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On the superconducting state in Ba0.6K0.4BiO3 perovskite oxide
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On the superconducting state in Ba 0.6K 0.4BiO3 perovskite oxide ⁎
D. Szcześniaka, A.Z. Kaczmarekb, E.A. Drzazgab, , K.A. Szewczyka, R. Szcześniaka,b a b
Institute of Physics, Jan Długosz University in Czȩstochowa, Ave. Armii Krajowej 13/15, 42200 Czȩstochowa, Poland Institute of Physics, Czȩstochowa University of Technology, Ave. Armii Krajowej 19, 42200 Czȩstochowa, Poland
A R T I C L E I N F O
A BS T RAC T
Keywords: Superconductivity Thermodynamic properties Perovskite oxide BKBO Oxygen isotope effect
We report study on the superconducting state in Ba 0.6K 0.4BiO3 (BKBO) perovskite oxide, motivated by the inconclusive results on the pairing mechanism in this compound. Our investigations are conducted within the Migdal-Eliashberg formalism, to account for the phonon-mediated superconducting phase. The considered doping level of the discussed material corresponds to the highest critical temperature in this compound, and allows simultaneous analysis of the oxygen isotope effect, for the O16 and O18 isotopes, respectively. We found that such effect is particularly visible for the critical values of the Coulomb pseudopotential (μC⋆ ), which equals to 0.18 for the O16 and 0.16 for the O18 isotope in BKBO. Moreover, we determine the size of the superconducting energy band gap (Δg ) and note that obtained values (9.68 meV and 9.55 meV for the O16 and O18 , respectively) are in good agreement with the experimental predictions which give Δg ∼ 8.68 meV . Finally, we calculate the characteristic dimensionless parameters, such as the zero-temperature energy gap to the critical temperature, the ratio for the specific heat, as well as the ratio associated with the zero-temperature thermodynamic critical field, which suggest occurrence of the strong-coupling and retardation effects within the phonon-mediated scenario in the analyzed material. Where possible the dimensionless ratios are compared to the experimental estimates, and agrees with these which account for the strong-coupling character of the BKBO superconductor.
1. Introduction The Ba 0.6K 0.4BiO3 (BKBO) is the first cooper free oxide perovskite superconductor with the transition temperature (TC ) near 30 K [1,2], which attracted high interest over last years [1,2]. In fact, in the terms of the TC value for the conventional superconductors, BKBO is only suppressed by the materials such as H3S [3,4], MgB2 [5,6], and members of the Csx RbyC60 family [7,8]. Although Ba1− xK xBiO3 was initially believed to be unconventional high-TC superconducting system [1], its cubic perovskite structure [2] and lack of any kind of cooperative magnetic behavior [9], suggested the phonon-mediated mechanism. This was later confirmed by the experimental results on the softening of phonon-modes [10] and the isotope effect of oxygen in BKBO (for two most popular oxygen isotopes, namely O16 and O18) [11]. Specifically, it is predicted that the electron-phonon pairing in BKBO is predominantly governed by the high-frequency modes, similarly like in the BaPbBiO3 [12] and MgB2 [2] systems, contrary to the majority of phonon-mediated superconductors. Moreover, the relatively high electron-phonon constant values (λ = 1.10 for the O16 and λ = 1.09 for the O18 oxygen isotope in BKBO) sets this material within the strong-coupling regime [13]. Herein, we present the comprehensive and complementary analysis of the BKBO material within the Eliashberg equations formalims [14].
⁎
In particular, the employed formalism constitute the many-body generalization of the mean-field BCS theory for the conventional superconductors [15,16], and allows the quantitative determination of selected thermodynamic parameters for the phonon-mediated superconductors within the strong-coupling regime. Therefore, we account for the phonon-mediated and strong-coupling character of the analyzed BKBO superconductors, and check whether corresponding experimental predictions of the selected thermodynamic properties can be confirmed within the employed theory. Specifically, we consider two oxygen isotopes present in the BKBO system, namely the O16 and O18 isotopes, which allows to additionally observe the isotope effect in analyzed superconducting material. By assuming the experimental values of TC for the superconductors of interest, we compute the physical value of the Coulomb pseudopotential, the superconducting energy band gap size, as well as the thermal behavior of the thermodynamic critical field and the specific heat for the superconducting state. Our analysis is supplemented by the calculations of the characteristic dimensionless ratios, familiar in the BCS theory [15,16], which correspond to the aforementioned thermodynamic parameters. Such calculations are important due to the fact that mentioned dimensionless ratios allow direct and convenient comparison of the obtained results with the experimental data. In
Corresponding author. E-mail addresses: [email protected] (D. Szcześniak), [email protected], [email protected] (E.A. Drzazga).
https://doi.org/10.1016/j.physb.2017.10.127 Received 15 June 2017; Received in revised form 24 October 2017; Accepted 31 October 2017 0921-4526/ © 2017 Elsevier B.V. All rights reserved.
Please cite this article as: Szczesniak, D., Physica B (2017), https://doi.org/10.1016/j.physb.2017.10.127
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summary, the estimated parameters characterize superconducting phase in BKBO and provide guidelines for further corresponding theoretical and experimental investigations.
basis of Eq. (4), the order parameter as a function of temperature can be given as [14,19]:
2. Theoretical model
In this context, the solutions on the real axis allows determination of the superconducting energy band gap at the Fermi level, which reads Δg = 2Δ(0), where Δ(0) ≃ Δ(T0 ).
Δ(T ) = Re[Δ(ω = Δ(T ), T )].
