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Superconductivity in α-polonium at the reduced volume
Journal of Physics and Chemistry of Solids 75 (2014) 224–229
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Journal of Physics and Chemistry of Solids journal homepage: www.elsevier.com/locate/jpcs
Superconductivity in
αpolonium at the reduced volume
́ iak, A.P. Durajski n, P.W. Pach R. Szcz˛esn Institute of Physics, Cz˛estochowa University of Technology, Ave. Armii Krajowej 19, 42-200 Cz˛estochowa, Poland
art ic l e i nf o
a b s t r a c t
Article history: Received 24 June 2013 Received in revised form 22 August 2013 Accepted 18 September 2013 Available online 25 September 2013
The paper discusses the thermodynamic properties of the superconducting state that gets induced in the α phase of polonium at the reduced volume (V=V exp ¼ 0:93). It has been shown that the critical temperature (TC) is equal to 7.11 K, if the assumed value of the Coulomb pseudopotential equals 0.1. Then, the thermodynamic critical field (HC) has been calculated, as well as the specific heat in the superconducting state (CS) and in the normal state (CN). It has been proven that the values of the dimensionless ratios RH T C C N ðT C Þ=H 2C ð0Þ and RC ΔCðT C Þ=C N ðT C Þ differ significantly from the expectations of the BCS theory. In particular, RH ¼ 0:147 and RC ¼ 2:34. In the next step, the order parameter (Δ) and the electron effective mass have been calculated. It has been found that the ratio of the energy gap to the critical temperature significantly exceeds the value predicted by the BCS model: RΔ 2Δð0Þ=kB T C ¼ 4:12. The electron effective mass is high and reaches its maximum equal to 2:191me at the critical temperature, where the parameter me denotes the electron band mass. Nevertheless, some recent papers show that αpolonium becomes unstable for V=V exp lower or equal to 0.97. In this case, our study relates to the unstable hypothetical phase. & 2013 Elsevier Ltd. All rights reserved.
Keywords: A. Polonium A. Superconductivity D. Thermodynamic properties
1. Introduction Polonium (Po) was discovered in 1898 by Maria SkłodowskaCurie and Pierre Curie [1–4]. Polonium is a radioactive metal, which is the potent alpha radiation emitter. However, polonium also produces very low gamma radiation [5–7]. There are 33 polonium isotopes, but none of them is stable. The most enduring and naturally occurring isotope of polonium is the isotope 210 with a half-life of 138.3 days. The isotope 210 Po decay product is a stable isotope of lead (Pb) 206. It is noteworthy that the most stable, but not naturally occurring isotope of Po is 209 – its half-life is equal to 103 years. The quantity of polonium naturally occurring on the Earth is so small that, for scientific or industrial purposes, it is produced by bombarding bismuth with neutrons or by alteration of uranium ore, where polonium appears as the product of the radioactive decay in the radioactive series of uranium 238 [8]. There are two crystalline phases in polonium – at lower temperatures: the simple-cubic structure (αPo), and at higher temperatures: the trigonal structure (β Po). Note that the melting point of polonium is relatively low as for metals and is about 527 K [9–11]. In the paper, we have discussed the results obtained when determining the thermodynamic properties of the superconducting state that gets induced in the phase αPo at the reduced volume
n
Corresponding author. E-mail address: [email protected] (A.P. Durajski).
0022-3697/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jpcs.2013.09.019
(V=V exp ¼ 0:93). In the considered case, the electron–phonon coupling constant is equal to 1.109 [12]. Thus, the strong-coupling and the retardation effects in αPo are relevant and necessary calculations should be conducted in the framework of the Eliashberg formalism [13]. Let us note that the papers [14–16] demonstrate that αpolonium becomes unstable at relatively low pressures of 1–3 GPa, i.e. at V =V exp ¼ 0:93–0:98 (using the experimental lattice constant of 3.345 Å [14–16]) or at V=V exp ¼ 0:92–0:97 (using the experimental lattice constant of 3.359 Å as in [12]). In this case our study would be rather academic, dealing with superconductivity of the hypothetical unstable phase.
2. The Eliashberg formalism on the imaginary axis The Eliashberg equations are the natural generalization of the BCS theory, which is valid only in the case of the weak electron– pffiffiffiffiffiffiffiffi phonon coupling [17,18]. On the imaginary axis (i 1) the Eliashberg equations can be written in the following form [19]:
ϕn ¼
π M λðiωn iωm Þ μ⋆ θðωc jωm jÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∑ ϕm ; β m ¼ M ω2 Z 2 þ ϕ 2 m m
ð1Þ
m
and Zn ¼ 1 þ
1 π
M
∑
ωn β m ¼ M
λðiωn iωm Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ωm Z m : ω2m Z 2m þ ϕ2m
ð2Þ
The solutions of the Eliashberg equations are denoted by two functions: ϕn ϕðiωn Þ and Z n Zðiωn Þ. The first one represents the
́ iak et al. / Journal of Physics and Chemistry of Solids 75 (2014) 224–229 R. Szcz˛esn
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order parameter function and the second one is the wave function renormalization factor. Let us note that the ratio ϕn =Z n determines the value of the order parameter (Δn ). The quantity ωn is the Matsubara frequency, which is defined by the formula: ωn ðπ =βÞð2n 1Þ. The inversed temperature β is given by the expression: β ðkB TÞ 1 , where kB is the Boltzmann constant. The electron–phonon interaction determines the form of the pairing kernel Z Ωmax Ω λðzÞ 2 dΩ 2 α2 FðΩÞ; ð3Þ 0 Ω z2
where the upper limit of the integral (Ωmax ), in the case of the αPo, is equal to 13.3 meV [12]. The quantity α2 FðΩÞ denotes the Eliashberg function, whose form has also been calculated in the work [12]. It should be underlined that the Eliashberg function used by us has been determined, inter alia, by using the local-density approximation method (LDA) [12]. However, the function α2 FðΩÞ can also be calculated in a more advanced way: LDA þSOC, where the abbreviation SOC denotes the spin–orbit coupling included in the second-variational scheme. Nevertheless, as it turned out, the influence of SOC on the superconductivity is small yet detrimental [12]. In particular, SOC reduces slightly the value of the electron– phonon coupling constant and the critical temperature. This fact results from the phonon hardening induced by SOC. Note that in the case of lead, the spin–orbit coupling contributes to the growth of the coupling constant by about 40%, because SOC softens the phonons [20,21]. We would like to underline that the difference between Po- and Pb-SOC behavior arises from the different structures of the Fermi-surface [20]. The depairing electron correlations in the Eliashberg equations have been determined with the help of the Coulomb pseudopotential (μ⋆ ) [22]. Additionally, the symbol θ represents the Heaviside unit function and ωc denotes the cut-off frequency: ωc ¼ 5Ωmax . In the presented study, we have assumed a typical value of the Coulomb pseudopotential: μ⋆ ¼ 0:1. It needs to be noted that the calculations have been based on the result that can be obtained using the Bennemann–Garland formula [23]:
μ⋆ C 0:26
ρð0Þ ; 1 þ ρð0Þ
Fig. 1. (A) The form of the order parameter on the imaginary axis for the selected values of the temperature. (B) The dependence of the maximum value of the order parameter on the temperature.
