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The high-pressure superconductivity in SiH4: The strong-coupling approach
Solid State Communications 172 (2013) 5–9
Contents lists available at ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
The high-pressure superconductivity in SiH4: The strongcoupling approach R. Szczęśniak, A.P. Durajski n Institute of Physics, Częstochowa University of Technology, Ave. Armii Krajowej 19, 42-200 Częstochowa, Poland
art ic l e i nf o
a b s t r a c t
Article history: Received 20 May 2013 Received in revised form 5 July 2013 Accepted 6 August 2013 by F. Peeters Available online 17 August 2013
In the paper, the thermodynamic parameters of the high-pressure superconducting state in the SiH4 compound have been determined ðp ¼ 250 GPaÞ. By using the Eliashberg equations in the mixed representation, the critical temperature, the energy gap, and the electron effective mass have been calculated. It has been stated that the critical temperature (TC) decreases from 51.65 K to 20.62 K, if the Coulomb pseudopotential increases ðμ⋆ A 〈0:1; 0:3〉Þ. The dimensionless ratio 2Δð0Þ=kB T C decreases from 4.10 to 3.84, where the symbol Δð0Þ denotes the value of the order parameter close to the zero temperature. The ratio of the electron effective mass to the band electron mass is high, and it reaches maximum equal to 1.95 for the critical temperature. & 2013 Elsevier Ltd. All rights reserved.
Keywords: A. Superconductivity A. Hydrogen-rich compounds D. Thermodynamic properties D. High-pressure effects.
The metallic state in the hydrogen is intensely studied since 1935 [1]. It has been shown that the hydrogen is metallic under the high compression ð 400 GPaÞ [2]. At the low value of the temperature, the metallic state of the hydrogen transforms into the superconducting state [3]. In the pressure (p) range from 400 GPa to 500 GPa, the value of the critical temperature ðT C Þ reaches maximum equal to 242 K ðp ¼ 450 GPaÞ [4,5]. For the pressure's values from 500 GPa to 800 GPa, the critical temperature increases: T C A 〈282; 360〉 K [6,7]. The extremely high value of the critical temperature has been predicted near 2000 GPa, where TC can assumes even 600 K [8,9]. Recently, it has been suggested that the hydrogen-rich compounds become metallic at the pressure's value lower than for the pure hydrogen. In particular, the following compounds have been taken under consideration: methane ðCH4 Þ [10], silane ðSiH4 ; SiH4 ðH2 Þ2 Þ [11–13], disilane ðSi2 H6 Þ [14], and germane ðGeH4 ; GeH4 ðH2 Þ2 Þ [15–17]. In the case of SiH4 ðH2 Þ2 under pressure at 270 GPa and Si2H6 at p ¼250 GPa, the theoretical results have predicted the existence of the high-temperature superconducting state: TC ¼ 129.83 K and TC ¼173.36 K [18,19]. We notice that the above values of the critical temperature are even higher than for the cuprates [20]. In the presented paper, we have studied the thermodynamic properties of the superconducting state in the SiH4 compound under the pressure at 250 GPa.
We notice that the experimental data have confirmed the metallization in SiH4 ðp C 50 GPaÞ [11,12]. Furthermore, the critical temperature increases with the pressure and assumes the maximum equal to 17.5 K at 96 GPa and at 120 GPa. In the SiH4 compound ðp ¼ 250 GPaÞ, the strong electron– phonon interaction has been predicted [21]. In particular, the electron–phonon coupling constant takes the value: λ ¼ 0:91. Taking into consideration the above result one should expect the high value of the critical temperature. In the considered case, the remaining thermodynamic parameters are probably beyond the BCS predictions [22,23]. For this reason, the numerical calculations have been made in the framework of the Eliashberg approach [24]. The Eliashberg equations defined both on the real and imaginary axis (the mixed representation) can be written in the following form [25,26]:
ϕðωÞ ¼
π M ½λðωiωm Þμ⋆ ðωm Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∑ ϕm β m ¼ M ω2 Z 2 þ ϕ2
þ iπ þ iπ
Z
m m
þ1
0
Z 0
þ1
m
dω′α2 Fðω′Þ½½Nðω′Þ þ f ðω′ωÞKðω; ω′Þϕðωω′Þ dω′α2 Fðω′Þ½½Nðω′Þ þ f ðω′þ ωÞKðω; ω′Þϕðω þ ω′Þ; ð1Þ
and n
Corresponding author. E-mail address: [email protected] (A.P. Durajski).
