⋆ e ∕me should be determined on the basis of the following formula:
Fig. 1. (a) The order parameter as a function of the temperature obtained in the framework of the Eliashberg equations on the imaginary axis (lines). Symbols represent the values of the order parameter obtained on the basis of Eq. (1). (b)–(c) Renormalized density of states for the selected values of temperature.
ory [18]. Additionally, 1100 Matsubara frequencies were taken into account, and the value of Coulomb pseudo-potential (𝜇 ⋆ ) equals 0.1. was adopted. The stability of the Eliashberg equations’ solutions was obtained for a temperature higher or equal to T0 = 20 K. In Fig. 1 (a) we have plotted a temperature dependence of the order parameter. It was obtained as part of the imaginary axis formalism and using the equation:
Δ(T ) = Re[Δ(𝜔 = Δ(T ))],
(1)
m⋆ e = Z (𝜔 = 0) . me
where function of the order parameter on the real axis (Δ (𝜔)) was obtained using the method of the analytical extension, which has been described in detail in the paper [27]. We have obtained the following critical temperature values: TCBiH = 103 K and TCBiH = 100 K. It 5
Graphs of the ratio of the electron effective mass to electron band mass in the temperature range from 0 to TC , were collected in the drawing Fig. 3. It can be seen that in both cases the effective mass takes values slightly higher than two. This result proves a strong renormalization of the electron band mass by the electron-phonon interaction. Additionally, the ratio m⋆ e ∕me is slightly dependent on the temperature, and its value at the critical temperature can be calculated with good [ ] accuracy using the formula: m⋆ e ∕me T =TC = 1 + 𝜆 [30]. Let us note [ ] that for T = TC the numerical calculations give m⋆ e ∕me BiH5 = 2.3 and ] [ ⋆ me ∕me BiH = 2.33, which is in a good agreement with the analytical 6 result. Basing on the solutions of the Eliashberg equations on the imaginary axis (Δn and Zn ) we have calculated the free energy difference between the superconducting state and the normal state:
6
means that BiH5 and BiH6 compounds should belong to the family of high-temperature superconductors. Let us consider the order parameter Δ(T0 ) = Δ (0) estimated on the basis of Eq. (1). Hence the dimensionless ratio RΔ = 2Δ (0) ∕kB TC equals 4.17 and 4.2, for BiH5 and BiH6 . respectively. Compared with the predictions of the BCS theory, these are very high values, because RΔBCS = 3.53 [28,29]. Please note that the above result comes from the existence of the significant retardation and strong-coupling effects. They can be characterized by the parameter ]r = kB TC ∕𝜔ln , where [
the quantity 𝜔ln = exp 2𝜆 ∫0 max dΩ 𝛼 FΩ(Ω) ln (Ω) denotes the logarithmic phonon frequency. For BiH5 and BiH6 compounds it was obtained r = 0.11. In the BCS limit we have r = 0 [30]. Basing on the function of the order parameter on the real axis, one Ω
(3)
2
Fig. 2. (a)–(f) The real and imaginary parts of the order parameter on the real axis for the selected temperature values. (g) and (h) The values of the order parameter on the complex plane. The curves correspond to the frequency range from 0 to 𝜔D . 28
M.W. Jarosik et al.
Solid State Communications 279 (2018) 27–29
wherein the specific heat of the normal state can be determined on the basis of the formula: CN (T )∕kB 𝜌(0) = 𝛾 kB T, where 𝛾 = 23 𝜋 2 (1 + 𝜆) denotes the Sommerfeld constant. Obtained results were collected in the drawing Fig. 4. It can be seen that the calculated thermodynamic functions assume similar values for BiH5 and BiH6 , compounds, which results from similar values of the electron-phonon coupling constants. Additionally, it should be emphasized that the values of the thermodynamic critical field and specific heat cannot be correctly calculated within the framework of the meanfield BCS theory. To find out about this, let us estimate the dimensionless parameters RC = ΔC (TC ) ∕CN (TC ) and RH = TC CN (TC ) ∕HC2 (0). For BiH5 compound we have received RC = 2.54 i RH = 0.146, while for BiH6 compound we have RC = 2.58 and RH = 0.146. Recall that the BCS theory gives accordingly 1.43 and 0.168 [28,29]. In summary, in the presented work we have determined the values of the thermodynamic parameters of the superconducting state in BiH5 and BiH6 compounds. The systems were located under the pressure at 200 GPa. We have shown that for the Coulomb pseudopotential’s value of 0.1 the superconducting state with a very high critical temperature of the order of 100 K is induced in both studied compounds. Additionally, within the framework of Eliashberg formalism, we have determined the values of: the energy gap, the electron effective mass, the thermodynamic critical field and the specific heat of the superconducting state. Performed calculations prove that the thermodynamic parameters of the superconducting state in BiH5 and BiH6 compounds cannot be calculated correctly in the framework of the mean-field BCS theory.
Fig. 3. The ratio of the electron effective mass to the electron band mass determined for BiH5 and BiH6 compounds as a part of the Eliashberg formalism of the imaginary axis (lines) and with the aid of formula (3) (stars).
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
Fig. 4. (a) The free energy as a function of the temperature (lower panel) and the thermodynamic critical field (upper panel). (b) The specific heat of the superconducting state and the normal state as a function of the temperature.
∑ ΔF = −2𝜋 kB T 𝜌 (0) n=1 M
(√
)
𝜔2n + Δ2n − ||𝜔n ||
|𝜔 | × (ZnS − ZnN √ | n | ), 𝜔2n + Δ2n
[18] [19] [20]
(4)
[21] [22] [23]
where 𝜌 (0) denotes the value of the electron density of states at the Fermi level, 𝜔n is the fermion Matsubara frequency, and symbols S and N refer respectively to the superconducting and normal (metallic) state. On this basis, the thermodynamic critical field can be estimated: H √ C = 𝜌 (0 )
√
[
−8𝜋 ΔF ∕𝜌 (0)
]
[24] [25] [26]
(5)
[27] [28] [29] [30]
and the difference in specific heat between the superconducting state and the normal state (ΔC = CS − CN ): [ ] 2 ΔC (T ) 1 d ΔF ∕𝜌 (0) , (6) =− 2 kB 𝜌 (0) 𝛽 d( k B T )
29
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