Thermodynamic properties of antiperovskite MgCNi3

Solid State Communications 203 (2015) 63–68

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Thermodynamic properties of antiperovskite MgCNi3 in superconducting phase R. Szczȩśniak a,b, A.P. Durajski a,n, Ł. Herok b a b

Institute of Physics, Czȩstochowa University of Technology, Ave. Armii Krajowej 19, 42-200 Czȩstochowa, Poland Institute of Physics, Jan Długosz University, Ave. Armii Krajowej 13/15, 42-200 Czȩstochowa, Poland

art ic l e i nf o

a b s t r a c t

Article history: Received 22 October 2014 Received in revised form 18 November 2014 Accepted 20 November 2014 by F. Peeters Available online 29 November 2014

The aim of the present work is to explore the physical properties of the transition-metal based antiperovskite MgCNi3 in superconducting state. In particular, the critical value of the Coulomb pseudopotential and temperature dependence of the energy gap, specific heat, thermodynamic critical field and London penetration depth are theoretically analyzed within the framework of the Eliashberg formalism. Moreover, we determined the dimensionless ratios which are related to the above thermodynamic functions: 2Δð0Þ=kB T C ¼ 4:19, ΔC ðT C Þ=C N ðT C Þ ¼ 2:27 and T C C N ðT C Þ=H 2C ð0Þ ¼ 0:141. Our calculations show that obtained results significantly diverge from the values predicted by the BCS model due to the strong-coupling corrections and retardation effect existing in investigated antiperovskite. & 2014 Elsevier Ltd. All rights reserved.

Keywords: A. Superconductors A. Antiperovskite MgCNi3 D. Thermodynamic properties

1. Introduction The non-oxide perovskite (antiperovskite) compounds have attracted considerable attention due to a wide range of unique physical properties, such as giant magnetoresistance [1], magnetostriction [2], large negative magnetocaloric effect [3], negative thermal expansion [4], non-Fermi liquid behaviour [5] and low temperature coefficient of resistivity [6]. The discovery of superconductivity near 8 K in the intermetallic antiperovskite compound MgCNi3 [7], containing a high concentration of Ni element which usually results in magnetism rather than superconductivity, has stimulated extensive research on the exact nature of the superconducting state and its microscopic origin [8,9]. The current experimental situation is controversial. On the one hand, evidence for conventional s-wave behaviour is found in specific heat measurements [10] and in the 13C NMR experiment [11,12], whereas tunnelling spectra and penetration depth measurements suggest that the possibility of unconventional pairing in MgCNi3 should not be excluded [13,14]. Additional experiments are needed to further clarification of the pairing symmetry of this superconductor. Nevertheless, the theoretical results give evidence for the phonon-mediated pairing mechanism in MgCNi3 with a high value of the electron–phonon coupling constant (λ Z1) [15–17]. Strong coupling is also suggested by measurements of the thermopower and thermal conductivity [18], specific heat in the normal and superconducting states [13,19–22] and the large energy gap

n

Corresponding author. E-mail address: [email protected] (A.P. Durajski).

http://dx.doi.org/10.1016/j.ssc.2014.11.018 0038-1098/& 2014 Elsevier Ltd. All rights reserved.

determined from tunneling experiments [13]. Similar situation is also observed in other antiperovskites e.g., Sc3TlB, Sc3InB and SrPt3P [8,23,24]. Due to above, in the present theoretical investigation, we have decided for a more detailed examination of the superconducting state induced in antiperovskite MgCNi3. In particular, the Coulomb pseudopotential, superconducting energy gap, free energy and entropy difference, specific heat jump, thermodynamic critical field and normalized London penetration depth were analysed within the framework of the strong-coupling Eliashberg theory of superconductivity [25] and compared with classical Bardeen– Cooper–Schrieffer (BCS) theory [26, 27] and with available experimental results. To investigate the origin of superconducting properties in transition-metal based antiperovskite MgCNi3 the Eliashberg phonon spectral function α2 FðωÞ was calculated in paper [16]. The calculations were based on the self-consistent density functional perturbation theory [28] using planewaves and ultrasoft pseudopotentials from Quantum-ESPRESSO package [29,16]. The resulting λ is 1.34, indicating that thermodynamic properties of superconducting MgCNi3 should be analysed within the framework of the Eliashberg formalism, because the materials with strong electron–phonon coupling are poorly described by the BCS theory [30–32]. The starting point of both the BCS theory and the Eliashberg formalism is the Fröhlich Hamiltonian which describes the interaction of the electrons with the phonons [33]. To obtain the Hamiltonian of the BCS theory [26] the canonical transformation should be used to elimination of the phonon degrees of freedom. Then, the traditional description of superconductivity is based on a mean-field approximation [26, 27] which can be successfully apply

