Superconductivity in α�polonium at the reduced volume

R. Szczesniak, A.P. Durajski n, P.W. PachInstitute of Physics, Czestochowa University of Technology, Ave. Armii Krajowej 19, 42-200 Czestochowa, Poland

a r t i c l e i n f o

Article history:Received 24 June 2013Received in revised form22 August 2013Accepted 18 September 2013Available online 25 September 2013

Keywords:A. PoloniumA. SuperconductivityD. Thermodynamic properties

a b s t r a c t

The paper discusses the thermodynamic properties of the superconducting state that gets induced in theα phase of polonium at the reduced volume (V=Vexp ¼ 0:93). It has been shown that the criticaltemperature (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 super-conducting state (CS) and in the normal state (CN). It has been proven that the values of the dimensionlessratios RH � TCC

NðTC Þ=H2C ð0Þ and RC �ΔCðTC Þ=CNðTC 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 electroneffective mass have been calculated. It has been found that the ratio of the energy gap to the criticaltemperature significantly exceeds the value predicted by the BCS model: RΔ � 2Δð0Þ=kBTC ¼ 4:12. Theelectron 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 thatα�polonium becomes unstable for V=Vexp lower or equal to 0.97. In this case, our study relates to theunstable hypothetical phase.

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1. Introduction

Polonium (Po) was discovered in 1898 by Maria Skłodowska-Curie and Pierre Curie [1–4].

Polonium is a radioactive metal, which is the potent alpharadiation emitter. However, polonium also produces very lowgamma radiation [5–7].

There are 33 polonium isotopes, but none of them is stable.The most enduring and naturally occurring isotope of polonium isthe isotope 210 with a half-life of 138.3 days. The isotope 210 Podecay product is a stable isotope of lead (Pb) 206. It is noteworthythat the most stable, but not naturally occurring isotope of Po is209 – its half-life is equal to 103 years.

The quantity of polonium naturally occurring on the Earth is sosmall that, for scientific or industrial purposes, it is produced bybombarding bismuth with neutrons or by alteration of uraniumore, where polonium appears as the product of the radioactivedecay in the radioactive series of uranium 238 [8].

There are two crystalline phases in polonium – at lower tempera-tures: the simple-cubic structure (α�Po), and at higher temperatures:the trigonal structure (β�Po). Note that the melting point of poloniumis relatively low as for metals and is about 527 K [9–11].

In the paper, we have discussed the results obtained whendetermining the thermodynamic properties of the superconductingstate that gets induced in the phase α�Po at the reduced volume

(V=Vexp ¼ 0:93). In the considered case, the electron–phonon couplingconstant is equal to 1.109 [12]. Thus, the strong-coupling and the retar-dation effects in α�Po are relevant and necessary calculations shouldbe conducted in the framework of the Eliashberg formalism [13].

Let us note that the papers [14–16] demonstrate thatα�polonium becomes unstable at relatively low pressures of1–3 GPa, i.e. at V=Vexp ¼ 0:93–0:98 (using the experimental latticeconstant of 3.345 Å [14–16]) or at V=Vexp ¼ 0:92–0:97 (using theexperimental lattice constant of 3.359 Å as in [12]). In this case ourstudy would be rather academic, dealing with superconductivityof the hypothetical unstable phase.

2. The Eliashberg formalism on the imaginary axis

The Eliashberg equations are the natural generalization of theBCS theory, which is valid only in the case of the weak electron–phonon coupling [17,18]. On the imaginary axis (i�

ffiffiffiffiffiffiffiffi�1

p) the

Eliashberg equations can be written in the following form [19]:

ϕn ¼πβ

∑M

m ¼ �M

λðiωn� iωmÞ�μ⋆θðωc�jωmjÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiω2

mZ2mþϕ2

m

q ϕm; ð1Þ

and

Zn ¼ 1þ 1ωn

πβ

∑M

m ¼ �M

λðiωn� iωmÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiω2

mZ2mþϕ2

m

q ωmZm: ð2Þ

The solutions of the Eliashberg equations are denoted by twofunctions: ϕn �ϕðiωnÞ and Zn � ZðiωnÞ. The first one represents the

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Journal of Physics and Chemistry of Solids

0022-3697/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.jpcs.2013.09.019

n Corresponding author.E-mail address: adurajski@wip.pcz.pl (A.P. Durajski).