In the present paper, the theoretical description of superconducting phase induced in the considered BKBO superconductors, with O16 and O18 oxygen isotopes, is conducted within the formalism of isotropic Eliashberg equations [14]. It is due to the relatively high electronphonon coupling constants and isotropic Fermi surface character in the discussed material, as it has been proved by the first-principle calculations presented in [13]. Specifically, herein, the Eliashberg equations are treated numerically within the computational packages used previously in [4,6,8], while adopting Eliashberg spectral functions (α 2 F(ω), where α 2 is an effective electron-phonon coupling function, F (ω) denotes phonon density of states and ω stands for the phonon frequency) presented in [13]. In the employed numerical procedures, the Eliashberg equations are solved on the imaginary axis, which later allows to obtain the real axis solutions on the basis of the analytical continuation technique. In particular, the Eliashberg equations on the imaginary axis for the order parameter function (ϕn ≡ ϕ(iωn )) and the wave function renormalization factor (Zn ≡ Z (iωn )) are assumed here to take the following form [14]:
ϕn =
π β
M
K (iωn − iωm ) − μ⋆ θ (ωc − |ωm|)
∑
ωm2Zm2 + ϕm2
m =−M
Zn = 1 +
1 π ωn β
M
∑ m =−M
K (iωn − iωm ) ωm2Zm2 + ϕm2
3. Numerical results 3.1. Coulomb pseudopotential We begin our analysis with the determination of the Coulomb pseudopotential (μ⋆ ), which models depairing electron correlations and plays pivotal role in the Eliashberg theory, since multiple thermodynamic properties of the superconducting state depends strongly on its value. In this context, Coulomb pseudopotential opposes the induction of the superconducting phase, and its proper initial determination is of crucial importance to the further analysis aimed at quantitative determination of the thermodynamic properties of interest. We note that usually μ⋆ is treated as a fitting parameter, however in the case when experimental value of the critical temperature is know, it should be possible to provide physically relevant value of the Coulomb pseudopotential. In this respect, the most meaningful is the critical value of the Coulomb pseudopotential (μC⋆ ) which corresponds to the situation when the maximum value of the order parameter (Δm=1 ) for the superconducting state takes value of 0 at the critical temperature, i.e. [Δm =1 ]T = T = 0 . Herein, we conduct such computations by solving C Eliashberg equations on the imaginary axis, assuming that TC equals to the experimental critical temperature values i.e. TC = 28 K for O16 isotope and TC = 27.55 K for O18 isotope. The experimental TC values are adopted from [1]. Specifically, In Fig. 1, we present the results of the conducted calculations, where Fig. 1 (A) and (B) depicts the order parameter function (Δm ) at the selected values of Coulomb pseudopotential for BKBO with O16 and O18 isotopes, respectively. Moreover, Fig. 1 (C) shows the full dependence of Δm=1 function on μ⋆ . The estimated value of μC⋆ for the O16 isotope is 0.18, whereas corresponding value for the O18 isotope equals to 0.16. The obtained result indicates that for considered BKBO material the differences in obtained values are caused by the isotope effect.
ϕm , (1)
ωmZm, (2)
where i is the imaginary unit and ωn denotes the n-th Matsubara frequency: ωn ≡ (π / β )(2n − 1), where β ≡ (kBT )−1, kB is the Boltzmann constant. In the present analysis, the numerical stability is ensured by taking into account 2201 Matsubara frequencies. Therefore, the quantitative predictions can be obtained for T ≥ T0 , where T0 = 2 K, assuming the cut-off frequency ωc = 10Ωmax , where Ωmax is the maximum phonon frequency and equals to 78.81 meV and 79.51 meV for O16 and O18 isotopes, respectively. Additionally, in Eqs. (1) and (2), μ⋆ denotes the so-called Coulomb pseudopotential, θ is the Heaviside function and K (z ) stands for the pairing kernel defined as:
K (z ) ≡ 2
∫0
+∞
dω
ω α 2(ω)F (ω). ω2 − z 2
(5)
(3)
Specifically, the Eliashberg equations on the imaginary axis, presented above, allows to determine thermodynamic properties such as: the order parameter of the superconducting state (Δn = ϕn / Zn ), the value of the critical temperature (TC ), the free energy difference between the normal and superconducting state (ΔF ), the thermodynamic critical field (HC ), and the specific heat difference between the normal and superconducting state (ΔC ). In what follows, the procedure of the analytical continuation of the order parameter on the real axis (Δn → Δ(ω)) is conducted here within the high-accuracy Padé analytic continuation method [17]. We note that other implementation of this method, presented in [18], is also possible and give exactly the same results as the technique given in [17]. Herein, the employed technique is chosen due to its more convenient integration within the used numerical codes. In particular, the following relation for Δ(ω) is adopted [17]:
Δ(ω) =
pΔ1 + pΔ2 ω + ⋯ + pΔr ωr −1 qΔ1 + qΔ2ω + ⋯ + qΔr ωr −1 + ωr
Fig. 1. The order parameter on the imaginary axis for selected values of the Coulomb pseudopotential for BKBO with (A) O16 and (B) O18 isotope. (C) The maximum value of the order parameter as a function of the Coulomb pseudopotential for two different analyzed oxygen isotopes.
, (4)
where pΔj and qΔj are the numerical coefficients and r = 550 . On the 2
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It is also important to comment on the magnitude of the calculated μC⋆ parameters. In particular, for both discussed oxygen isotopes, Coulomb pseudopotential presents relatively high values, therefore it suggests strong electron depairing effects in these material. However, it is not possible to explain the magnitude of calculated μC⋆ in a simple manner on the basis of the Morel-Anderson model [20]. In fact, the high values of μC⋆ can be understood only when certain corrections to the Morel-Anderson model are considered, as proposed by Bauer et al. [21]. In particular, Morel-Anderson model takes into account only first order corrections to the Coulomb repulsion, whereas Bauer et al. include higher order corrections to μC⋆ . The main outcome drawn in [21] is that the retardation effects lead to the reduction of μC⋆ also in the approach of Bauer et al., but not so efficiently as in the case of the Morel-Anderson model. In our opinion, this fact is the most probable scenario that explains the magnitude of the calculated μC⋆ values for the discussed BKBO superconductors. We note, that above discussion has qualitative character and additional in-depth investigations devoted only to this issue can be interesting. Fig. 3. The wave function renormalization factor as a function of the number m at the selected values of temperature for BKBO with (A) O16 and (B) O18 isotopes. (B) The dependence of Z m=1 on the temperature for two different analyzed oxygen isotopes.