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R. Szczȩśniak et al. / Solid State Communications 203 (2015) 63–68

also to other models [34]. On the other side, in order to derive the Eliashberg equations, the Fröhlich Hamiltonian should be rewrite with the use of the Nambu spinors [35], which allows us to obtain the Dyson equation [36]. Then, the set of two coupled non-linear integral Eliashberg equations is determined in a self-consistent way [32]. On the imaginary frequency axis the equations for the order parameter function ϕn ϕðiωn Þ and for the wave function renormalization factor Z n Z ðiωn Þ take the following form [25, 32]:

ϕn ¼

π M λðiωn iωm Þ μ⋆ θðωc jωm jÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∑ ϕm β m ¼ M ω2 Z 2 þ ϕ 2

ð1Þ

m

m m

and Zn ¼ 1 þ

1 π M λðiωn iωm Þ ∑ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω Z : ωn β m ¼ M ω2 Z 2 þ ϕ2 m m m m

ð2Þ

m

Here ωn stands for the Matsubara frequency: ωn π =β ð2n 1Þ, with n ¼ 0; 7 1; 7 2; …; 7 M, where M¼1100. Symbol β is associated with the Boltzmann constant kB in the following way: β ðkB T Þ 1 , where T is the temperature. The order parameter is defined as follows: Δn ϕn =Z n . The pairing kernel for the electron– phonon interaction takes the following form: Z Ωmax Ω ð3Þ λðiωn iωm Þ 2 dΩ α2 F Ω : 2 0 ðωn ωm Þ2 þ Ω The value of the maximum phonon frequency (Ωmax ) is equal to 83.54 meV [16]. Symbol μ⋆ represents the Coulomb pseudopotential, which models the depairing interaction between the electrons. The quantity θ denotes the Heaviside function, and ωc is a cut-off frequency, we chosen three times the maximum phonon frequency: ωc ¼ 3Ωmax . The physical values of the functions ϕ and Z can be determined using the Eliashberg equations in the mixed representation (defined both on the real and imaginary frequency axis) [37]. In the present work, for investigated the superconducting properties of antiperovskite MgCNi3 within experimental accuracy, we adopted the numerical methods described and tested in papers [38–40].

2. Results and discussion

the half-width of the Δm and Zm functions is decreasing when the temperature increasing. It means, that together with the growth of temperature, less successive Matsubara frequencies commit a relevant contribution to the Eliashberg equations. On the other hand, the maximum value of these frequencies is inversely proportional to the time delay (retardation) an effective electron–electron interaction in the superconducting condensate. Thus, a decrease in halfwidth of these functions is directly linked to the rise of the time of retardation. From the physical point of view, the most interesting is the dependence of studied functions for the first Matsubara frequency on the temperature, because the Δm function with a good approximation reproduces the influence of the temperature on the value of the energy gap at the Fermi surface and the Zm function presents influence of the electron-phonon interaction on the electron effective mass. The obtained results are presented in Fig. 2(B) and (D), respectively. The exact value of the order parameter and the electron effective mass can be calculated on the basis of the Eliashberg equations in the mixed representation [37]: λðω iωm Þ μ⋆ θðωc jωm jÞ π M qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ϕðωÞ ¼ ∑ ϕm β m ¼ M ω2m Z 2m þ ϕ2m Z þ1 þ iπ dω0 α2 F ðω0 Þ N ðω0 Þ þ f ðω0 ωÞ 0 K ðω; ω0 Þϕðω ω0 Þ Z þ1 dω0 α2 F ðω0 Þ N ðω0 Þ þ f ðω0 þ ωÞ þ iπ 0 ð4Þ K ðω; ω0 Þϕðω þ ω0 Þ ; and Z ð ωÞ ¼ 1 þ þ