ð4Þ
where ρð0Þ denotes the electron density of states at the Fermi level. For αPo the following has been obtained: ρð0Þ ¼ 0:57 states=eV (at V=V exp ¼ 0:93, LDA þSOC calculation – see Table 1 in [12]). Thus, a simple estimation gives the value μ⋆ C 0:094. The Eliashberg equations set has been solved with the help of the iterative method presented in the works [24–28]. It has been assumed: M ¼1100. In the presented case the solutions ϕn and Zn are stable for T Z T 0 ¼ 1 K. Figs. 1(A) and 2(A) present the form of the solutions of the Eliashberg equations on the imaginary axis for the selected values of the temperature. It is easy to see that the functions Δn and Zn reach the maximum for m ¼1. For higher Matsubara frequencies their values decline relatively fast, and are saturable for m 80. The temperature dependence of the order parameter and the wave function renormalization factor can be traced in the most convenient way by plotting the functions Δm ¼ 1 ðTÞ and Z m ¼ 1 ðTÞ. Taking into account the obtained results (Figs. 1(B) and 2(B)), it has been found that the values of Δm ¼ 1 are typical patterns expected for the order parameter, whereas the values of Z m ¼ 1 slightly increase with the increasing temperature to obtain the maximum for the critical temperature (T C ¼ 7:11 K). It should be noted that in the case of αPo, the value of the critical temperature determined by us is slightly higher than the value predicted in the work [12], where T C ¼ 6:28 K. The difference
Fig. 2. (A) The form of the renormalization factor on the imaginary axis for the selected values of the temperature. (B) The dependence of the maximum value of the renormalization factor on the temperature.
results from the fact that the authors of the work [12] have used the approximate analytical formula, which lowers the value of TC. It is also worth mentioning that the estimated critical temperature for αPo is very close to the critical temperature measured experimentally for lead (T C ¼ 7:19 K) [29–31]. Having at our disposal the explicit form of the order parameter and the wave function renormalization factor, we have been able to calculate the free energy difference between the superconducting and the normal state [32]:
ΔF 2π M ¼ ∑ ρð0Þ β n¼1 0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω2n þ Δ2n jωn j 1
jωn j C B @Z Sn Z N n qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA; ω2n þ Δ2n
ð5Þ
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226
where the symbols ZSn and ZN n denote the wave function renormalization factor for the superconducting state (S) and the normal state (N), respectively. The shape of the function ΔF=ρð0Þ has been presented in the lower panel in Fig. 3. Next, the thermodynamic critical field has been calculated [33]: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi HC pffiffiffiffiffiffiffiffiffi ¼ 8π ½ΔF=ρð0Þ: ð6Þ ρð0Þ The upper panel in Fig. 3 presents the influence of the pffiffiffiffiffiffiffiffiffi temperature on the ratio H C = ρð0Þ. It can be easily seen that the thermodynamic critical field decreases with the increasing temperature, taking the zero value for T ¼ T C . Let us notice that the pffiffiffiffiffiffiffiffiffi maximum value of the considered function equals: H C ð0Þ= ρð0Þ ¼ 5:95 meV, where H C ð0Þ H C ðT 0 Þ. The specific heat difference between the superconducting and the normal state ðΔC C S C N Þ should be calculated on the basis of the formula [33]:
ΔC 1 d ½ΔF=ρð0Þ ¼ : kB ρð0Þ β dðkB TÞ2 2
ð7Þ
On the other hand, the estimation of the specific heat in the normal state requires the use of the simple formula [33]: CN γ ¼ ; kB ρð0Þ β
ð8Þ
where γ is the Sommerfeld constant: γ ð2=3Þπ 2 ð1 þ λÞ. The obtained results have been presented in Fig. 4. It can be clearly seen that with the increasing temperature the specific heat of the superconducting state grows strongly, reaching its maximum at the critical temperature. Then, it undergoes the characteristic jump between CS and CN. In the case of αPo, the value of ΔC=kB ρð0Þ equals 19.79 meV. On the basis of the determined thermodynamic functions, one can calculate the values of the dimensionless ratios: RH
T C C N ðT C Þ H 2C ð0Þ
and
RC
ΔCðT C Þ C N ðT C Þ
:
ð9Þ
In the limit of the weak electron–phonon coupling (the BCS theory), their values are equal to RH ¼ 0:168 and RC ¼ 1:43 [18], respectively. On the other hand, the estimations made for polonium gave the following values: RH ¼ 0:147 and RC ¼ 2:34. Based on the presented results, it can be seen that the properties of the superconducting state in αPo cannot be properly described in the framework of the BCS theory. The reason for this fact is associated with the occurrence of the strong-coupling (λ ¼ 1:109) and retardation effects in the considered physical system.
Fig. 4. The dependence of the specific heat for the superconducting and the normal state on the temperature.
We notice that, from the physical point of view, the retardation effects in the phonon-induced superconductivity exist due to the sluggishness of the phonon response (the temporal domain). Therefore, in the Eliashberg approach, these effects are encoded in the frequency dependence of the self-energy. In particular, the explicit form of the phonon propagator must be taken into account. In the Eliashberg formalism, the strength of the strongcoupling and retardation effects can be characterized quantitatively with the help of the parameter: r kB T C =ωln . For αPo, it has been achieved: r ¼0.092. The BCS theory predicts r ¼0 [19].