Journal of Physics and Chemistry of Solids 75 (2014) 224–229

order parameter function and the second one is the wave functionrenormalization factor. Let us note that the ratio ϕn=Zn determinesthe value of the order parameter (Δn). The quantity ωn is theMatsubara frequency, which is defined by the formula:ωn � ðπ=βÞð2n�1Þ. The inversed temperature β is given by theexpression: β� ðkBTÞ�1, where kB is the Boltzmann constant.

The electron–phonon interaction determines the form of thepairing kernel

λðzÞ � 2Z Ωmax

0dΩ

Ω

Ω2�z2α2FðΩÞ; ð3Þ

where the upper limit of the integral (Ωmax), in the case of theα�Po, is equal to 13.3 meV [12]. The quantity α2FðΩÞ denotesthe Eliashberg function, whose form has also been calculated inthe work [12].

It should be underlined that the Eliashberg function used by ushas been determined, inter alia, by using the local-density approx-imation method (LDA) [12]. However, the function α2FðΩÞ can alsobe calculated in a more advanced way: LDAþSOC, where theabbreviation SOC denotes the spin–orbit coupling included in thesecond-variational scheme. Nevertheless, as it turned out, the influ-ence 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 factresults from the phonon hardening induced by SOC.

Note that in the case of lead, the spin–orbit coupling contri-butes to the growth of the coupling constant by about 40%,because SOC softens the phonons [20,21]. We would like tounderline that the difference between Po- and Pb-SOC behaviorarises from the different structures of the Fermi-surface [20].

The depairing electron correlations in the Eliashberg equationshave been determined with the help of the Coulomb pseudopo-tential (μ⋆) [22]. Additionally, the symbol θ represents the Heavisideunit function and ωc denotes the cut-off frequency: ωc ¼ 5Ωmax.

In the presented study, we have assumed a typical value of theCoulomb pseudopotential: μ⋆ ¼ 0:1. It needs to be noted that thecalculations have been based on the result that can be obtainedusing the Bennemann–Garland formula [23]:

μ⋆C0:26ρð0Þ

1þρð0Þ ; ð4Þ

where ρð0Þ denotes the electron density of states at the Fermi level.For α�Po the following has been obtained: ρð0Þ ¼ 0:57 states=eV(at V=Vexp ¼ 0:93, LDAþSOC calculation – see Table 1 in [12]).Thus, a simple estimation gives the value μ⋆C0:094.

The Eliashberg equations set has been solved with the help ofthe iterative method presented in the works [24–28]. It has beenassumed: M¼1100. In the presented case the solutions ϕn and Znare stable for TZT0 ¼ 1 K.

Figs. 1(A) and 2(A) present the form of the solutions of theEliashberg equations on the imaginary axis for the selected valuesof the temperature. It is easy to see that the functions Δn and Znreach the maximum for m¼1. For higher Matsubara frequenciestheir values decline relatively fast, and are saturable for m� 80.

The temperature dependence of the order parameter and thewave function renormalization factor can be traced in the mostconvenient way by plotting the functions Δm ¼ 1ðTÞ and Zm ¼ 1ðTÞ.Taking into account the obtained results (Figs. 1(B) and 2(B)), it hasbeen found that the values of Δm ¼ 1 are typical patterns expectedfor the order parameter, whereas the values of Zm ¼ 1 slightlyincrease with the increasing temperature to obtain the maximumfor the critical temperature (TC ¼ 7:11 K).

It should be noted that in the case of α�Po, the value of thecritical temperature determined by us is slightly higher than thevalue predicted in the work [12], where TC ¼ 6:28 K. The difference

results from the fact that the authors of the work [12] have usedthe approximate analytical formula, which lowers the value of TC.It is also worth mentioning that the estimated critical temperaturefor α�Po is very close to the critical temperature measuredexperimentally for lead (TC ¼ 7:19 K) [29–31].