3.2. Order parameter function: solutions on the imaginary axis In the next step we provide the supplementary solutions of the Eliashberg equations on the imaginary axis for the order parameter function of the superconducting state in BKBO. The solutions in this section are presented in order to prove the correctness and numerical accuracy of the previous calculations for the Coulomb pseudopotential, as well as to provide the estimates of electron effective mass in the analyzed material. In this respect, herein, our analysis is conducted by using the previously determined critical values of the Coulomb pseudopotential. In particular, Fig. 2 present in detail the remaining characteristics of the order parameter function (Δm ), whereas Fig. 3 give the thermal behavior of the wave-function renormalization factor (Zm ). We remind that latter property is defined by the Eq. (2) and constitute partial solutions of the Eliashberg equations on the imaginary axis, which are required for the determination of the Δm function. On its own, the wave-function renormalization factor represents the behavior of the aforementioned electron effective mass in superconductors (me⋆ ) [19]. The Δm parameter as a function of the number m for selected values of the temperature is plotted in Fig. 2 (A) and (B) for BKBO with O16 and O18 isotope, respectively. The presented results are lorenzian in form, where Δm function has its maximum for m = 1 and shows suppression of the Eliashberg solutions for m > 200 , in both cases of
the oxygen isotope. The latter observation proves numerical accuracy of our computations, which are conducted for m = 1100 , a value which greatly exceeds the aforementioned saturation point. It should be also noted that the negative values of the order parameter function, in Fig. 2 (A) and (B), correspond to the fact that the Coulomb pseudopotential takes on non-zero value. Moreover, the decrease of the Δm together with the increase of temperature, means that fewer Matsubara frequencies contribute to the solutions of the Eliashberg equations. In Fig. 2 (C) we depict the functional behavior of the Δm=1 parameter on the temperature, when assuming μ⋆ equal to the previously determined μC⋆ values. Therein, the Δm=1 function decreased along with the increase of the temperature and takes value of zero for the experimental values of TC . We note, that such functional behavior of the Δm=1 function is typical for the phonon-mediated superconductors [19]. Moreover, the fact that Δm =1 (TC ) = 0 , confirms the correctness of our calculations for the Coulomb pseudopotential. On the other hand, the wave function renormalization factor for the discrete values of m is presented in Fig. 3 (A) for the O16 isotope and in Fig. 3 (B) for the O18 isotope. Similarly as in the case of the order parameter, the Zm function has the characteristic lorenzian shape. However, the temperature dependence of Zm=1 is exactly inverse to the functional behavior of the maximum value of the order parameter, as presented in Fig. 3 (C). As already mentioned the wave function renormalization factor has particular physical meaning. Specifically, the Zm=1 function can describe in the first approximation the value of the electron effective mass (me⋆ ): me⋆ ≃ Zm =1me , where me denotes the band electron mass. The obtained values of me⋆ at T = TC , for the discussed BKBO material, are equal to 2.10 me and 2.09 me for O16 and O18 isotope, respectively. Note, that reported values can be obtained even in a simpler way due to the following expression: Zm =1(TC ) = 1 + λ , which gives values of 2.1 and 2.09 for O16 and O18, respectively. Therefore, calculated results are exactly the same as the solutions of Eliashberg equations on the imaginary axis. In summary, we state that on the basis of results presented in Fig. 3 (C) the effective mass of electrons in the BKBO compounds is relatively high within the entire range of considered temperatures. 3.3. Order parameter function: solutions on the real axis The Eliashberg equations on the imaginary axis are considered to give less accurate results than their real axis counterparts, especially in the terms of the energy band gap at the Fermi level (Δg ). It is mainly due to the fact that real axis frequencies directly correspond to the quasi-particle energies [22]. Therefore, it is reasonable to determine Δg
Fig. 2. The order parameter as a function of the number m at the selected values of temperature for BKBO with (A) O16 and (B) O18 isotopes. (B) The dependence of Δm=1 on the temperature for two different analyzed oxygen isotopes.
3
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Fig. 4. The real (Re[Δ(ω)]) and imaginary (Im[Δ(ω)]) part of the order parameter on the real axis for the selected values of temperature. First row depicts results for the O16 isotope of BKBO, whereas second row corresponds to the O18 isotope. The obtained results are imposed over the rescaled Eliashberg function (α 2 F(ω)) presented in grey color.
all these quoted values of RΔ ratio, our theoretical estimates are in almost perfect agreement with two particular experimental results presented in [25,29], which suggest RΔ = 4.0 and RΔ = 4.1, respectively. In our opinion, studies presented in [25,29] seems to be the most consistent ones, due to the fact that they have been conducted within the same research group by using two different methods, namely the superconductor-insulator-normal metal (SIN) tunneling measurements and the magnetization data analysis, respectively. Moreover, to our knowledge, they are one of the most recent experimental investigations of RΔ ratio in the literature.
function by solving Eliashberg equations on the real axis. However, the direct treating of the Eliashberg equations on the real axis is computationally expensive. To reduce the computational needs it is convenient to use approximate methods which can still maintain the accuracy of exact real axis equations. As already mentioned, herein, we employ the technique which allows analytical continuation of the Δm and Zm functions on the real axis (ω) in the framework of Padé technique, as presented in [17]. Specifically, this method based on the procedure where rational polynomial function is fitted to a set of input points within the single matrix inversion (see Eq. (4)), and is accomplished by using a high-accuracy symbolic computation algorithm (for more details see [17]). In Fig. 4, we plot the real (Re[Δ(ω)]) and imaginary (Im[Δ(ω)]) part of the order parameter on the real axis for the selected values of temperature. Moreover, the rescaled Eliashberg function is presented for the comparison purposes. In fact, we note that within the lowfrequency range the Eliashberg function is visibly correlated with the shapes of Re[Δ(ω)] and Im[Δ(ω)] functions. Moreover, such mapping is stronger for lower values of the temperature than higher ones. We also observe the absence of the damping effects in the considered material due to the non-zero values of Re[Δ(ω)] at low ω [23]. Next, the quantitative predictions of the superconducting energy band gap at the Fermi level can be made, according to the Eqs. (4) and (5). The obtained Δg values for O16 and O18 isotopes are equal to 9.68 meV and 9.55 meV , respectively. Both values can be compared with the experimental result given in [24], which predicts superconducting energy gap to be of the size of ∼8.68 meV for BKBO with O16 . Therefore we observe that our theoretical predictions are in good agreement with the experimental ones, assuring high accuracy of our analysis and favoring the phonon-mediated mechanism in BKBO. Moreover, above estimates allows us to determine the characteristic dimensionless ratio for the energy band gap, which reads RΔ ≡ 2Δ(0)/ kBTC , and equals to 4.01 for O16 and 4.02 for O18 isotope. We observe that these results deviate strongly from the predictions of the BCS theory, which gives RΔ = 3.53 [15,16], and suggest existence of the strong-coupling and retardation effects in the analyzed material. Herein, we note, that multiple experimental predictions of RΔ ratio for BKBO with O16 isotope exist in the literature [24–34]. These studies have been conducted on the basis of various techniques and their predictions of the RΔ ratio values range from 3.5 to 4.8, indicating weak- to strong-coupling regime for the BKBO superconductor. From
3.4. Thermodynamic critical field and specific heat for superconducting state In this section, we supplement above results by the determination of other important thermodynamic properties of the superconducting state in BKBO with O16 and O18 isotopes. It is done to additionally confirm the strong-coupling character of the superconducting phase in the analyzed material. In particular, herein, we concentrate ourselves on the analysis of thermodynamic critical field and specific heat for superconducting state in BKBO. For this purpose, it is first required to estimate the normalized dependence of the free energy on the temperature, what is done the framework of the following formula [35]:
ΔF 2π = − ρ(0) β
M
∑
( ωm2 + Δm2 − ωm )
m =1
⎛ ωm × ⎜ZmS − ZmN ⎜ 2 ωm + Δm2 ⎝
⎞ ⎟, ⎟ ⎠
(6)
where ΔF denotes the free energy difference between the superconducting and normal state, ρ(0) stands for the value of the electronic density of states at the Fermi energy, whereas ZmS and ZmN are the wave function renormalization factors for the superconducting (S) state and the normal (N) state, respectively. In the bottom part of Fig. 5 (A) we present the dependence of the ratio ΔF / ρ(0) on the temperature for both considered oxygen isotopes. We note that the negative values of ΔF / ρ(0) function along the entire temperature range, confirm the thermodynamic stability of the superconducting state in BKBO. In the next step the ΔF / ρ(0) function can be employed to determine the dependence of the normalized thermodynamic critical field 4
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ratio is estimated to take on the values of 1.98 for the O16 and 1.99 for O18 isotope. We note that BCS theory predicts that RH = 0.168 and RC = 1.43 [15,16]. Therefore, obtained results reinforce our previous observation that BKBO with both the O16 and O18 isotopes falls under the strong-coupling regime. The comparison of our results for RH and RC with the corresponding experimental predictions can be another crucial test to the pairing mechanism in BKBO, as well as final verification of our calculations. However, to our knowledge, it is not yet possible to calculate the value of RH from the experimental data. In particular, such calculations should be done on the basis of relation presented in [16]:
⎛ T ⎞2 RH ≡ γ ⎜ C ⎟ . ⎝ HC (0) ⎠
Nonetheless, the value of HC (T ) at T = 0 K, which is pivotal for such calculation, has been only reported in [25], and it does not provides physically relevant results within above equation. In fact, results reported in [25] for HC (0) are one magnitude lower than expected. Despite this fact, the full thermal behavior of HC / ρ(0) function in Fig. 5 (A) and calculated value of RH can be of future use when necessary data will be available. Situation is different in the case of RC ratio, which also can be calculated from the experimental data on the basis of relation from [16]:
Fig. 5. (A) The dependence of the normalized dependence of the free energy (ΔF /ρ(0)) and thermodynamic critical field (HC / ρ(0) ) on the temperature for BKBO with O16 and (B) O18 isotopes. Moreover, the specific heats for the superconducting (C S ) and normal state (C N ) are presented in (B) and (C) for the O16 and (B) O18 isotopes, respectively.