3. The Eliashberg equations in the mixed representation The physical value of the order parameter and the wave function renormalization factor need to be determined on the basis of the courses of Δ and Z on the real axis (ω). The form of the functions in question is normally obtained analytically by extending the solutions of the Eliashberg equations from the imaginary axis: Δn -ΔðωÞ and Z n -ΔðωÞ. In the presented study, the values of the order parameter function and the wave function renormalization factor on the real axis have been obtained with the help of the Eliashberg equations in the mixed representation [34] 2
ϕðω þ iδÞ ¼
π M 6 ϕm ffi ∑ 4λðω iωm Þ μ⋆ θðωc jωm jÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi β m ¼ M ω2 Z 2 þ ϕ2 þ iπ
Z
2
þ1
m m
m
dω′α2 Fðω′Þ4½Nðω′Þ þ f ðω′ ωÞ
0
ϕðω ω′ þ iδÞ
3
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 2 ðω ω′Þ2 Z 2 ðω ω′ þ iδÞ ϕ ðω ω′ þ iδÞ 2 Z þ1 dω′α2 Fðω′Þ4½Nðω′Þ þ f ðω′ þ ωÞ þ iπ 0
ϕðω þ ω′ þ iδÞ
3
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5; 2 ðω þ ω′Þ2 Z 2 ðω þ ω′ þ iδÞ ϕ ðω þ ω′ þ iδÞ and Zðω þiδÞ ¼ 1 þ
Fig. 3. (Lower panel) The dependence of the free energy difference on the temperature. (Upper panel) The dependence of the thermodynamic critical field on the temperature.
þ
i π
M
∑
ω β m ¼ M
iπ
ω
Z
þ1 0
ωm Z m λðω iωm Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω2m Z 2m þ ϕ2m 2
dω′ α Fðω′Þ4½Nðω′Þ þ f ðω′ ωÞ 2
ð10Þ
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3 ðω ω′ÞZðω ω′þ iδÞ 7 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5 2 2 2 ðω ω′Þ Z ðω ω′þ iδÞ ϕ ðω ω′ þiδÞ 2 Z iπ þ 1 2 þ dω′ α Fðω′Þ4½Nðω′Þ
ω
0
3 ðω þ ω′ÞZðω þ ω′ þ iδÞ 7 þ f ðω′þ ωÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5; 2 2 2 ðω þ ω′Þ Z ðω þ ω′ þ iδÞ ϕ ðω þ ω′ þ iδÞ ð11Þ where NðωÞ and f ðωÞ denote the Bose–Einstein and Fermi–Dirac functions respectively. Note that the solutions of the Eliashberg equations in the mixed representation have been determined in the temperature range from T0 to TC. The numerical methods tested and discussed in the works [35–39] have been used. The course of the order parameter on the real axis for the selected values of the temperature has been plotted in Fig. 5. The frequencies of the range from 0 to Ωmax have been taken into account. It has been found that for the lower ω, the non-zero values are only reached by the real part of the order parameter. Note that, from the physical point of view, the zero values of the imaginary part of the order parameter indicate the absence of the damping effects in the relevant frequency range [40]. Above 4–5 meV, both functions (Re½ΔðωÞ and Im½ΔðωÞ) are characterized by the complicated course, wherein, for ω Ωmax , the real part of the order parameter begins to decrease, which is linked to the rapid loss of the Eliashberg function. The values of the order parameter are also worth plotting on the complex plane (see Fig. 6). The characteristic deformed spirals have been obtained in the considered case. The radii of those spirals decrease with the increasing temperature. Additionally, it should be noted that for ω A 〈0; Ωmax 〉, the values of the function ΔðωÞ are situated on the positive complex half-plane. This fact means that in the frequency range for which the Eliashberg function takes the non-zero values, the effective electron–phonon interaction is pairing in its nature [40]. Fig. 7 presents the form of the wave function renormalization factor on the real axis. The values of ω in the range from 0 to Ωmax have been taken into account. As it was in the case of the order parameter, the non-zero values for lower frequencies are taken only by the real part of the function ZðωÞ. Let us notice that in the
227
range of frequencies, in which the Eliashberg function takes high values, the real part and the imaginary part of the renormalization factor is characterized by the complicated course. For ω Ωmax both functions (Re½ZðωÞ and Im½ZðωÞ) begin to decrease together with the growth of ω. On the basis of the solutions of the Eliashberg equations in the mixed representation, the dependence of the order parameter and the ratio of the electron effective mass (m⋆ e ) to the electron band mass (me) on the temperature have been determined. In the case of the order parameter its physical values have been calculated with the help of the equation below [33]
ΔðTÞ ¼ Re½Δðω ¼ ΔðTÞÞ:
ð12Þ
On the other hand, the ratio of the effective mass to the electron band mass is represented by the formula [33]: m⋆ e ¼ Re½Zð0Þ: me
ð13Þ
The obtained results have been presented in Fig. 8(A) and (B). It can be noted that the function ΔðTÞ has a typical course of the order parameter. In the case of the electron effective mass, it has been found that its value is high throughout the whole range of the existence of the superconducting state. Additionally, it should be noted that the maximum value of the ratio m⋆ e =me is equal to ZðT C Þ ¼ 2:191 C 1 þ λ. While analyzing the numerical results presented in Fig. 8 (A) and (B), we have found that it is possible to reproduce them
Fig. 6. The values of the order parameter on the complex plane for the selected temperatures. The symbols determine the area of the frequencies from 0 to Ωmax . The lines without symbols have been obtained for ωA ðΩmax ; ωc 〉.
Fig. 5. The real part and the imaginary part of the order parameter on the real axis for the selected values of the temperature. The rescaled Eliashberg function (5α2 FðωÞ) has also been plotted.
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228
Fig. 7. The form of the wave function renormalization factor on the real axis. The rescaled Eliashberg function (4α2 FðωÞ) has been additionally plotted.