0038-1098/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ssc.2013.08.007
Z ðωÞ ¼ 1 þ
M iπ λðωiωm Þωm ∑ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZ ωβ m ¼ M ω2 Z 2 þ ϕ2 m m m
m
6
R. Szczęśniak, A.P. Durajski / Solid State Communications 172 (2013) 5–9
iπ
Z
þ1
the equations [24]:
dω′α2 F ðω′Þ N ðω′Þ þf ðω′ωÞ ω 0 K ðω; ω′Þðωω′ÞZ ðωω′Þ þ
iπ
Z
þ1
dω′α2 F ðω′Þ N ðω′Þ þf ðω′ þ ωÞ ω 0 Kðω; ω′Þðω þ ω′ÞZðω þ ω′Þ; þ
ϕn ¼
m m
ð2Þ
where 1 K ðω; ω′Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi: 2 2 2 ðω þ ω′Þ Z ðω þ ω′Þϕ ðω þ ω′Þ
ð3Þ
The symbols ϕðωÞ and ZðωÞ denote the order parameter function and the wave function renormalization factor on the real axis ðωÞ, respectively; ϕm ϕðiωm Þ and Z m Zðiωm Þ represent pffiffiffiffiffiffiffi the values of these functions on the imaginary axis ði 1Þ. The Matsubara frequency is given by ωm ðπ =βÞð2m1Þ, where β ðkB TÞ1 ðkB is the Boltzmann constant). Both on the real and imaginary axis, the order parameter is defined as Δ ϕ=Z. The electron–phonon pairing kernel has the form:
λðzÞ 2
Z Ωmax 0
dΩ
Ω
Ω
2
z2
α2 F Ω ;
π M λðiωn iωm Þμ⋆ ðωm Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∑ ϕm ; β m ¼ M ω2 Z 2 þ ϕ 2
ð4Þ
where the Eliashberg function ðα2 FðΩÞÞ has been calculated in the paper [21]; the maximum phonon frequency ðΩmax Þ is equal to 426.1 meV. The function μ⋆ ðωm Þ μ⋆ θðωc jωm jÞ describes the electron depairing interaction; μ⋆ is the Coulomb pseudopotential. Due to the absence of the experimental value of the critical temperature, the Coulomb pseudopotential is unknown. For this reason, we have assumed: μ⋆ A 〈0:1; 0:3〉. The symbol θ denotes the Heaviside unit function and ωc is the cut-off frequency ðωc ¼ 3Ωmax Þ. The quantities NðωÞ and fðωÞ denote the Bose–Einstein and Fermi–Dirac functions, respectively. The order parameter function and the wave function renormalization factor on the imaginary axis have been calculated by using
Zn ¼ 1 þ
1π
M
∑
ωn β m ¼ M
ð5Þ
m
λðiωn iωm Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ωm Z m : ω2m Z 2m þ ϕ2m
ð6Þ
The Eliashberg set has been solved for M ¼1100. We have used the numerical methods presented in the papers [27] and [28]. In the considered case, the solutions of the Eliashberg equations are stable for T Z T 0 ¼ 4:64 K. In Fig. 1, the form of the order parameter on the real axis has been presented for the selected values of the temperature and μ⋆ ¼ 0:1. Moreover, the rescaled Eliashberg function ð10α2 FðΩÞÞ has been also plotted. It is easy to see that for the low frequencies, the non-zero values are taken only by the real part of the order parameter. The obtained result indicates that in the considered range of frequencies the damping effects related with Im½ΔðωÞ do not exist. Additionally, it has been stated that the shapes of the functions Re½ΔðωÞ and Im½ΔðωÞ are correlated with the complicated form of the Eliashberg function. The physical value of the order parameter has been calculated by using the equation:
ΔðTÞ ¼ Re½Δðω ¼ ΔðTÞÞ:
ð7Þ
In Fig. 2(A), the dependence of the order parameter on the temperature for the selected values of the Coulomb pseudopotential has been presented. We notice that the value ΔðT; Þ can be parameterized by using the formula: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi β T ΔðT Þ ¼ ΔðT 0 Þ 1 ; ð8Þ TC where β ¼ 3:4, and
ΔðT 0 Þ ¼ 91:93ðμ⋆ Þ2 64:73μ⋆ þ 14:47 meV:
ð9Þ
Fig. 1. (Color online) The real and imaginary part of the order parameter for selected values of the temperature and μ⋆ ¼ 0:1. In the figure, the rescaled Eliashberg function has been also plotted.