(HC / ρ(0) ) on the temperature, by using the following relation: HC ρ(0)
=
−8π[ΔF / ρ(0)] ,
RC ≡ γ (7)
S
where ΔC ≡ C − C reads:
γ CN = . kBρ(0) β
N
ΔC (TC ) . TC
(11)
where we assume γ = 1.5 mJ/mol K2 , after [37]. In this manner, we calculated the RC values by adopting the ΔC (TC )TC ratios from different experimental reports. Again, the closest results were obtained for results presented in [25], which suggest that RC values range from 1.71 to 2.06 for BKBO with O16 isotope. Therefore, our theoretical estimates are in close agreement with the higher values predicted in [25], and once again suggest strong-coupling character of superconducting phase in BKBO.
as well as the thermal behavior of the specific heat for the superconducting state (C S) from the difference of the specific heat between the superconducting (S) and normal (N) state (ΔC ):
ΔC 1 d 2[ΔF / ρ(0)] =− , kBρ(0) β d (kBT )2
(10)
(8)
and the specific heat for the normal state (C N )
4. Summary
(9)
In summary, we have presented the in-depth calculations of the thermodynamic parameters for superconducting phase induced in BKBO, while considering two popular oxygen isotopes, namely O16 and O18. We have conducted our calculations within the Eliashberg formalism to provide quantitative results for the above, and account for the strong-coupling phonon-mediated superconducting phase in BKBO. The most crucial to the presented analysis are values obtained for the Coulomb pseudopotential and the size of superconducting energy gap in BKBO. In particular, we have estimated that μC⋆ = 0.18 and μC⋆ = 0.16 for the O16 and O18 isotope, respectively, and exhibits indirectly the isotope effect of TC . On the other hand, determined energy gap size equals to 9.68 meV for O16 and 9.55 meV for O16 isotope. When compared to the experimental value of ∼8.68 meV for BKBO with O16 , given in [24], one can notice that our theoretical predictions are in good agreement, and prove that the assumed phonon-mediated mechanism may be appropriate for the considered material. Moreover, we have calculated dimensionless characteristic ratios, familiar in the BCS theory. Presented results suggest that BKBO with O16 and O18 isotopes is a strong-coupling superconductor and that the retardation effects play a significant role in this phase. Moreover, where possible the mentioned dimensionless ratios were compared to the experimental estimates, and also agreed with these which suggest the strong-coupling character of the BKBO superconductor.
In Eq. (9), γ is the Sommerfeld constant. We note that the detailed derivation of formulas (7) and (8) can be found in [36,19]. All the supplementary parameters are plotted in Fig. 5 (A)–(C). In particular, the upper panel of Fig. 5 (A) presents the thermal behavior of the normalized critical thermodynamic field. It clearly shows that the HC / ρ(0) function decreases together with the increase of the temperature and goes to zero at the TC . We also note, that above T ∼ 10 K the thermal decrease of HC / ρ(0) function is almost linear. Above observations are in agreement with the behavior observed for conventional phonon-mediated superconductors [19]. On the other hand, Fig. 5 (B) and (C) depict the dependence of the normalized specific heat for the superconducting and normal state. We observe that both functions increase with the temperature, whereas this increase is linear in the case of C N function and exponential for the C S function. Moreover, the normalized specific heat for superconducting state presents characteristic jump for T = TC . Again described behavior of the normalized specific heat functions is in agreement with the characteristics predicted for the phonon-mediated superconductors [19]. In what follows, results plotted in Fig. 5 (A)–(C) additionally suggests the electron-phonon pairing mechanism in the analyzed BKBO superconductors. The result presented in this section allow to calculate two remaining dimensionless characteristic ratios, given in the BCS theory [15,16]. In particular, the RH ≡ TCC N (TC )/ HC2(0) ratio is equal to 0.148 for both considered BKBO oxygen isotopes, whereas the RC ≡ ΔC (TC )/ C N (TC ) 5
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Acknowledgments Some calculations have been conducted on the Czȩstochowa University of Technology cluster, built in the framework of the PLATON project, no. POIG.02.03.00-00-028/08 - the service of the campus calculations U3. References
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G.M. Eliashberg, Sov. Phys. JETP 11 (1960) 696. J. Bardeen, L.N. Cooper, J.R. Schrieffer, Phys. Rev. 106 (1957) 162. J. Bardeen, L.N. Cooper, J.R. Schrieffer, Phys. Rev. 108 (1957) 1175. K.S.D. Beach, R.J. Gooding, F. Marsiglio, Phys. Rev. B 61 (2000) 5147. H.J. Vidberg, J.W. Serene, J. Low Temp. Phys. 29 (1977) 179. J.P. Carbotte, Rev. Mod. Phys. 62 (1990) 1027. P. Morel, P.W. Anderson, Phys. Rev. B 125 (1962) 1263. J. Bauer, J.E. Han, O. Gunnarsson, J. Phys. Condens. Matter 24 (2012) 492202. F. Marsiglio, M. Schossmann, J.P. Carbotte, Phys. Rev. B 37 (1988) 4965. G. Varelogiannis, Z. Phys. B Condens. Matter 104 (1997) 411. Z. Schlesinger, R.T. Collins, J.A. Calise, D.G. Hinks, A.W. Mitchell, Y. Zheng, B. Dabrowski, N.E. Bickers, D.J. Scalapino, Phys. Rev. B 40 (1989) 6862. S.N. Barilo, S.V. Shiryaev, V.I. Gatalskaya, J.W. Lynn, M. Baran, H. Szymczak, R. Szymczak, D. Dew-Hughes, Phys. Rev. B 58 (1998) 12355. L.A.K.V.F. Gantmakher, N.V. Barkowskii, G.E. Tsadynzhapov, S. Wieger, A.K. Geim, Phys. Rev. B 54 (1996) 6133. G. Zhao, D.E. Morris, Phys. Rev. B 51 (1995) 12848. M.S. Pambianchi, S.N. Anlage, E.S. Hellman, E.H. Hartford, M. Bruns, S.Y. Lee, Appl. Phys. Lett. 64 (1994) 244. P. Samuely, N.L. Bobrov, A. Jansen, P. Wyder, S.N. Barilo, S.V. Shiryaev, Phys. Rev. B 48 (1993) 13904. F. Sharifi, A. Pargellis, R.C. Dynes, B. Miller, E.S. Hellman, J. Rosamilia, E.H. Hartford, Phys. Rev. B 44 (1991) 12521. B.M. Moon, C.E. Platt, R.A. Schweinfurth, D.J.V. Harlingen, Appl. Phys. Lett. 59 (1991) 1905. Q. Huang, J.F. Zasadzinski, N. Tralshawala, K.E. Gray, D.G. Hinks, J.L. Peng, R.L. Greene, Nature 347 (1990) 369. H. Sato, H. Takagi, S.S. Uchida, Phys. C 169 (1990) 391. J.F. Zasadzinski, N. Tralshawala, D.G. Hinks, B. Dabrowski, A.N. Mitchell, D.R. Richards, Phys. C 158 (1989) 519. J. Bardeen, M. Stephen, Phys. Rev. 136 (1964) A1485. J. Blezius, J.P. Carbotte, Phys. Rev. B 36 (1987) 3622. J.E. Graebner, L.F. Schneemeyer, J.K. Thomas, Phys. Rev. B 39 (1989) 9682.