Fig. 8. (A) The dependence of the physical value of the order parameter on the temperature. (B) The dependence of the ratio of the electron effective mass to the electron band mass on the temperature. In both cases the symbols represent the numerical results. Lines have been obtained on the basis of Eqs. (14) and (15).
with the help of the simple formulas: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi β T ΔðTÞ ¼ Δð0Þ 1 ; TC
Acknowledgments
ð14Þ
and β m⋆ T e ¼ Zð0Þ þ ½ZðT C Þ Zð0Þ ; TC me
ð15Þ
where Δð0Þ ΔðT 0 Þ ¼ 1:26 meV and Zð0Þ ZðT 0 Þ ¼ 2:053. Moreover, the value of the exponent β is equal to 3.8. The dimensionless ratio has been calculated as the last step: RΔ
2Δð0Þ : kB T C
where T C C 4 K. We underline that at the ambient pressure, the values of the thermodynamic parameters of polonium differ less from the BCS predictions than the ones for α–Po at the reduced volume (V/Vexp=0.93). This fact results from the increase of the electron–phonon coupling constant with the increasing of pressure [12]. Then the temperature dependence of the thermodynamic critical field, the specific heat and the order parameter have been determined. The designated thermodynamic functions allowed the calculation of the characteristic dimensionless ratios. In particular, the following has been obtained: RH ¼ 0:147, RC ¼ 2:34, and RΔ ¼ 4:12. In the last step, it has been proved that the electron effective mass in the superconducting state is high and at TC it reaches its maximum equal to 2:191me . Let us note that, according to papers [14–16], a simple cubic αpolonium is unstable for V=V exp lower or equal to 0.97. However, it cannot be excluded that it could be stabilized by some external agents, e.g. in a form of a thin film on an appropriate substrate.
ð16Þ
In the case of αPo, it has been obtained: RΔ ¼ 4:12. Let us notice that the BCS theory predicts a much lower value: 3.53 [18].
4. Summary The properties of the superconducting state of polonium α phase at the reduced volume have been determined in the presented study. It has been proved that the values of the thermodynamic parameters determining the superconducting condensate cannot be properly estimated by the BCS theory due to the existence of the strong-coupling and retardation effects. For this reason, the required numerical calculations have been performed in the framework of the Eliashberg formalism. In the first step, it has been found that for μ⋆ ¼ 0:1, the critical temperature equals 7.11 K. Note that the obtained value is higher than the value of the critical temperature at the ambient pressure,
The authors wish to thank Prof. K. Dziliński for providing excellent working conditions and the financial support. All numerical calculations have been based on the Eliashberg function for polonium sent to us by Prof. B.I. Min and Prof. ChangJong Kang to whom we are very thankful. Additionally, we are grateful to the Cz˛estochowa 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] [12] [13] [14] [15] [16] [17] [18] [19]
M. Curie, Comptes Rendus 126 (1898) 1101. P. Curie, M. Curie, Comptes Rendus 127 (1898) 175. M. Pfutzner, Acta Physica Polonica B 30 (1999) 1197. J.P. Adloff, Radiochimica Acta 91 (2003) 681. Data from the text Polonium made by the Argonne National Laboratory, 2009. Nuclear Data Center, Korea Atomic Energy Research Institute, 2000. G. Gilmore, J. Hemingway, Practical Gamma-Ray Spectrometry, John Wiley & Sons, Chichester, 1995. J. Emsley, Nature's Building Blocks: An A–Z Guide to the Elements, Oxford University Press, New York, 2011. R.J. DeSando, R.C. Lange, Journal of Inorganic and Nuclear Chemistry 28 (1966) 1837. W.H. Beamer, C.R. Maxwell, Journal of Chemical Physics 14 (1946) 569. M.A. Rollier, S.B. Hendricks, L.R. Maxwell, Journal of Chemical Physics 4 (1936) 648. Ch.-J. Kang, K. Kim, B.I. Min, Physical Review B 86 (2012) 054115. G.M. Eliashberg, Soviet Physics JETP 11 (1960) 696. D. Legut, M. Friak, M. Sob, Physical Review Letters 99 (2007) 016402. D. Legut, M. Friak, M. Sob, Physical Review B 81 (2010) 214118. A. Zaoui, A. Belabbes, R. Ahuja, M. Ferhat, Physics Letters A 375 (2011) 1695. J. Bardeen, L.N. Cooper, J.R. Schrieffer, Physical Review 106 (1957) 162. J. Bardeen, L.N. Cooper, J.R. Schrieffer, Physical Review 108 (1957) 1175. J.P. Carbotte, F. Marsiglio, in: K.H. Bennemann, J.B. Ketterson (Eds.), The Physics of Superconductors, vol. 1, Springer, Berlin, 2003, p. 223.
́ iak et al. / Journal of Physics and Chemistry of Solids 75 (2014) 224–229 R. Szcz˛esn
[20] M.J. Verstraete, M. Torrent, F. Jollet, G. Zerah, X. Gonze, Physical Review B 78 (2008) 045119. [21] R. Heid, K.-P. Bohnen, I.Y. Sklyadneva, E.V. Chulkov, Physical Review B 81 (2010) 174527. [22] P. Morel, P.W. Anderson, Physical Review 125 (1962) 1263. [23] K.H. Bennemann, J.W. Garland, AIP Conference Proceedings 41 (1972) 103. ́ iak, A.P. Durajski, P.W. Pach, Physica Scripta 91 (2013) 025704. [24] R. Szcz˛esn ́ iak, A.P. Durajski, M.W. Jarosik, Modern Physics Letters B 26 (2012) [25] R. Szcze˛ sn 1250050. ́ iak, E.A. Drzazga, Solid State Sciences 19 (2013) 167. [26] R. Szcze˛ sn ́ iak, M.W. Jarosik, Acta Physica Polonica B 121 (2012) 841. [27] R. Szcz˛esn ́ ́ iak, Physica Status Solidi B 249 (2012) 2194. [28] R. Szcze˛ sniak, D. Szcz˛esn [29] B. Mitrovic, H.G. Zarate, J.P. Carbotte, Physical Review B 29 (1984) 184. ́ iak, Physica Status Solidi B 244 (2007) 2538. [30] R. Szcze˛ sn
[31] [32] [33] [34] [35] [36] [37] [38] [39] [40]
229
́ iak, Solid State Communications 144 (2007) 27. R. Szcz˛esn J. Bardeen, M. Stephen, Physical Review 136 (1964) A1485. J.P. Carbotte, Reviews of Modern Physics 62 (1990) 1027. F. Marsiglio, M. Schossmann, J.P. Carbotte, Physical Review B 37 (1988) 4965. ́ iak, M.W. Jarosik, Phase Transitions 85 (2012) 727. A.P. Durajski, R. Szcz˛esn ́ iak, A.P. Durajski, P.W. Pach, Journal of Low Temperature Physics 171 R. Szcze˛ sn (2013) 769. ́ iak, A.P. Durajski, Solid State Communications 153 (2013) 26. R. Szcze˛ sn ́ iak, A.P. Durajski, Journal of Physics and Chemistry of Solids 74 R. Szcz˛esn (2013) 641. ́ iak, A.P. Durajski, D. Szcz˛esn ́ iak, Solid State Communications 165 R. Szcz˛esn (2013) 39. G. Varelogiannis, Zeitschrift für Physik B 104 (1997) 411.