R. Szczęśniak, A.P. Durajski / Solid State Communications 172 (2013) 5–9
7
Fig. 2. (Color online) (A) The dependence of the order parameter on the temperature for selected values of the Coulomb pseudopotential. (B) The critical temperature as a function of the Coulomb pseudopotential. The circles represent the results obtained by using the Eliashberg equations. The squares and triangles are related to the classical Allen-Dynes and McMillan expression. The solid line has been achieved with the help of Eq. (10).
Fig. 3. (Color online) The dependence of the ratio RΔ on the value of the Coulomb pseudopotential. In the inset, the influence of μ⋆ on Δð0Þ has been presented.
Next, we have determined the dependence of the critical temperature on the Coulomb pseudopotential. The results have been presented in Fig. 2(B). It is easy to notice that the value of TC is high in the whole range of the considered values of μ⋆ . In particular, T C A 〈51:65; 20:62〉 K. Additionally, we underline that for large value of the Coulomb pseudopotential, the critical temperature cannot be precisely estimated by the classical Allen–Dynes or McMillan formula [29,30]. However, the modified Allen–Dynes formula derived originally for SiH4 ðH2 Þ2 compound works very well [18]:
ω 1:125ð1 þ λÞ kB T C ¼ f 1 f 2 ln exp ; ð10Þ ⋆ 1:37 λμ where f1 and f2 denote the functions [29]: pffiffiffiffiffiffiffi ω2 2 " 3=2 #1=3 1 λ λ ωln ; and f 2 1 þ : f 1 1þ Λ1 λ2 þ Λ22
Fig. 4. (Color online) The dependence of the total normalized density of states on the frequency for T ¼T0. The selected values of the Coulomb pseudopotential have been assumed.
The second moment of the normalized weight function and the logarithmic phonon frequency is given by
ω2
2
Z Ωmax
λ
0
and
ωln exp
dΩα2 F Ω Ω;
" Z 2 Ωmax
λ
0
ð12Þ
#
dΩ
α2 FðΩÞ ln Ω : Ω
ð13Þ
The electron–phonon coupling constant has the form: ð11Þ
The quantities Λ1 and Λ2 take the form: Λ1 ¼ 20:14μ⋆ , and pffiffiffiffiffiffiffi Λ2 ¼ ð0:27 þ 10μ⋆ Þð ω2 =ωln Þ.
λ2
Z Ωmax 0
α2 FðΩÞ : Ω
dΩ
ð14Þ
For SiH4 under the pressure at 250 GPa, it has been achieved: pffiffiffiffiffiffiffi ω2 ¼ 167:64 meV and ωln ¼ 72:42 meV.