Having at our disposal the explicit form of the order parameterand the wave function renormalization factor, we have been able tocalculate the free energy difference between the superconductingand the normal state [32]:

ΔFρð0Þ ¼ � 2π

β∑M

n ¼ 1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiω2

nþΔ2n

q�jωnj

� �

� ZSn�ZN

njωnjffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiω2

nþΔ2n

q0B@

1CA; ð5Þ

Fig. 1. (A) The form of the order parameter on the imaginary axis for the selectedvalues of the temperature. (B) The dependence of the maximum value of the orderparameter on the temperature.

Fig. 2. (A) The form of the renormalization factor on the imaginary axis for theselected values of the temperature. (B) The dependence of the maximum value ofthe renormalization factor on the temperature.

R. Szczesniak et al. / Journal of Physics and Chemistry of Solids 75 (2014) 224–229 225

where the symbols ZSn and ZNn denote the wave function renorma-lization factor for the superconducting state (S) and the normalstate (N), respectively.

The shape of the function ΔF=ρð0Þ has been presented in thelower panel in Fig. 3.

Next, the thermodynamic critical field has been calculated [33]:

HCffiffiffiffiffiffiffiffiffiρð0Þ

p ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�8π½ΔF=ρð0Þ�

q: ð6Þ

The upper panel in Fig. 3 presents the influence of thetemperature on the ratio HC=

ffiffiffiffiffiffiffiffiffiρð0Þ

p. It can be easily seen that

the thermodynamic critical field decreases with the increasingtemperature, taking the zero value for T ¼ TC . Let us notice that themaximum value of the considered function equals: HCð0Þ=

ffiffiffiffiffiffiffiffiffiρð0Þ

p¼

5:95 meV, where HCð0Þ �HCðT0Þ.The specific heat difference between the superconducting and

the normal state ðΔC � CS�CNÞ should be calculated on the basisof the formula [33]:

ΔCkBρð0Þ

¼ � 1βd2½ΔF=ρð0Þ�

dðkBTÞ2: ð7Þ

On the other hand, the estimation of the specific heat in thenormal state requires the use of the simple formula [33]:

CN

kBρð0Þ¼ γβ; ð8Þ

where γ is the Sommerfeld constant: γ � ð2=3Þπ2ð1þλÞ.The obtained results have been presented in Fig. 4. It can be

clearly seen that with the increasing temperature the specific heatof the superconducting state grows strongly, reaching its max-imum at the critical temperature. Then, it undergoes the char-acteristic jump between CS and CN. In the case of α�Po, the valueof ΔC=kBρð0Þ equals 19.79 meV.

On the basis of the determined thermodynamic functions, onecan calculate the values of the dimensionless ratios:

RH � TCCNðTCÞ

H2Cð0Þ

and RC �ΔCðTCÞCNðTCÞ

: ð9Þ

In the limit of the weak electron–phonon coupling (the BCStheory), their values are equal to RH ¼ 0:168 and RC ¼ 1:43 [18],respectively. On the other hand, the estimations made for polo-nium gave the following values: RH ¼ 0:147 and RC ¼ 2:34.

Based on the presented results, it can be seen that the proper-ties of the superconducting state in α�Po cannot be properlydescribed in the framework of the BCS theory. The reason for thisfact is associated with the occurrence of the strong-coupling(λ¼ 1:109) and retardation effects in the considered physicalsystem.

We notice that, from the physical point of view, the retardationeffects in the phonon-induced superconductivity exist due to thesluggishness of the phonon response (the temporal domain).Therefore, in the Eliashberg approach, these effects are encodedin the frequency dependence of the self-energy. In particular,the explicit form of the phonon propagator must be taken intoaccount.

In the Eliashberg formalism, the strength of the strong-coupling and retardation effects can be characterized quantita-tively with the help of the parameter: r� kBTC=ωln. For α�Po, ithas been achieved: r¼0.092. The BCS theory predicts r¼0 [19].