Contents lists available at ScienceDirect
Journal of Physics and Chemistry of Solids journal homepage: www.elsevier.com/locate/jpcs
Superconductivity in
αpolonium at the reduced volume
́ iak, A.P. Durajski n, P.W. Pach R. Szcz˛esn Institute of Physics, Cz˛estochowa University of Technology, Ave. Armii Krajowej 19, 42-200 Cz˛estochowa, Poland
art ic l e i nf o
a b s t r a c t
Article history: Received 24 June 2013 Received in revised form 22 August 2013 Accepted 18 September 2013 Available online 25 September 2013
The paper discusses the thermodynamic properties of the superconducting state that gets induced in the α phase of polonium at the reduced volume (V=V exp ¼ 0:93). It has been shown that the critical temperature (TC) is equal to 7.11 K, if the assumed value of the Coulomb pseudopotential equals 0.1. Then, the thermodynamic critical field (HC) has been calculated, as well as the specific heat in the superconducting state (CS) and in the normal state (CN). It has been proven that the values of the dimensionless ratios RH T C C N ðT C Þ=H 2C ð0Þ and RC ΔCðT C Þ=C N ðT C Þ differ significantly from the expectations of the BCS theory. In particular, RH ¼ 0:147 and RC ¼ 2:34. In the next step, the order parameter (Δ) and the electron effective mass have been calculated. It has been found that the ratio of the energy gap to the critical temperature significantly exceeds the value predicted by the BCS model: RΔ 2Δð0Þ=kB T C ¼ 4:12. The electron effective mass is high and reaches its maximum equal to 2:191me at the critical temperature, where the parameter me denotes the electron band mass. Nevertheless, some recent papers show that αpolonium becomes unstable for V=V exp lower or equal to 0.97. In this case, our study relates to the unstable hypothetical phase. & 2013 Elsevier Ltd. All rights reserved.
Keywords: A. Polonium A. Superconductivity D. Thermodynamic properties
1. Introduction Polonium (Po) was discovered in 1898 by Maria SkłodowskaCurie and Pierre Curie [1–4]. Polonium is a radioactive metal, which is the potent alpha radiation emitter. However, polonium also produces very low gamma radiation [5–7]. There are 33 polonium isotopes, but none of them is stable. The most enduring and naturally occurring isotope of polonium is the isotope 210 with a half-life of 138.3 days. The isotope 210 Po decay product is a stable isotope of lead (Pb) 206. It is noteworthy that the most stable, but not naturally occurring isotope of Po is 209 – its half-life is equal to 103 years. The quantity of polonium naturally occurring on the Earth is so small that, for scientific or industrial purposes, it is produced by bombarding bismuth with neutrons or by alteration of uranium ore, where polonium appears as the product of the radioactive decay in the radioactive series of uranium 238 [8]. There are two crystalline phases in polonium – at lower temperatures: the simple-cubic structure (αPo), and at higher temperatures: the trigonal structure (β Po). Note that the melting point of polonium is relatively low as for metals and is about 527 K [9–11]. In the paper, we have discussed the results obtained when determining the thermodynamic properties of the superconducting state that gets induced in the phase αPo at the reduced volume
n
Corresponding author. E-mail address: [email protected] (A.P. Durajski).
0022-3697/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jpcs.2013.09.019
(V=V exp ¼ 0:93). In the considered case, the electron–phonon coupling constant is equal to 1.109 [12]. Thus, the strong-coupling and the retardation effects in αPo are relevant and necessary calculations should be conducted in the framework of the Eliashberg formalism [13]. Let us note that the papers [14–16] demonstrate that αpolonium becomes unstable at relatively low pressures of 1–3 GPa, i.e. at V =V exp ¼ 0:93–0:98 (using the experimental lattice constant of 3.345 Å [14–16]) or at V=V exp ¼ 0:92–0:97 (using the experimental lattice constant of 3.359 Å as in [12]). In this case our study would be rather academic, dealing with superconductivity of the hypothetical unstable phase.
2. The Eliashberg formalism on the imaginary axis The Eliashberg equations are the natural generalization of the BCS theory, which is valid only in the case of the weak electron– pffiffiffiffiffiffiffiffi phonon coupling [17,18]. On the imaginary axis (i 1) the Eliashberg equations can be written in the following form [19]:
ϕn ¼
π M λðiωn iωm Þ μ⋆ θðωc jωm jÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∑ ϕm ; β m ¼ M ω2 Z 2 þ ϕ 2 m m
ð1Þ
m
and Zn ¼ 1 þ
1 π
M
∑
ωn β m ¼ M
λðiωn iωm Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ωm Z m : ω2m Z 2m þ ϕ2m
ð2Þ
The solutions of the Eliashberg equations are denoted by two functions: ϕn ϕðiωn Þ and Z n Zðiωn Þ. The first one represents the
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order parameter function and the second one is the wave function renormalization factor. Let us note that the ratio ϕn =Z n determines the value of the order parameter (Δn ). The quantity ωn is the Matsubara frequency, which is defined by the formula: ωn ðπ =βÞð2n 1Þ. The inversed temperature β is given by the expression: β ðkB TÞ 1 , where kB is the Boltzmann constant. The electron–phonon interaction determines the form of the pairing kernel Z Ωmax Ω λðzÞ 2 dΩ 2 α2 FðΩÞ; ð3Þ 0 Ω z2
where the upper limit of the integral (Ωmax ), in the case of the αPo, is equal to 13.3 meV [12]. The quantity α2 FðΩÞ denotes the Eliashberg function, whose form has also been calculated in the work [12]. It should be underlined that the Eliashberg function used by us has been determined, inter alia, by using the local-density approximation method (LDA) [12]. However, the function α2 FðΩÞ can also be calculated in a more advanced way: LDA þSOC, where the abbreviation SOC denotes the spin–orbit coupling included in the second-variational scheme. Nevertheless, as it turned out, the influence of SOC on the superconductivity is small yet detrimental [12]. In particular, SOC reduces slightly the value of the electron– phonon coupling constant and the critical temperature. This fact results from the phonon hardening induced by SOC. Note that in the case of lead, the spin–orbit coupling contributes to the growth of the coupling constant by about 40%, because SOC softens the phonons [20,21]. We would like to underline that the difference between Po- and Pb-SOC behavior arises from the different structures of the Fermi-surface [20]. The depairing electron correlations in the Eliashberg equations have been determined with the help of the Coulomb pseudopotential (μ⋆ ) [22]. Additionally, the symbol θ represents the Heaviside unit function and ωc denotes the cut-off frequency: ωc ¼ 5Ωmax . In the presented study, we have assumed a typical value of the Coulomb pseudopotential: μ⋆ ¼ 0:1. It needs to be noted that the calculations have been based on the result that can be obtained using the Bennemann–Garland formula [23]:
μ⋆ C 0:26
ρð0Þ ; 1 þ ρð0Þ
Fig. 1. (A) The form of the order parameter on the imaginary axis for the selected values of the temperature. (B) The dependence of the maximum value of the order parameter on the temperature.