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On the superconducting state in Ba 0.6K 0.4BiO3 perovskite oxide ⁎
D. Szcześniaka, A.Z. Kaczmarekb, E.A. Drzazgab, , K.A. Szewczyka, R. Szcześniaka,b a b
Institute of Physics, Jan Długosz University in Czȩstochowa, Ave. Armii Krajowej 13/15, 42200 Czȩstochowa, Poland Institute of Physics, Czȩstochowa University of Technology, Ave. Armii Krajowej 19, 42200 Czȩstochowa, Poland
A R T I C L E I N F O
A BS T RAC T
Keywords: Superconductivity Thermodynamic properties Perovskite oxide BKBO Oxygen isotope effect
We report study on the superconducting state in Ba 0.6K 0.4BiO3 (BKBO) perovskite oxide, motivated by the inconclusive results on the pairing mechanism in this compound. Our investigations are conducted within the Migdal-Eliashberg formalism, to account for the phonon-mediated superconducting phase. The considered doping level of the discussed material corresponds to the highest critical temperature in this compound, and allows simultaneous analysis of the oxygen isotope effect, for the O16 and O18 isotopes, respectively. We found that such effect is particularly visible for the critical values of the Coulomb pseudopotential (μC⋆ ), which equals to 0.18 for the O16 and 0.16 for the O18 isotope in BKBO. Moreover, we determine the size of the superconducting energy band gap (Δg ) and note that obtained values (9.68 meV and 9.55 meV for the O16 and O18 , respectively) are in good agreement with the experimental predictions which give Δg ∼ 8.68 meV . Finally, we calculate the characteristic dimensionless parameters, such as the zero-temperature energy gap to the critical temperature, the ratio for the specific heat, as well as the ratio associated with the zero-temperature thermodynamic critical field, which suggest occurrence of the strong-coupling and retardation effects within the phonon-mediated scenario in the analyzed material. Where possible the dimensionless ratios are compared to the experimental estimates, and agrees with these which account for the strong-coupling character of the BKBO superconductor.
1. Introduction The Ba 0.6K 0.4BiO3 (BKBO) is the first cooper free oxide perovskite superconductor with the transition temperature (TC ) near 30 K [1,2], which attracted high interest over last years [1,2]. In fact, in the terms of the TC value for the conventional superconductors, BKBO is only suppressed by the materials such as H3S [3,4], MgB2 [5,6], and members of the Csx RbyC60 family [7,8]. Although Ba1− xK xBiO3 was initially believed to be unconventional high-TC superconducting system [1], its cubic perovskite structure [2] and lack of any kind of cooperative magnetic behavior [9], suggested the phonon-mediated mechanism. This was later confirmed by the experimental results on the softening of phonon-modes [10] and the isotope effect of oxygen in BKBO (for two most popular oxygen isotopes, namely O16 and O18) [11]. Specifically, it is predicted that the electron-phonon pairing in BKBO is predominantly governed by the high-frequency modes, similarly like in the BaPbBiO3 [12] and MgB2 [2] systems, contrary to the majority of phonon-mediated superconductors. Moreover, the relatively high electron-phonon constant values (λ = 1.10 for the O16 and λ = 1.09 for the O18 oxygen isotope in BKBO) sets this material within the strong-coupling regime [13]. Herein, we present the comprehensive and complementary analysis of the BKBO material within the Eliashberg equations formalims [14].
⁎
In particular, the employed formalism constitute the many-body generalization of the mean-field BCS theory for the conventional superconductors [15,16], and allows the quantitative determination of selected thermodynamic parameters for the phonon-mediated superconductors within the strong-coupling regime. Therefore, we account for the phonon-mediated and strong-coupling character of the analyzed BKBO superconductors, and check whether corresponding experimental predictions of the selected thermodynamic properties can be confirmed within the employed theory. Specifically, we consider two oxygen isotopes present in the BKBO system, namely the O16 and O18 isotopes, which allows to additionally observe the isotope effect in analyzed superconducting material. By assuming the experimental values of TC for the superconductors of interest, we compute the physical value of the Coulomb pseudopotential, the superconducting energy band gap size, as well as the thermal behavior of the thermodynamic critical field and the specific heat for the superconducting state. Our analysis is supplemented by the calculations of the characteristic dimensionless ratios, familiar in the BCS theory [15,16], which correspond to the aforementioned thermodynamic parameters. Such calculations are important due to the fact that mentioned dimensionless ratios allow direct and convenient comparison of the obtained results with the experimental data. In
Corresponding author. E-mail addresses: [email protected] (D. Szcześniak), [email protected], [email protected] (E.A. Drzazga).
https://doi.org/10.1016/j.physb.2017.10.127 Received 15 June 2017; Received in revised form 24 October 2017; Accepted 31 October 2017 0921-4526/ © 2017 Elsevier B.V. All rights reserved.
Please cite this article as: Szczesniak, D., Physica B (2017), https://doi.org/10.1016/j.physb.2017.10.127
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summary, the estimated parameters characterize superconducting phase in BKBO and provide guidelines for further corresponding theoretical and experimental investigations.
basis of Eq. (4), the order parameter as a function of temperature can be given as [14,19]:
2. Theoretical model
In this context, the solutions on the real axis allows determination of the superconducting energy band gap at the Fermi level, which reads Δg = 2Δ(0), where Δ(0) ≃ Δ(T0 ).