3. The Eliashberg equations in the mixed representation

The physical value of the order parameter and the wavefunction renormalization factor need to be determined on thebasis of the courses of Δ and Z on the real axis (ω). The form of thefunctions in question is normally obtained analytically by extend-ing the solutions of the Eliashberg equations from the imaginaryaxis: Δn-Δ ωð Þ and Zn-Δ ωð Þ. In the presented study, the values ofthe order parameter function and the wave function renormaliza-tion factor on the real axis have been obtained with the help of theEliashberg equations in the mixed representation [34]

ϕðωþ iδÞ ¼ πβ

∑M

m ¼ �Mλðω� iωmÞ�μ⋆θðωc�jωmjÞ�

ϕmffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiω2

mZ2mþϕ2

m

q264

þ iπZ þ1

0dω′α2Fðω′Þ

24½Nðω′Þþ f ω′�ωÞ�ð

� ϕðω�ω′þ iδÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðω�ω′Þ2Z2ðω�ω′þ iδÞ�ϕ2ðω�ω′þ iδÞ

q35

þ iπZ þ1

0dω′α2Fðω′Þ

24½Nðω′Þþ f ðω′þωÞ�

� ϕðωþω′þ iδÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðωþω′Þ2Z2ðωþω′þ iδÞ�ϕ2ðωþω′þ iδÞ

q35; ð10Þ

and

Zðωþ iδÞ ¼ 1þ iωπβ

∑M

m ¼ �Mλðω� iωmÞ

ωmZmffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiω2

mZ2mþϕ2

m

q

þ iπω

Z þ1

0dω′ α2Fðω′Þ

24½Nðω′Þþ f ðω′�ωÞ�Fig. 3. (Lower panel) The dependence of the free energy difference on the

temperature. (Upper panel) The dependence of the thermodynamic critical fieldon the temperature.

Fig. 4. The dependence of the specific heat for the superconducting and the normalstate on the temperature.

R. Szczesniak et al. / Journal of Physics and Chemistry of Solids 75 (2014) 224–229226

� ðω�ω′ÞZðω�ω′þ iδÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðω�ω′Þ2Z2ðω�ω′þ iδÞ�ϕ2ðω�ω′þ iδÞ

q375

þ iπω

Z þ1

0dω′ α2Fðω′Þ

24½Nðω′Þ

þ f ðω′þωÞ� ðωþω′ÞZðωþω′þ iδÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðωþω′Þ2Z2ðωþω′þ iδÞ�ϕ2ðωþω′þ iδÞ

q375;

ð11Þ

where NðωÞ and f ðωÞ denote the Bose–Einstein and Fermi–Diracfunctions respectively.

Note that the solutions of the Eliashberg equations in the mixedrepresentation have been determined in the temperature rangefrom T0 to TC. The numerical methods tested and discussed in theworks [35–39] have been used.

The course of the order parameter on the real axis for theselected values of the temperature has been plotted in Fig. 5. Thefrequencies of the range from 0 to Ωmax have been taken intoaccount. It has been found that for the lower ω, the non-zerovalues are only reached by the real part of the order parameter.Note that, from the physical point of view, the zero values of theimaginary part of the order parameter indicate the absence of thedamping effects in the relevant frequency range [40].

Above 4–5 meV, both functions (Re½ΔðωÞ� and Im½ΔðωÞ�) arecharacterized by the complicated course, wherein, for ω�Ωmax,the real part of the order parameter begins to decrease, which islinked to the rapid loss of the Eliashberg function.

The values of the order parameter are also worth plotting onthe complex plane (see Fig. 6). The characteristic deformed spiralshave been obtained in the considered case. The radii of thosespirals decrease with the increasing temperature. Additionally, itshould be noted that for ωA ⟨0;Ωmax⟩, the values of the functionΔðωÞ are situated on the positive complex half-plane. This factmeans that in the frequency range for which the Eliashbergfunction takes the non-zero values, the effective electron–phononinteraction is pairing in its nature [40].

Fig. 7 presents the form of the wave function renormalizationfactor on the real axis. The values of ω in the range from 0 toΩmax

have been taken into account. As it was in the case of the orderparameter, the non-zero values for lower frequencies are takenonly by the real part of the function ZðωÞ. Let us notice that in the

range of frequencies, in which the Eliashberg function takes highvalues, the real part and the imaginary part of the renormalizationfactor is characterized by the complicated course. For ω�Ωmax

both functions (Re½ZðωÞ� and Im½ZðωÞ�) begin to decrease togetherwith the growth of ω.