8
R. Szczęśniak, A.P. Durajski / Solid State Communications 172 (2013) 5–9
Fig. 5. (Color online) The dependence of the real and imaginary part of the wave function renormalization factor on the frequency for T ¼ T0 and T ¼TC. In the figure, the rescaled Eliashberg function has been also plotted ð5α2 FðΩÞÞ. The results have been obtained for μ⋆ ¼ 0:1.
Now, we consider the low-temperature value of the order parameter at the Fermi level ðΔð0Þ ΔðT 0 ÞÞ. In particular, we have Δð0Þ A 〈4:56; 1:71〉 meV for μ⋆ A 〈0:1; 0:3〉. The full dependence of Δð0Þ on the Coulomb pseudopotential has been shown in the inset in Fig. 3. Taking into consideration the obtained results, we have calculated the dimensionless ratio RΔ 2Δð0Þ=kB T C . It has been achieved: RΔ A 〈4:10; 3:84〉 (see also Fig. 3). We notice that the values of RΔ for SiH4 compound differ significantly from the prediction of the BCS model, where RΔ ¼ 3:53 [22,23]. The order parameter function on the real axis allows to calculate the total normalized density of states [24]: 2 3 N S ð ωÞ jωiΓ j 6 7 ¼ Re4qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5; N N ð ωÞ 2 ðωiΓ Þ2 Δ ðωÞ
ð15Þ
where the symbols NS and NN denote the density in the superconducting and normal states, respectively. The pair breaking parameter Γ is equal to 0.15 meV. The plot of the total normalized density of states has been presented in Fig. 4. We have assumed T¼ T0 and the selected values of μ⋆ . It is easy to see the reduction of the function N S ðωÞ=N N ðωÞ by the depairing electron correlations. In Fig. 5, the real and imaginary part of the wave function renormalization factor on the real axis has been presented (T ¼ T 0 and T ¼ T C ). On the basis of the obtained results, it has been stated that ZðωÞ weakly depends on the temperature in comparison with the order parameter. We can also observe the weak correlation between the wave function renormalization factor and the shape of the Eliashberg function. The dependence of the electron effective mass ðm⋆ e Þ on the temperature has been determined by using the expression: m⋆ e ¼ Re½Zð0Þme , where me denotes the band electron mass. In our case, the results prove that the electron effective mass takes the high value in the whole range of the superconducting phase's existence. The maximum of m⋆ e has been observed for T ¼TC, where m⋆ e ¼ 1:95me . We notice that the Coulomb pseudopotential does not influence on the value of ½m⋆ e max . In the paper, the Eliashberg equations for the SiH4 compound under the pressure at 250 GPa have been solved. We have shown that the value of the critical temperature decreases from 51.65 K to 20.62 K if μ⋆ A 〈0:1; 0:3〉. The dimensionless ratio RΔ takes the values beyond the prediction of the BCS model: RΔ A 〈4:10; 3:84〉. Additionally, it has been stated that the electron effective mass takes the high values and ½m⋆ e max is equal to 1.95me for the critical temperature.
All numerical calculations have been based on the Eliashberg function sent to us by Prof. Xiao-Jia Chen for whom we are very thankful. The authors are grateful to the Czestochowa University of Technology – MSK CzestMAN for granting access to the computing infrastructure built in the project No. POIG.02.03.00-00-028/08 “PLATON – Science Services Platform”. References [1] [2] [3] [4]
[5] [6] [7] [8] [9] [10] [11]
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[21] [22] [23] [24]
Acknowledgments The authors wish to thank Prof. K. Dziliński for providing excellent working conditions and the financial support.