Δ(T ) = Re[Δ(ω = Δ(T ), T )].
In the present paper, the theoretical description of superconducting phase induced in the considered BKBO superconductors, with O16 and O18 oxygen isotopes, is conducted within the formalism of isotropic Eliashberg equations [14]. It is due to the relatively high electronphonon coupling constants and isotropic Fermi surface character in the discussed material, as it has been proved by the first-principle calculations presented in [13]. Specifically, herein, the Eliashberg equations are treated numerically within the computational packages used previously in [4,6,8], while adopting Eliashberg spectral functions (α 2 F(ω), where α 2 is an effective electron-phonon coupling function, F (ω) denotes phonon density of states and ω stands for the phonon frequency) presented in [13]. In the employed numerical procedures, the Eliashberg equations are solved on the imaginary axis, which later allows to obtain the real axis solutions on the basis of the analytical continuation technique. In particular, the Eliashberg equations on the imaginary axis for the order parameter function (ϕn ≡ ϕ(iωn )) and the wave function renormalization factor (Zn ≡ Z (iωn )) are assumed here to take the following form [14]:
ϕn =
π β
M
K (iωn − iωm ) − μ⋆ θ (ωc − |ωm|)
∑
ωm2Zm2 + ϕm2
m =−M
Zn = 1 +
1 π ωn β
M
∑ m =−M
K (iωn − iωm ) ωm2Zm2 + ϕm2
3. Numerical results 3.1. Coulomb pseudopotential We begin our analysis with the determination of the Coulomb pseudopotential (μ⋆ ), which models depairing electron correlations and plays pivotal role in the Eliashberg theory, since multiple thermodynamic properties of the superconducting state depends strongly on its value. In this context, Coulomb pseudopotential opposes the induction of the superconducting phase, and its proper initial determination is of crucial importance to the further analysis aimed at quantitative determination of the thermodynamic properties of interest. We note that usually μ⋆ is treated as a fitting parameter, however in the case when experimental value of the critical temperature is know, it should be possible to provide physically relevant value of the Coulomb pseudopotential. In this respect, the most meaningful is the critical value of the Coulomb pseudopotential (μC⋆ ) which corresponds to the situation when the maximum value of the order parameter (Δm=1 ) for the superconducting state takes value of 0 at the critical temperature, i.e. [Δm =1 ]T = T = 0 . Herein, we conduct such computations by solving C Eliashberg equations on the imaginary axis, assuming that TC equals to the experimental critical temperature values i.e. TC = 28 K for O16 isotope and TC = 27.55 K for O18 isotope. The experimental TC values are adopted from [1]. Specifically, In Fig. 1, we present the results of the conducted calculations, where Fig. 1 (A) and (B) depicts the order parameter function (Δm ) at the selected values of Coulomb pseudopotential for BKBO with O16 and O18 isotopes, respectively. Moreover, Fig. 1 (C) shows the full dependence of Δm=1 function on μ⋆ . The estimated value of μC⋆ for the O16 isotope is 0.18, whereas corresponding value for the O18 isotope equals to 0.16. The obtained result indicates that for considered BKBO material the differences in obtained values are caused by the isotope effect.
ϕm , (1)
ωmZm, (2)
where i is the imaginary unit and ωn denotes the n-th Matsubara frequency: ωn ≡ (π / β )(2n − 1), where β ≡ (kBT )−1, kB is the Boltzmann constant. In the present analysis, the numerical stability is ensured by taking into account 2201 Matsubara frequencies. Therefore, the quantitative predictions can be obtained for T ≥ T0 , where T0 = 2 K, assuming the cut-off frequency ωc = 10Ωmax , where Ωmax is the maximum phonon frequency and equals to 78.81 meV and 79.51 meV for O16 and O18 isotopes, respectively. Additionally, in Eqs. (1) and (2), μ⋆ denotes the so-called Coulomb pseudopotential, θ is the Heaviside function and K (z ) stands for the pairing kernel defined as:
K (z ) ≡ 2
∫0
+∞
dω
ω α 2(ω)F (ω). ω2 − z 2
(5)
(3)
Specifically, the Eliashberg equations on the imaginary axis, presented above, allows to determine thermodynamic properties such as: the order parameter of the superconducting state (Δn = ϕn / Zn ), the value of the critical temperature (TC ), the free energy difference between the normal and superconducting state (ΔF ), the thermodynamic critical field (HC ), and the specific heat difference between the normal and superconducting state (ΔC ). In what follows, the procedure of the analytical continuation of the order parameter on the real axis (Δn → Δ(ω)) is conducted here within the high-accuracy Padé analytic continuation method [17]. We note that other implementation of this method, presented in [18], is also possible and give exactly the same results as the technique given in [17]. Herein, the employed technique is chosen due to its more convenient integration within the used numerical codes. In particular, the following relation for Δ(ω) is adopted [17]:
Δ(ω) =
pΔ1 + pΔ2 ω + ⋯ + pΔr ωr −1 qΔ1 + qΔ2ω + ⋯ + qΔr ωr −1 + ωr
Fig. 1. The order parameter on the imaginary axis for selected values of the Coulomb pseudopotential for BKBO with (A) O16 and (B) O18 isotope. (C) The maximum value of the order parameter as a function of the Coulomb pseudopotential for two different analyzed oxygen isotopes.
, (4)
where pΔj and qΔj are the numerical coefficients and r = 550 . On the 2
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It is also important to comment on the magnitude of the calculated μC⋆ parameters. In particular, for both discussed oxygen isotopes, Coulomb pseudopotential presents relatively high values, therefore it suggests strong electron depairing effects in these material. However, it is not possible to explain the magnitude of calculated μC⋆ in a simple manner on the basis of the Morel-Anderson model [20]. In fact, the high values of μC⋆ can be understood only when certain corrections to the Morel-Anderson model are considered, as proposed by Bauer et al. [21]. In particular, Morel-Anderson model takes into account only first order corrections to the Coulomb repulsion, whereas Bauer et al. include higher order corrections to μC⋆ . The main outcome drawn in [21] is that the retardation effects lead to the reduction of μC⋆ also in the approach of Bauer et al., but not so efficiently as in the case of the Morel-Anderson model. In our opinion, this fact is the most probable scenario that explains the magnitude of the calculated μC⋆ values for the discussed BKBO superconductors. We note, that above discussion has qualitative character and additional in-depth investigations devoted only to this issue can be interesting. Fig. 3. The wave function renormalization factor as a function of the number m at the selected values of temperature for BKBO with (A) O16 and (B) O18 isotopes. (B) The dependence of Z m=1 on the temperature for two different analyzed oxygen isotopes.