On the basis of the solutions of the Eliashberg equations in themixed representation, the dependence of the order parameter andthe ratio of the electron effective mass (m⋆

e ) to the electron bandmass (me) on the temperature have been determined. In the caseof the order parameter its physical values have been calculatedwith the help of the equation below [33]

ΔðTÞ ¼ Re½Δðω¼ΔðTÞÞ�: ð12Þ

On the other hand, the ratio of the effective mass to theelectron band mass is represented by the formula [33]:

m⋆e

me¼ Re½Zð0Þ�: 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 theorder parameter. In the case of the electron effective mass, it hasbeen found that its value is high throughout the whole range ofthe existence of the superconducting state. Additionally, it shouldbe noted that the maximum value of the ratio m⋆

e =me is equal toZðTCÞ ¼ 2:191C1þλ.

While analyzing the numerical results presented in Fig. 8(A) and (B), we have found that it is possible to reproduce them

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 (5α2FðωÞ) hasalso been plotted.

Fig. 6. The values of the order parameter on the complex plane for the selectedtemperatures. The symbols determine the area of the frequencies from 0 to Ωmax.The lines without symbols have been obtained for ωA ðΩmax ;ωc⟩.

R. Szczesniak et al. / Journal of Physics and Chemistry of Solids 75 (2014) 224–229 227

with the help of the simple formulas:

ΔðTÞ ¼Δð0Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� T

TC

� �βs

; ð14Þ

and

m⋆e

me¼ Zð0Þþ½ZðTCÞ�Zð0Þ� T

TC

� �β

; ð15Þ

where Δð0Þ �ΔðT0Þ ¼ 1:26 meV and Zð0Þ � ZðT0Þ ¼ 2:053. More-over, the value of the exponent β is equal to 3.8.

The dimensionless ratio has been calculated as the last step:

RΔ � 2Δð0ÞkBTC

: ð16Þ

In the case of α�Po, it has been obtained: RΔ ¼ 4:12. Let usnotice that the BCS theory predicts a much lower value: 3.53 [18].

4. Summary

The properties of the superconducting state of polonium αphase at the reduced volume have been determined in thepresented study.

It has been proved that the values of the thermodynamicparameters determining the superconducting condensate cannotbe properly estimated by the BCS theory due to the existence ofthe strong-coupling and retardation effects. For this reason, therequired numerical calculations have been performed in theframework of the Eliashberg formalism.

In the first step, it has been found that for μ⋆ ¼ 0:1, the criticaltemperature equals 7.11 K. Note that the obtained value is higherthan the value of the critical temperature at the ambient pressure,

where TCC4 K. We underline that at the ambient pressure, thevalues of the thermodynamic parameters of polonium differ lessfrom the BCS predictions than the ones for α–Po at the reducedvolume (V/Vexp=0.93). This fact results from the increase of theelectron–phonon coupling constant with the increasing of pres-sure [12].

Then the temperature dependence of the thermodynamiccritical field, the specific heat and the order parameter have beendetermined. The designated thermodynamic functions allowed thecalculation of the characteristic dimensionless ratios. In particular,the following has been obtained: RH ¼ 0:147, RC ¼ 2:34, andRΔ ¼ 4:12.

In the last step, it has been proved that the electron effectivemass in the superconducting state is high and at TC it reaches itsmaximum equal to 2:191me.

Let us note that, according to papers [14–16], a simple cubicα�polonium is unstable for V=Vexp lower or equal to 0.97. However,it cannot be excluded that it could be stabilized by some externalagents, e.g. in a form of a thin film on an appropriate substrate.

Acknowledgments

The authors wish to thank Prof. K. Dziliński for providingexcellent working conditions and the financial support.

All numerical calculations have been based on the Eliashbergfunction for polonium sent to us by Prof. B.I. Min and Prof. Chang-Jong Kang to whom we are very thankful.

Additionally, we are grateful to the Czestochowa University ofTechnology – MSK CzestMAN for granting access to the computinginfrastructure built in the project No. POIG.02.03.00-00-028/08“PLATON – Science Services Platform”.

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R. Szczesniak et al. / Journal of Physics and Chemistry of Solids 75 (2014) 224–229 229