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[27] R. Szczęśniak, M.W. Jarosik, D. Szczęśniak, Physica B 405 (2010) 4897–4902; M.W. Jarosik, R. Szczęśniak, D. Szczęśniak, Acta Physica Polonica A 118 (2010) 1031–1033.R. Szczęśniak, D. Szczęśniak, Solid State Communications 152 (2012) 779–783; R. Szczęśniak, D. Szczęśniak, Physica Status Solidi B 249 (2012) 2194–2201. [28] A.P. Durajski, R. Szczęśniak, M.W. Jarosik, Phase Transitions 85 (2012) 727–734;
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R. Szczęśniak, A.P. Durajski, M.W. Jarosik, Modern Physics Letters B 26 (2012) 1250050-1–1250050-7; R. Szczęśniak, A.P. Durajski, P.W. Pach, Journal of Low Temperature Physics 171 (2013) 769–778. [29] P.B. Allen, R.C. Dynes, Physical Review B 12 (1975) 905–922. [30] W.L. McMillan, Physical Review 167 (1968) 331–344.
Contents lists available at ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
The high-pressure superconductivity in SiH4: The strongcoupling approach R. Szczęśniak, A.P. Durajski n Institute of Physics, Częstochowa University of Technology, Ave. Armii Krajowej 19, 42-200 Częstochowa, Poland
art ic l e i nf o
a b s t r a c t
Article history: Received 20 May 2013 Received in revised form 5 July 2013 Accepted 6 August 2013 by F. Peeters Available online 17 August 2013
In the paper, the thermodynamic parameters of the high-pressure superconducting state in the SiH4 compound have been determined ðp ¼ 250 GPaÞ. By using the Eliashberg equations in the mixed representation, the critical temperature, the energy gap, and the electron effective mass have been calculated. It has been stated that the critical temperature (TC) decreases from 51.65 K to 20.62 K, if the Coulomb pseudopotential increases ðμ⋆ A 〈0:1; 0:3〉Þ. The dimensionless ratio 2Δð0Þ=kB T C decreases from 4.10 to 3.84, where the symbol Δð0Þ denotes the value of the order parameter close to the zero temperature. The ratio of the electron effective mass to the band electron mass is high, and it reaches maximum equal to 1.95 for the critical temperature. & 2013 Elsevier Ltd. All rights reserved.
Keywords: A. Superconductivity A. Hydrogen-rich compounds D. Thermodynamic properties D. High-pressure effects.
The metallic state in the hydrogen is intensely studied since 1935 [1]. It has been shown that the hydrogen is metallic under the high compression ð 400 GPaÞ [2]. At the low value of the temperature, the metallic state of the hydrogen transforms into the superconducting state [3]. In the pressure (p) range from 400 GPa to 500 GPa, the value of the critical temperature ðT C Þ reaches maximum equal to 242 K ðp ¼ 450 GPaÞ [4,5]. For the pressure's values from 500 GPa to 800 GPa, the critical temperature increases: T C A 〈282; 360〉 K [6,7]. The extremely high value of the critical temperature has been predicted near 2000 GPa, where TC can assumes even 600 K [8,9]. Recently, it has been suggested that the hydrogen-rich compounds become metallic at the pressure's value lower than for the pure hydrogen. In particular, the following compounds have been taken under consideration: methane ðCH4 Þ [10], silane ðSiH4 ; SiH4 ðH2 Þ2 Þ [11–13], disilane ðSi2 H6 Þ [14], and germane ðGeH4 ; GeH4 ðH2 Þ2 Þ [15–17]. In the case of SiH4 ðH2 Þ2 under pressure at 270 GPa and Si2H6 at p ¼250 GPa, the theoretical results have predicted the existence of the high-temperature superconducting state: TC ¼ 129.83 K and TC ¼173.36 K [18,19]. We notice that the above values of the critical temperature are even higher than for the cuprates [20]. In the presented paper, we have studied the thermodynamic properties of the superconducting state in the SiH4 compound under the pressure at 250 GPa.