3.2. Order parameter function: solutions on the imaginary axis In the next step we provide the supplementary solutions of the Eliashberg equations on the imaginary axis for the order parameter function of the superconducting state in BKBO. The solutions in this section are presented in order to prove the correctness and numerical accuracy of the previous calculations for the Coulomb pseudopotential, as well as to provide the estimates of electron effective mass in the analyzed material. In this respect, herein, our analysis is conducted by using the previously determined critical values of the Coulomb pseudopotential. In particular, Fig. 2 present in detail the remaining characteristics of the order parameter function (Δm ), whereas Fig. 3 give the thermal behavior of the wave-function renormalization factor (Zm ). We remind that latter property is defined by the Eq. (2) and constitute partial solutions of the Eliashberg equations on the imaginary axis, which are required for the determination of the Δm function. On its own, the wave-function renormalization factor represents the behavior of the aforementioned electron effective mass in superconductors (me⋆ ) [19]. The Δm parameter as a function of the number m for selected values of the temperature is plotted in Fig. 2 (A) and (B) for BKBO with O16 and O18 isotope, respectively. The presented results are lorenzian in form, where Δm function has its maximum for m = 1 and shows suppression of the Eliashberg solutions for m > 200 , in both cases of
the oxygen isotope. The latter observation proves numerical accuracy of our computations, which are conducted for m = 1100 , a value which greatly exceeds the aforementioned saturation point. It should be also noted that the negative values of the order parameter function, in Fig. 2 (A) and (B), correspond to the fact that the Coulomb pseudopotential takes on non-zero value. Moreover, the decrease of the Δm together with the increase of temperature, means that fewer Matsubara frequencies contribute to the solutions of the Eliashberg equations. In Fig. 2 (C) we depict the functional behavior of the Δm=1 parameter on the temperature, when assuming μ⋆ equal to the previously determined μC⋆ values. Therein, the Δm=1 function decreased along with the increase of the temperature and takes value of zero for the experimental values of TC . We note, that such functional behavior of the Δm=1 function is typical for the phonon-mediated superconductors [19]. Moreover, the fact that Δm =1 (TC ) = 0 , confirms the correctness of our calculations for the Coulomb pseudopotential. On the other hand, the wave function renormalization factor for the discrete values of m is presented in Fig. 3 (A) for the O16 isotope and in Fig. 3 (B) for the O18 isotope. Similarly as in the case of the order parameter, the Zm function has the characteristic lorenzian shape. However, the temperature dependence of Zm=1 is exactly inverse to the functional behavior of the maximum value of the order parameter, as presented in Fig. 3 (C). As already mentioned the wave function renormalization factor has particular physical meaning. Specifically, the Zm=1 function can describe in the first approximation the value of the electron effective mass (me⋆ ): me⋆ ≃ Zm =1me , where me denotes the band electron mass. The obtained values of me⋆ at T = TC , for the discussed BKBO material, are equal to 2.10 me and 2.09 me for O16 and O18 isotope, respectively. Note, that reported values can be obtained even in a simpler way due to the following expression: Zm =1(TC ) = 1 + λ , which gives values of 2.1 and 2.09 for O16 and O18, respectively. Therefore, calculated results are exactly the same as the solutions of Eliashberg equations on the imaginary axis. In summary, we state that on the basis of results presented in Fig. 3 (C) the effective mass of electrons in the BKBO compounds is relatively high within the entire range of considered temperatures. 3.3. Order parameter function: solutions on the real axis The Eliashberg equations on the imaginary axis are considered to give less accurate results than their real axis counterparts, especially in the terms of the energy band gap at the Fermi level (Δg ). It is mainly due to the fact that real axis frequencies directly correspond to the quasi-particle energies [22]. Therefore, it is reasonable to determine Δg
Fig. 2. The order parameter as a function of the number m at the selected values of temperature for BKBO with (A) O16 and (B) O18 isotopes. (B) The dependence of Δm=1 on the temperature for two different analyzed oxygen isotopes.
3
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Fig. 4. The real (Re[Δ(ω)]) and imaginary (Im[Δ(ω)]) part of the order parameter on the real axis for the selected values of temperature. First row depicts results for the O16 isotope of BKBO, whereas second row corresponds to the O18 isotope. The obtained results are imposed over the rescaled Eliashberg function (α 2 F(ω)) presented in grey color.
all these quoted values of RΔ ratio, our theoretical estimates are in almost perfect agreement with two particular experimental results presented in [25,29], which suggest RΔ = 4.0 and RΔ = 4.1, respectively. In our opinion, studies presented in [25,29] seems to be the most consistent ones, due to the fact that they have been conducted within the same research group by using two different methods, namely the superconductor-insulator-normal metal (SIN) tunneling measurements and the magnetization data analysis, respectively. Moreover, to our knowledge, they are one of the most recent experimental investigations of RΔ ratio in the literature.
function by solving Eliashberg equations on the real axis. However, the direct treating of the Eliashberg equations on the real axis is computationally expensive. To reduce the computational needs it is convenient to use approximate methods which can still maintain the accuracy of exact real axis equations. As already mentioned, herein, we employ the technique which allows analytical continuation of the Δm and Zm functions on the real axis (ω) in the framework of Padé technique, as presented in [17]. Specifically, this method based on the procedure where rational polynomial function is fitted to a set of input points within the single matrix inversion (see Eq. (4)), and is accomplished by using a high-accuracy symbolic computation algorithm (for more details see [17]). In Fig. 4, we plot the real (Re[Δ(ω)]) and imaginary (Im[Δ(ω)]) part of the order parameter on the real axis for the selected values of temperature. Moreover, the rescaled Eliashberg function is presented for the comparison purposes. In fact, we note that within the lowfrequency range the Eliashberg function is visibly correlated with the shapes of Re[Δ(ω)] and Im[Δ(ω)] functions. Moreover, such mapping is stronger for lower values of the temperature than higher ones. We also observe the absence of the damping effects in the considered material due to the non-zero values of Re[Δ(ω)] at low ω [23]. Next, the quantitative predictions of the superconducting energy band gap at the Fermi level can be made, according to the Eqs. (4) and (5). The obtained Δg values for O16 and O18 isotopes are equal to 9.68 meV and 9.55 meV , respectively. Both values can be compared with the experimental result given in [24], which predicts superconducting energy gap to be of the size of ∼8.68 meV for BKBO with O16 . Therefore we observe that our theoretical predictions are in good agreement with the experimental ones, assuring high accuracy of our analysis and favoring the phonon-mediated mechanism in BKBO. Moreover, above estimates allows us to determine the characteristic dimensionless ratio for the energy band gap, which reads RΔ ≡ 2Δ(0)/ kBTC , and equals to 4.01 for O16 and 4.02 for O18 isotope. We observe that these results deviate strongly from the predictions of the BCS theory, which gives RΔ = 3.53 [15,16], and suggest existence of the strong-coupling and retardation effects in the analyzed material. Herein, we note, that multiple experimental predictions of RΔ ratio for BKBO with O16 isotope exist in the literature [24–34]. These studies have been conducted on the basis of various techniques and their predictions of the RΔ ratio values range from 3.5 to 4.8, indicating weak- to strong-coupling regime for the BKBO superconductor. From