ð4Þ
where ρð0Þ denotes the electron density of states at the Fermi level. For αPo the following has been obtained: ρð0Þ ¼ 0:57 states=eV (at V=V exp ¼ 0:93, LDA þSOC calculation – see Table 1 in [12]). Thus, a simple estimation gives the value μ⋆ C 0:094. The Eliashberg equations set has been solved with the help of the iterative method presented in the works [24–28]. It has been assumed: M ¼1100. In the presented case the solutions ϕn and Zn are stable for T Z T 0 ¼ 1 K. Figs. 1(A) and 2(A) present the form of the solutions of the Eliashberg equations on the imaginary axis for the selected values of the temperature. It is easy to see that the functions Δn and Zn reach the maximum for m ¼1. For higher Matsubara frequencies their values decline relatively fast, and are saturable for m 80. The temperature dependence of the order parameter and the wave function renormalization factor can be traced in the most convenient way by plotting the functions Δm ¼ 1 ðTÞ and Z m ¼ 1 ðTÞ. Taking into account the obtained results (Figs. 1(B) and 2(B)), it has been found that the values of Δm ¼ 1 are typical patterns expected for the order parameter, whereas the values of Z m ¼ 1 slightly increase with the increasing temperature to obtain the maximum for the critical temperature (T C ¼ 7:11 K). It should be noted that in the case of αPo, the value of the critical temperature determined by us is slightly higher than the value predicted in the work [12], where T C ¼ 6:28 K. The difference
Fig. 2. (A) The form of the renormalization factor on the imaginary axis for the selected values of the temperature. (B) The dependence of the maximum value of the renormalization factor on the temperature.
results from the fact that the authors of the work [12] have used the approximate analytical formula, which lowers the value of TC. It is also worth mentioning that the estimated critical temperature for αPo is very close to the critical temperature measured experimentally for lead (T C ¼ 7:19 K) [29–31]. Having at our disposal the explicit form of the order parameter and the wave function renormalization factor, we have been able to calculate the free energy difference between the superconducting and the normal state [32]:
ΔF 2π M ¼ ∑ ρð0Þ β n¼1 0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω2n þ Δ2n jωn j 1
jωn j C B @Z Sn Z N n qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA; ω2n þ Δ2n
ð5Þ
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where the symbols ZSn and ZN n denote the wave function renormalization factor for the superconducting state (S) and the normal state (N), respectively. The shape of the function ΔF=ρð0Þ has been presented in the lower panel in Fig. 3. Next, the thermodynamic critical field has been calculated [33]: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi HC pffiffiffiffiffiffiffiffiffi ¼ 8π ½ΔF=ρð0Þ: ð6Þ ρð0Þ The upper panel in Fig. 3 presents the influence of the pffiffiffiffiffiffiffiffiffi temperature on the ratio H C = ρð0Þ. It can be easily seen that the thermodynamic critical field decreases with the increasing temperature, taking the zero value for T ¼ T C . Let us notice that the pffiffiffiffiffiffiffiffiffi maximum value of the considered function equals: H C ð0Þ= ρð0Þ ¼ 5:95 meV, where H C ð0Þ H C ðT 0 Þ. The specific heat difference between the superconducting and the normal state ðΔC C S C N Þ should be calculated on the basis of the formula [33]:
ΔC 1 d ½ΔF=ρð0Þ ¼ : kB ρð0Þ β dðkB TÞ2 2
ð7Þ
On the other hand, the estimation of the specific heat in the normal state requires the use of the simple formula [33]: CN γ ¼ ; kB ρð0Þ β
ð8Þ
where γ is the Sommerfeld constant: γ ð2=3Þπ 2 ð1 þ λÞ. The obtained results have been presented in Fig. 4. It can be clearly seen that with the increasing temperature the specific heat of the superconducting state grows strongly, reaching its maximum at the critical temperature. Then, it undergoes the characteristic jump between CS and CN. In the case of αPo, the value of ΔC=kB ρð0Þ equals 19.79 meV. On the basis of the determined thermodynamic functions, one can calculate the values of the dimensionless ratios: RH
T C C N ðT C Þ H 2C ð0Þ
and
RC
ΔCðT C Þ C N ðT C Þ
:
ð9Þ
In the limit of the weak electron–phonon coupling (the BCS theory), their values are equal to RH ¼ 0:168 and RC ¼ 1:43 [18], respectively. On the other hand, the estimations made for polonium gave the following values: RH ¼ 0:147 and RC ¼ 2:34. Based on the presented results, it can be seen that the properties of the superconducting state in αPo cannot be properly described in the framework of the BCS theory. The reason for this fact is associated with the occurrence of the strong-coupling (λ ¼ 1:109) and retardation effects in the considered physical system.
Fig. 4. The dependence of the specific heat for the superconducting and the normal state on the temperature.
We notice that, from the physical point of view, the retardation effects in the phonon-induced superconductivity exist due to the sluggishness of the phonon response (the temporal domain). Therefore, in the Eliashberg approach, these effects are encoded in the frequency dependence of the self-energy. In particular, the explicit form of the phonon propagator must be taken into account. In the Eliashberg formalism, the strength of the strongcoupling and retardation effects can be characterized quantitatively with the help of the parameter: r kB T C =ωln . For αPo, it has been achieved: r ¼0.092. The BCS theory predicts r ¼0 [19].