We notice that the experimental data have confirmed the metallization in SiH4 ðp C 50 GPaÞ [11,12]. Furthermore, the critical temperature increases with the pressure and assumes the maximum equal to 17.5 K at 96 GPa and at 120 GPa. In the SiH4 compound ðp ¼ 250 GPaÞ, the strong electron– phonon interaction has been predicted [21]. In particular, the electron–phonon coupling constant takes the value: λ ¼ 0:91. Taking into consideration the above result one should expect the high value of the critical temperature. In the considered case, the remaining thermodynamic parameters are probably beyond the BCS predictions [22,23]. For this reason, the numerical calculations have been made in the framework of the Eliashberg approach [24]. The Eliashberg equations defined both on the real and imaginary axis (the mixed representation) can be written in the following form [25,26]:
ϕðωÞ ¼
π M ½λðωiωm Þμ⋆ ðωm Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∑ ϕm β m ¼ M ω2 Z 2 þ ϕ2
þ iπ þ iπ
Z
m m
þ1
0
Z 0
þ1
m
dω′α2 Fðω′Þ½½Nðω′Þ þ f ðω′ωÞKðω; ω′Þϕðωω′Þ dω′α2 Fðω′Þ½½Nðω′Þ þ f ðω′þ ωÞKðω; ω′Þϕðω þ ω′Þ; ð1Þ
and n
Corresponding author. E-mail address: [email protected] (A.P. Durajski).
0038-1098/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ssc.2013.08.007
Z ðωÞ ¼ 1 þ
M iπ λðωiωm Þωm ∑ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZ ωβ m ¼ M ω2 Z 2 þ ϕ2 m m m
m
6
R. Szczęśniak, A.P. Durajski / Solid State Communications 172 (2013) 5–9
iπ
Z
þ1
the equations [24]:
dω′α2 F ðω′Þ N ðω′Þ þf ðω′ωÞ ω 0 K ðω; ω′Þðωω′ÞZ ðωω′Þ þ
iπ
Z
þ1
dω′α2 F ðω′Þ N ðω′Þ þf ðω′ þ ωÞ ω 0 Kðω; ω′Þðω þ ω′ÞZðω þ ω′Þ; þ
ϕn ¼
m m
ð2Þ
where 1 K ðω; ω′Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi: 2 2 2 ðω þ ω′Þ Z ðω þ ω′Þϕ ðω þ ω′Þ
ð3Þ
The symbols ϕðωÞ and ZðωÞ denote the order parameter function and the wave function renormalization factor on the real axis ðωÞ, respectively; ϕm ϕðiωm Þ and Z m Zðiωm Þ represent pffiffiffiffiffiffiffi the values of these functions on the imaginary axis ði 1Þ. The Matsubara frequency is given by ωm ðπ =βÞð2m1Þ, where β ðkB TÞ1 ðkB is the Boltzmann constant). Both on the real and imaginary axis, the order parameter is defined as Δ ϕ=Z. The electron–phonon pairing kernel has the form:
λðzÞ 2
Z Ωmax 0
dΩ
Ω
Ω
2
z2
α2 F Ω ;
π M λðiωn iωm Þμ⋆ ðωm Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∑ ϕm ; β m ¼ M ω2 Z 2 þ ϕ 2
ð4Þ
where the Eliashberg function ðα2 FðΩÞÞ has been calculated in the paper [21]; the maximum phonon frequency ðΩmax Þ is equal to 426.1 meV. The function μ⋆ ðωm Þ μ⋆ θðωc jωm jÞ describes the electron depairing interaction; μ⋆ is the Coulomb pseudopotential. Due to the absence of the experimental value of the critical temperature, the Coulomb pseudopotential is unknown. For this reason, we have assumed: μ⋆ A 〈0:1; 0:3〉. The symbol θ denotes the Heaviside unit function and ωc is the cut-off frequency ðωc ¼ 3Ωmax Þ. The quantities NðωÞ and fðωÞ denote the Bose–Einstein and Fermi–Dirac functions, respectively. The order parameter function and the wave function renormalization factor on the imaginary axis have been calculated by using
Zn ¼ 1 þ
1π
M
∑
ωn β m ¼ M
ð5Þ
m
λðiωn iωm Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ωm Z m : ω2m Z 2m þ ϕ2m
ð6Þ