M iπ λðω iωm Þωm ∑ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z ωβ m ¼ M ω2 Z 2 þ ϕ2 m

iπ

Z

ω

0

ω

0

m m

þ1

m

dω α F ðω Þ N ðω0 Þ þ f ðω0 ωÞ 0

2

0

K ðω; ω0 Þðω ω0 ÞZ ðω ω0 Þ Z iπ þ 1 þ dω0 α2 F ðω0 Þ N ðω0 Þ þ f ðω0 þ ωÞ K ðω; ω0 Þðω þ ω0 ÞZ ðω þ ω0 Þ;

Due to the lack of knowledge of an appropriate critical value for the Coulomb pseudopotential (μ⋆ C ), the accurate prediction of the thermodynamic properties of the superconducting state is difficult. For weak coupling materials, an empirical value of μ⋆ C ¼ 0:1 is often adopted [41]. However, in many materials μ⋆ C is quite complex and outside of 0.1 are necessary for explaining experimental results. In a simple way, we can determine the approximate critical value of Coulomb pseudopotential using an analytical formula μ⋆ C ¼ NðE f Þ=ð1 þ NðE f ÞÞ [42], where NðEf Þ is the density of states at the Fermi level. For MgCNi3, NðEf Þ ¼ 5:34 states/ eV [43] and μ⋆ C is equal to 0.22. However, in the Eliashberg theory, the exact value of μ⋆ C is defined as the Coulomb pseudopotential at which the order parameter vanishes at critical temperature. In particular, on the basis of the expression: Δm ¼ 1 μ⋆ ¼ μ⋆ ¼ 0 at C ⋆ TC ¼ 8 K, we obtained μC ¼ 0:29 for MgCNi3 . In Fig. 1(A) and (B), the dependence of the order parameter on the number m for the selected values of the Coulomb pseudopotential and the full dependence of Δm ¼ 1 ðμ⋆ Þ are presented, respectively. The high critical value of the Coulomb pseudopotential suggests that the critical temperature of MgCNi3 cannot be correctly evaluated by means of the analytical Allen-Dynes and McMillan [44, 45]. In particular, these equations give the following results: 4.37 K and 4.12 K, respectively. In Fig. 2(A) and (C), we have plotted the dependence of the order parameter and the wave function renormalization factor on the number m for selected values of temperature. It can be noticed that

ð5Þ

where: 1 K ðω; ω0 Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi: 2 2 2 ðω þ ω0 Þ Z ðω þ ω0 Þ ϕ ðω þ ω0 Þ

ð6Þ

The symbols N ðωÞ and f ðωÞ denote the statistical function of the bosons and the fermions, respectively. The solutions of the Eliashberg equations on the imaginary frequency axis (iωn) are used as the input parameters to the Eliashberg equations in the mixed representation. The form of the order parameter on the real frequency axis for selected values of temperature is plotted in Fig. 3(A). On this basis and from the relation: Δð0Þ ¼ Re Δ ω ¼ Δð0Þ ; we can conclude that the maximum value of energy gap at the Fermi surface (2Δð0Þ ¼ 2ΔðT 0 Þ) is equal to 2.89 meV, where T0 denotes a minimum value of temperature for which convergence of the solutions was obtained. In our case, solutions are stable from T 0 ¼ 1 K to TC. Temperature dependence of the energy gapq can be by the simple analytical ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi modelled α formula: 2ΔðT Þ ¼ 2ΔðT 0 Þ 1 T=T C , where the parameter α is equal to 3.6. In Fig. 3(B), the dependence of the normalized energy gap on the temperature is presented. Moreover, the obtained results are compared with prediction of the BCS theory. Next, in Fig. 3(C) the real and imaginary part of the wave function renormalization factor on the real frequency axis for selected values of temperature is presented. On this basis, the dependence of the ratio mne =me on the temperature was determined using the


65

Fig. 1. (A) The order parameter on the imaginary frequency axis for the selected values of the Coulomb pseudopotential. (B) The full dependence of the maximum value of the order parameter on the Coulomb pseudopotential.