3.4. Thermodynamic critical field and specific heat for superconducting state In this section, we supplement above results by the determination of other important thermodynamic properties of the superconducting state in BKBO with O16 and O18 isotopes. It is done to additionally confirm the strong-coupling character of the superconducting phase in the analyzed material. In particular, herein, we concentrate ourselves on the analysis of thermodynamic critical field and specific heat for superconducting state in BKBO. For this purpose, it is first required to estimate the normalized dependence of the free energy on the temperature, what is done the framework of the following formula [35]:
ΔF 2π = − ρ(0) β
M
∑
( ωm2 + Δm2 − ωm )
m =1
⎛ ωm × ⎜ZmS − ZmN ⎜ 2 ωm + Δm2 ⎝
⎞ ⎟, ⎟ ⎠
(6)
where ΔF denotes the free energy difference between the superconducting and normal state, ρ(0) stands for the value of the electronic density of states at the Fermi energy, whereas ZmS and ZmN are the wave function renormalization factors for the superconducting (S) state and the normal (N) state, respectively. In the bottom part of Fig. 5 (A) we present the dependence of the ratio ΔF / ρ(0) on the temperature for both considered oxygen isotopes. We note that the negative values of ΔF / ρ(0) function along the entire temperature range, confirm the thermodynamic stability of the superconducting state in BKBO. In the next step the ΔF / ρ(0) function can be employed to determine the dependence of the normalized thermodynamic critical field 4
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ratio is estimated to take on the values of 1.98 for the O16 and 1.99 for O18 isotope. We note that BCS theory predicts that RH = 0.168 and RC = 1.43 [15,16]. Therefore, obtained results reinforce our previous observation that BKBO with both the O16 and O18 isotopes falls under the strong-coupling regime. The comparison of our results for RH and RC with the corresponding experimental predictions can be another crucial test to the pairing mechanism in BKBO, as well as final verification of our calculations. However, to our knowledge, it is not yet possible to calculate the value of RH from the experimental data. In particular, such calculations should be done on the basis of relation presented in [16]:
⎛ T ⎞2 RH ≡ γ ⎜ C ⎟ . ⎝ HC (0) ⎠
Nonetheless, the value of HC (T ) at T = 0 K, which is pivotal for such calculation, has been only reported in [25], and it does not provides physically relevant results within above equation. In fact, results reported in [25] for HC (0) are one magnitude lower than expected. Despite this fact, the full thermal behavior of HC / ρ(0) function in Fig. 5 (A) and calculated value of RH can be of future use when necessary data will be available. Situation is different in the case of RC ratio, which also can be calculated from the experimental data on the basis of relation from [16]:
Fig. 5. (A) The dependence of the normalized dependence of the free energy (ΔF /ρ(0)) and thermodynamic critical field (HC / ρ(0) ) on the temperature for BKBO with O16 and (B) O18 isotopes. Moreover, the specific heats for the superconducting (C S ) and normal state (C N ) are presented in (B) and (C) for the O16 and (B) O18 isotopes, respectively.
(HC / ρ(0) ) on the temperature, by using the following relation: HC ρ(0)
=
−8π[ΔF / ρ(0)] ,
RC ≡ γ (7)
S
where ΔC ≡ C − C reads:
γ CN = . kBρ(0) β
N
ΔC (TC ) . TC
(11)
where we assume γ = 1.5 mJ/mol K2 , after [37]. In this manner, we calculated the RC values by adopting the ΔC (TC )TC ratios from different experimental reports. Again, the closest results were obtained for results presented in [25], which suggest that RC values range from 1.71 to 2.06 for BKBO with O16 isotope. Therefore, our theoretical estimates are in close agreement with the higher values predicted in [25], and once again suggest strong-coupling character of superconducting phase in BKBO.
as well as the thermal behavior of the specific heat for the superconducting state (C S) from the difference of the specific heat between the superconducting (S) and normal (N) state (ΔC ):
ΔC 1 d 2[ΔF / ρ(0)] =− , kBρ(0) β d (kBT )2
(10)
(8)
and the specific heat for the normal state (C N )
4. Summary
(9)
In summary, we have presented the in-depth calculations of the thermodynamic parameters for superconducting phase induced in BKBO, while considering two popular oxygen isotopes, namely O16 and O18. We have conducted our calculations within the Eliashberg formalism to provide quantitative results for the above, and account for the strong-coupling phonon-mediated superconducting phase in BKBO. The most crucial to the presented analysis are values obtained for the Coulomb pseudopotential and the size of superconducting energy gap in BKBO. In particular, we have estimated that μC⋆ = 0.18 and μC⋆ = 0.16 for the O16 and O18 isotope, respectively, and exhibits indirectly the isotope effect of TC . On the other hand, determined energy gap size equals to 9.68 meV for O16 and 9.55 meV for O16 isotope. When compared to the experimental value of ∼8.68 meV for BKBO with O16 , given in [24], one can notice that our theoretical predictions are in good agreement, and prove that the assumed phonon-mediated mechanism may be appropriate for the considered material. Moreover, we have calculated dimensionless characteristic ratios, familiar in the BCS theory. Presented results suggest that BKBO with O16 and O18 isotopes is a strong-coupling superconductor and that the retardation effects play a significant role in this phase. Moreover, where possible the mentioned dimensionless ratios were compared to the experimental estimates, and also agreed with these which suggest the strong-coupling character of the BKBO superconductor.
In Eq. (9), γ is the Sommerfeld constant. We note that the detailed derivation of formulas (7) and (8) can be found in [36,19]. All the supplementary parameters are plotted in Fig. 5 (A)–(C). In particular, the upper panel of Fig. 5 (A) presents the thermal behavior of the normalized critical thermodynamic field. It clearly shows that the HC / ρ(0) function decreases together with the increase of the temperature and goes to zero at the TC . We also note, that above T ∼ 10 K the thermal decrease of HC / ρ(0) function is almost linear. Above observations are in agreement with the behavior observed for conventional phonon-mediated superconductors [19]. On the other hand, Fig. 5 (B) and (C) depict the dependence of the normalized specific heat for the superconducting and normal state. We observe that both functions increase with the temperature, whereas this increase is linear in the case of C N function and exponential for the C S function. Moreover, the normalized specific heat for superconducting state presents characteristic jump for T = TC . Again described behavior of the normalized specific heat functions is in agreement with the characteristics predicted for the phonon-mediated superconductors [19]. In what follows, results plotted in Fig. 5 (A)–(C) additionally suggests the electron-phonon pairing mechanism in the analyzed BKBO superconductors. The result presented in this section allow to calculate two remaining dimensionless characteristic ratios, given in the BCS theory [15,16]. In particular, the RH ≡ TCC N (TC )/ HC2(0) ratio is equal to 0.148 for both considered BKBO oxygen isotopes, whereas the RC ≡ ΔC (TC )/ C N (TC ) 5
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Acknowledgments Some calculations have been conducted on the Czȩstochowa University of Technology cluster, built in the framework of the PLATON project, no. POIG.02.03.00-00-028/08 - the service of the campus calculations U3. References
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