The Eliashberg set has been solved for M ¼1100. We have used the numerical methods presented in the papers [27] and [28]. In the considered case, the solutions of the Eliashberg equations are stable for T Z T 0 ¼ 4:64 K. In Fig. 1, the form of the order parameter on the real axis has been presented for the selected values of the temperature and μ⋆ ¼ 0:1. Moreover, the rescaled Eliashberg function ð10α2 FðΩÞÞ has been also plotted. It is easy to see that for the low frequencies, the non-zero values are taken only by the real part of the order parameter. The obtained result indicates that in the considered range of frequencies the damping effects related with Im½ΔðωÞ do not exist. Additionally, it has been stated that the shapes of the functions Re½ΔðωÞ and Im½ΔðωÞ are correlated with the complicated form of the Eliashberg function. The physical value of the order parameter has been calculated by using the equation:
ΔðTÞ ¼ Re½Δðω ¼ ΔðTÞÞ:
ð7Þ
In Fig. 2(A), the dependence of the order parameter on the temperature for the selected values of the Coulomb pseudopotential has been presented. We notice that the value ΔðT; Þ can be parameterized by using the formula: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi β T ΔðT Þ ¼ ΔðT 0 Þ 1 ; ð8Þ TC where β ¼ 3:4, and
ΔðT 0 Þ ¼ 91:93ðμ⋆ Þ2 64:73μ⋆ þ 14:47 meV:
ð9Þ
Fig. 1. (Color online) The real and imaginary part of the order parameter for selected values of the temperature and μ⋆ ¼ 0:1. In the figure, the rescaled Eliashberg function has been also plotted.
R. Szczęśniak, A.P. Durajski / Solid State Communications 172 (2013) 5–9
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Fig. 2. (Color online) (A) The dependence of the order parameter on the temperature for selected values of the Coulomb pseudopotential. (B) The critical temperature as a function of the Coulomb pseudopotential. The circles represent the results obtained by using the Eliashberg equations. The squares and triangles are related to the classical Allen-Dynes and McMillan expression. The solid line has been achieved with the help of Eq. (10).
Fig. 3. (Color online) The dependence of the ratio RΔ on the value of the Coulomb pseudopotential. In the inset, the influence of μ⋆ on Δð0Þ has been presented.
Next, we have determined the dependence of the critical temperature on the Coulomb pseudopotential. The results have been presented in Fig. 2(B). It is easy to notice that the value of TC is high in the whole range of the considered values of μ⋆ . In particular, T C A 〈51:65; 20:62〉 K. Additionally, we underline that for large value of the Coulomb pseudopotential, the critical temperature cannot be precisely estimated by the classical Allen–Dynes or McMillan formula [29,30]. However, the modified Allen–Dynes formula derived originally for SiH4 ðH2 Þ2 compound works very well [18]:
ω 1:125ð1 þ λÞ kB T C ¼ f 1 f 2 ln exp ; ð10Þ ⋆ 1:37 λμ where f1 and f2 denote the functions [29]: pffiffiffiffiffiffiffi ω2 2 " 3=2 #1=3 1 λ λ ωln ; and f 2 1 þ : f 1 1þ Λ1 λ2 þ Λ22
Fig. 4. (Color online) The dependence of the total normalized density of states on the frequency for T ¼T0. The selected values of the Coulomb pseudopotential have been assumed.
The second moment of the normalized weight function and the logarithmic phonon frequency is given by
ω2
2
Z Ωmax
λ
0
and
ωln exp
dΩα2 F Ω Ω;
" Z 2 Ωmax
λ
0
ð12Þ
#
dΩ
α2 FðΩÞ ln Ω : Ω
ð13Þ
The electron–phonon coupling constant has the form: ð11Þ
The quantities Λ1 and Λ2 take the form: Λ1 ¼ 20:14μ⋆ , and pffiffiffiffiffiffiffi Λ2 ¼ ð0:27 þ 10μ⋆ Þð ω2 =ωln Þ.