3. The Eliashberg equations in the mixed representation The physical value of the order parameter and the wave function renormalization factor need to be determined on the basis of the courses of Δ and Z on the real axis (ω). The form of the functions in question is normally obtained analytically by extending the solutions of the Eliashberg equations from the imaginary axis: Δn -ΔðωÞ and Z n -ΔðωÞ. In the presented study, the values of the order parameter function and the wave function renormalization factor on the real axis have been obtained with the help of the Eliashberg equations in the mixed representation [34] 2
ϕðω þ iδÞ ¼
π M 6 ϕm ffi ∑ 4λðω iωm Þ μ⋆ θðωc jωm jÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi β m ¼ M ω2 Z 2 þ ϕ2 þ iπ
Z
2
þ1
m m
m
dω′α2 Fðω′Þ4½Nðω′Þ þ f ðω′ ωÞ
0
ϕðω ω′ þ iδÞ
3
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 2 ðω ω′Þ2 Z 2 ðω ω′ þ iδÞ ϕ ðω ω′ þ iδÞ 2 Z þ1 dω′α2 Fðω′Þ4½Nðω′Þ þ f ðω′ þ ωÞ þ iπ 0
ϕðω þ ω′ þ iδÞ
3
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5; 2 ðω þ ω′Þ2 Z 2 ðω þ ω′ þ iδÞ ϕ ðω þ ω′ þ iδÞ and Zðω þiδÞ ¼ 1 þ
Fig. 3. (Lower panel) The dependence of the free energy difference on the temperature. (Upper panel) The dependence of the thermodynamic critical field on the temperature.
þ
i π
M
∑
ω β m ¼ M
iπ
ω
Z
þ1 0
ωm Z m λðω iωm Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω2m Z 2m þ ϕ2m 2
dω′ α Fðω′Þ4½Nðω′Þ þ f ðω′ ωÞ 2
ð10Þ
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3 ðω ω′ÞZðω ω′þ iδÞ 7 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5 2 2 2 ðω ω′Þ Z ðω ω′þ iδÞ ϕ ðω ω′ þiδÞ 2 Z iπ þ 1 2 þ dω′ α Fðω′Þ4½Nðω′Þ
ω
0
3 ðω þ ω′ÞZðω þ ω′ þ iδÞ 7 þ f ðω′þ ωÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5; 2 2 2 ðω þ ω′Þ Z ðω þ ω′ þ iδÞ ϕ ðω þ ω′ þ iδÞ ð11Þ where NðωÞ and f ðωÞ denote the Bose–Einstein and Fermi–Dirac functions respectively. Note that the solutions of the Eliashberg equations in the mixed representation have been determined in the temperature range from T0 to TC. The numerical methods tested and discussed in the works [35–39] have been used. The course of the order parameter on the real axis for the selected values of the temperature has been plotted in Fig. 5. The frequencies of the range from 0 to Ωmax have been taken into account. It has been found that for the lower ω, the non-zero values are only reached by the real part of the order parameter. Note that, from the physical point of view, the zero values of the imaginary part of the order parameter indicate the absence of the damping effects in the relevant frequency range [40]. Above 4–5 meV, both functions (Re½ΔðωÞ and Im½ΔðωÞ) are characterized by the complicated course, wherein, for ω Ωmax , the real part of the order parameter begins to decrease, which is linked to the rapid loss of the Eliashberg function. The values of the order parameter are also worth plotting on the complex plane (see Fig. 6). The characteristic deformed spirals have been obtained in the considered case. The radii of those spirals decrease with the increasing temperature. Additionally, it should be noted that for ω A 〈0; Ωmax 〉, the values of the function ΔðωÞ are situated on the positive complex half-plane. This fact means that in the frequency range for which the Eliashberg function takes the non-zero values, the effective electron–phonon interaction is pairing in its nature [40]. Fig. 7 presents the form of the wave function renormalization factor on the real axis. The values of ω in the range from 0 to Ωmax have been taken into account. As it was in the case of the order parameter, the non-zero values for lower frequencies are taken only by the real part of the function ZðωÞ. Let us notice that in the
227
range of frequencies, in which the Eliashberg function takes high values, the real part and the imaginary part of the renormalization factor is characterized by the complicated course. For ω Ωmax both functions (Re½ZðωÞ and Im½ZðωÞ) begin to decrease together with the growth of ω. On the basis of the solutions of the Eliashberg equations in the mixed representation, the dependence of the order parameter and the ratio of the electron effective mass (m⋆ e ) to the electron band mass (me) on the temperature have been determined. In the case of the order parameter its physical values have been calculated with the help of the equation below [33]
ΔðTÞ ¼ Re½Δðω ¼ ΔðTÞÞ:
ð12Þ
On the other hand, the ratio of the effective mass to the electron band mass is represented by the formula [33]: m⋆ e ¼ Re½Zð0Þ: me
ð13Þ
The obtained results have been presented in Fig. 8(A) and (B). It can be noted that the function ΔðTÞ has a typical course of the order parameter. In the case of the electron effective mass, it has been found that its value is high throughout the whole range of the existence of the superconducting state. Additionally, it should be noted that the maximum value of the ratio m⋆ e =me is equal to ZðT C Þ ¼ 2:191 C 1 þ λ. While analyzing the numerical results presented in Fig. 8 (A) and (B), we have found that it is possible to reproduce them
Fig. 6. The values of the order parameter on the complex plane for the selected temperatures. The symbols determine the area of the frequencies from 0 to Ωmax . The lines without symbols have been obtained for ωA ðΩmax ; ωc 〉.
Fig. 5. The real part and the imaginary part of the order parameter on the real axis for the selected values of the temperature. The rescaled Eliashberg function (5α2 FðωÞ) has also been plotted.
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Fig. 7. The form of the wave function renormalization factor on the real axis. The rescaled Eliashberg function (4α2 FðωÞ) has been additionally plotted.
Fig. 8. (A) The dependence of the physical value of the order parameter on the temperature. (B) The dependence of the ratio of the electron effective mass to the electron band mass on the temperature. In both cases the symbols represent the numerical results. Lines have been obtained on the basis of Eqs. (14) and (15).
with the help of the simple formulas: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi β T ΔðTÞ ¼ Δð0Þ 1 ; TC
Acknowledgments
ð14Þ
and β m⋆ T e ¼ Zð0Þ þ ½ZðT C Þ Zð0Þ ; TC me
ð15Þ
where Δð0Þ ΔðT 0 Þ ¼ 1:26 meV and Zð0Þ ZðT 0 Þ ¼ 2:053. Moreover, the value of the exponent β is equal to 3.8. The dimensionless ratio has been calculated as the last step: RΔ
2Δð0Þ : kB T C
where T C C 4 K. We underline that at the ambient pressure, the values of the thermodynamic parameters of polonium differ less from the BCS predictions than the ones for α–Po at the reduced volume (V/Vexp=0.93). This fact results from the increase of the electron–phonon coupling constant with the increasing of pressure [12]. Then the temperature dependence of the thermodynamic critical field, the specific heat and the order parameter have been determined. The designated thermodynamic functions allowed the calculation of the characteristic dimensionless ratios. In particular, the following has been obtained: RH ¼ 0:147, RC ¼ 2:34, and RΔ ¼ 4:12. In the last step, it has been proved that the electron effective mass in the superconducting state is high and at TC it reaches its maximum equal to 2:191me . Let us note that, according to papers [14–16], a simple cubic αpolonium is unstable for V=V exp lower or equal to 0.97. However, it cannot be excluded that it could be stabilized by some external agents, e.g. in a form of a thin film on an appropriate substrate.