λ2
Z Ωmax 0
α2 FðΩÞ : Ω
dΩ
ð14Þ
For SiH4 under the pressure at 250 GPa, it has been achieved: pffiffiffiffiffiffiffi ω2 ¼ 167:64 meV and ωln ¼ 72:42 meV.
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R. Szczęśniak, A.P. Durajski / Solid State Communications 172 (2013) 5–9
Fig. 5. (Color online) The dependence of the real and imaginary part of the wave function renormalization factor on the frequency for T ¼ T0 and T ¼TC. In the figure, the rescaled Eliashberg function has been also plotted ð5α2 FðΩÞÞ. The results have been obtained for μ⋆ ¼ 0:1.
Now, we consider the low-temperature value of the order parameter at the Fermi level ðΔð0Þ ΔðT 0 ÞÞ. In particular, we have Δð0Þ A 〈4:56; 1:71〉 meV for μ⋆ A 〈0:1; 0:3〉. The full dependence of Δð0Þ on the Coulomb pseudopotential has been shown in the inset in Fig. 3. Taking into consideration the obtained results, we have calculated the dimensionless ratio RΔ 2Δð0Þ=kB T C . It has been achieved: RΔ A 〈4:10; 3:84〉 (see also Fig. 3). We notice that the values of RΔ for SiH4 compound differ significantly from the prediction of the BCS model, where RΔ ¼ 3:53 [22,23]. The order parameter function on the real axis allows to calculate the total normalized density of states [24]: 2 3 N S ð ωÞ jωiΓ j 6 7 ¼ Re4qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5; N N ð ωÞ 2 ðωiΓ Þ2 Δ ðωÞ
ð15Þ
where the symbols NS and NN denote the density in the superconducting and normal states, respectively. The pair breaking parameter Γ is equal to 0.15 meV. The plot of the total normalized density of states has been presented in Fig. 4. We have assumed T¼ T0 and the selected values of μ⋆ . It is easy to see the reduction of the function N S ðωÞ=N N ðωÞ by the depairing electron correlations. In Fig. 5, the real and imaginary part of the wave function renormalization factor on the real axis has been presented (T ¼ T 0 and T ¼ T C ). On the basis of the obtained results, it has been stated that ZðωÞ weakly depends on the temperature in comparison with the order parameter. We can also observe the weak correlation between the wave function renormalization factor and the shape of the Eliashberg function. The dependence of the electron effective mass ðm⋆ e Þ on the temperature has been determined by using the expression: m⋆ e ¼ Re½Zð0Þme , where me denotes the band electron mass. In our case, the results prove that the electron effective mass takes the high value in the whole range of the superconducting phase's existence. The maximum of m⋆ e has been observed for T ¼TC, where m⋆ e ¼ 1:95me . We notice that the Coulomb pseudopotential does not influence on the value of ½m⋆ e max . In the paper, the Eliashberg equations for the SiH4 compound under the pressure at 250 GPa have been solved. We have shown that the value of the critical temperature decreases from 51.65 K to 20.62 K if μ⋆ A 〈0:1; 0:3〉. The dimensionless ratio RΔ takes the values beyond the prediction of the BCS model: RΔ A 〈4:10; 3:84〉. Additionally, it has been stated that the electron effective mass takes the high values and ½m⋆ e max is equal to 1.95me for the critical temperature.
All numerical calculations have been based on the Eliashberg function sent to us by Prof. Xiao-Jia Chen for whom we are very thankful. The authors are grateful to the Czestochowa University of Technology – MSK CzestMAN for granting access to the computing infrastructure built in the project No. POIG.02.03.00-00-028/08 “PLATON – Science Services Platform”. References [1] [2] [3] [4]
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Acknowledgments The authors wish to thank Prof. K. Dziliński for providing excellent working conditions and the financial support.
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