ð16Þ
In the case of αPo, it has been obtained: RΔ ¼ 4:12. Let us notice that the BCS theory predicts a much lower value: 3.53 [18].
4. Summary The properties of the superconducting state of polonium α phase at the reduced volume have been determined in the presented study. It has been proved that the values of the thermodynamic parameters determining the superconducting condensate cannot be properly estimated by the BCS theory due to the existence of the strong-coupling and retardation effects. For this reason, the required numerical calculations have been performed in the framework of the Eliashberg formalism. In the first step, it has been found that for μ⋆ ¼ 0:1, the critical temperature equals 7.11 K. Note that the obtained value is higher than the value of the critical temperature at the ambient pressure,
The authors wish to thank Prof. K. Dziliński for providing excellent working conditions and the financial support. All numerical calculations have been based on the Eliashberg function for polonium sent to us by Prof. B.I. Min and Prof. ChangJong Kang to whom we are very thankful. Additionally, we are grateful to the Cz˛estochowa 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] [12] [13] [14] [15] [16] [17] [18] [19]
M. Curie, Comptes Rendus 126 (1898) 1101. P. Curie, M. Curie, Comptes Rendus 127 (1898) 175. M. Pfutzner, Acta Physica Polonica B 30 (1999) 1197. J.P. Adloff, Radiochimica Acta 91 (2003) 681. Data from the text Polonium made by the Argonne National Laboratory, 2009. Nuclear Data Center, Korea Atomic Energy Research Institute, 2000. G. Gilmore, J. Hemingway, Practical Gamma-Ray Spectrometry, John Wiley & Sons, Chichester, 1995. J. Emsley, Nature's Building Blocks: An A–Z Guide to the Elements, Oxford University Press, New York, 2011. R.J. DeSando, R.C. Lange, Journal of Inorganic and Nuclear Chemistry 28 (1966) 1837. W.H. Beamer, C.R. Maxwell, Journal of Chemical Physics 14 (1946) 569. M.A. Rollier, S.B. Hendricks, L.R. Maxwell, Journal of Chemical Physics 4 (1936) 648. Ch.-J. Kang, K. Kim, B.I. Min, Physical Review B 86 (2012) 054115. G.M. Eliashberg, Soviet Physics JETP 11 (1960) 696. D. Legut, M. Friak, M. Sob, Physical Review Letters 99 (2007) 016402. D. Legut, M. Friak, M. Sob, Physical Review B 81 (2010) 214118. A. Zaoui, A. Belabbes, R. Ahuja, M. Ferhat, Physics Letters A 375 (2011) 1695. J. Bardeen, L.N. Cooper, J.R. Schrieffer, Physical Review 106 (1957) 162. J. Bardeen, L.N. Cooper, J.R. Schrieffer, Physical Review 108 (1957) 1175. J.P. Carbotte, F. Marsiglio, in: K.H. Bennemann, J.B. Ketterson (Eds.), The Physics of Superconductors, vol. 1, Springer, Berlin, 2003, p. 223.
́ iak et al. / Journal of Physics and Chemistry of Solids 75 (2014) 224–229 R. Szcz˛esn
[20] M.J. Verstraete, M. Torrent, F. Jollet, G. Zerah, X. Gonze, Physical Review B 78 (2008) 045119. [21] R. Heid, K.-P. Bohnen, I.Y. Sklyadneva, E.V. Chulkov, Physical Review B 81 (2010) 174527. [22] P. Morel, P.W. Anderson, Physical Review 125 (1962) 1263. [23] K.H. Bennemann, J.W. Garland, AIP Conference Proceedings 41 (1972) 103. ́ iak, A.P. Durajski, P.W. Pach, Physica Scripta 91 (2013) 025704. [24] R. Szcz˛esn ́ iak, A.P. Durajski, M.W. Jarosik, Modern Physics Letters B 26 (2012) [25] R. Szcze˛ sn 1250050. ́ iak, E.A. Drzazga, Solid State Sciences 19 (2013) 167. [26] R. Szcze˛ sn ́ iak, M.W. Jarosik, Acta Physica Polonica B 121 (2012) 841. [27] R. Szcz˛esn ́ ́ iak, Physica Status Solidi B 249 (2012) 2194. [28] R. Szcze˛ sniak, D. Szcz˛esn [29] B. Mitrovic, H.G. Zarate, J.P. Carbotte, Physical Review B 29 (1984) 184. ́ iak, Physica Status Solidi B 244 (2007) 2538. [30] R. Szcze˛ sn
[31] [32] [33] [34] [35] [36] [37] [38] [39] [40]
229
́ iak, Solid State Communications 144 (2007) 27. R. Szcz˛esn J. Bardeen, M. Stephen, Physical Review 136 (1964) A1485. J.P. Carbotte, Reviews of Modern Physics 62 (1990) 1027. F. Marsiglio, M. Schossmann, J.P. Carbotte, Physical Review B 37 (1988) 4965. ́ iak, M.W. Jarosik, Phase Transitions 85 (2012) 727. A.P. Durajski, R. Szcz˛esn ́ iak, A.P. Durajski, P.W. Pach, Journal of Low Temperature Physics 171 R. Szcze˛ sn (2013) 769. ́ iak, A.P. Durajski, Solid State Communications 153 (2013) 26. R. Szcze˛ sn ́ iak, A.P. Durajski, Journal of Physics and Chemistry of Solids 74 R. Szcz˛esn (2013) 641. ́ iak, A.P. Durajski, D. Szcz˛esn ́ iak, Solid State Communications 165 R. Szcz˛esn (2013) 39. G. Varelogiannis, Zeitschrift für Physik B 104 (1997) 411.