Fig. 2. (A) The order parameter on the imaginary frequency axis for the selected temperatures. (B) The dependence of the maximum value of the order parameter on the temperature. (C) The wave function renormalization factor on the imaginary axis for the selected temperatures. (D) The dependence of the maximum value of the wave function renormalization factor on the temperature.

expression: mne ¼ Re½Z ð0Þme , where the symbols mne and me denote the electron effective mass and electron band mass, respectively. The obtained results are presented in Fig. 3(D). We shown that the electron effective mass is large in the entire range, in which the superconducting state exists, and reaches a maximum value equal to 2.55 at TC. Based on the imaginary frequency axis solutions of the Eliashberg equations the free energy difference between the superconducting and the normal state (ΔF F S F N ) can be calculated using the following formula: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ΔFðTÞ 2π M ¼ ∑ ω2n þ Δ2n jωn j ρð0Þ β n¼1 0 1 jωn j C B @Z Sn Z N n qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA: ω2n þ Δ2n

ð7Þ

where ρð0Þ is the value of the electron density of states at the Fermi energy level. Symbols ZSn and ZN n denote the wave function

renormalization factors state, respectively. The function determine the ducting and the normal d ΔF=ρð0Þ ΔSðTÞ ¼ dðkB T Þ kB ρð0Þ

for the superconducting and the normal first and the second derivative of ΔFðTÞ entropy difference between the superconstate ðΔS SS SN Þ : ð8Þ

and the specific heat difference between the superconducting and the normal state ðΔC C S C N Þ :

2 ΔCðTÞ 1 d ΔF=ρð0Þ ¼ : kB ρð0Þ β dðkB T Þ2

ð9Þ

The obtained results are presented in Fig. 4. It can be noticed that the values of ΔF quickly increase with the temperature. From the physical point of view, the negative values of ΔFðTÞ determines the thermodynamic stability of the superconducting state in the temperature range from T0 to TC. It is also connected with the entropy of

66


Fig. 3. The real and imaginary part of the order parameter (A) and the wave function renormalization factor (C) on the real frequency axis for selected values of temperature. The dependence of the normalized energy gap (B) and the ratio mne =me (D) on the temperature.

the superconducting state, which is lower than for the normal state owing to the formation of the Cooper pairs [46] (see Fig. 4(B)). The temperature dependence on the specific heat for the superconducting and the normal state is presented in Fig. 4(C). We can note that with the increase of temperature the specific heat of the superconducting state grows strongly, reaching its maximum at the critical temperature. The characteristic specific heat jump at TC is marked with the vertical line. It is worth mentioning that the specific heat in normal state is calculated from: C N ðTÞ ¼ γ =β, where the Sommerfeld constant (γ) has the form: γ ð2=3Þπ 2 1 þ λ kB ρð0Þ. In the next step, the thermodynamic critical field HC(T) and the deviation function D(T) were obtained using the following equations: H ðTÞ pCffiffiffiffiffiffiffiffiffi ¼ ρð0Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8π ΔF=ρð0Þ

ð10Þ

and

" 2 # H c ðT Þ T DðTÞ ¼ 1 : H c ð0Þ Tc

ð11Þ

pffiffiffiffiffiffiffiffiffi Fig. 5(A) presents the temperature dependencies of H C = ρð0Þ. It can be seen that the thermodynamic critical field decreases with the increasing temperature, taking the zero value at TC. Let us notice that the maximum values of the considered function H C ð0Þ H C ðT 0 Þ is equal to 7.2 meV. In Fig. 5(B), the deviation of the thermodynamic critical field function of MgCNi3 is compared with prediction of the BCS theory. The obtained results confirms that MgCNi3 is non-BCS superconductor. Moreover, in Fig. 5(C) we supplement our results with the calculated London penetration depth (λL ) as a function of the


67

Fig. 4. (A) The free energy difference, (B) specific heat, and (C) entropy difference as a function of the temperature for antiperovskite MgCNi3.

Fig. 5. Temperature dependence of (A) the thermodynamic critical field, (B) deviation of the thermodynamic critical field and (C) the normalized London penetration depth.

temperature:

the experimental measurements are very divergent: ½RΔ exp A h3:2; 4:6i [11,13,22]. From the physical point of view, the differences between the BCS results and the data obtained for MgCNi3 are connected with the existence of the strong-coupling and retardation effects in the analysed compound. In the Eliashberg formalism these effects are described by the ratio kB T C =ωln , where the logarithmic phonon frequency (ωln ) is defined by " Z # α2 F Ω 2 Ωmax ωln exp dΩ ln Ω : ð13Þ

1 e2 v2F

ρð0Þλ

2 L ðT Þ

¼

4π M ∑ 3β n ¼1

h

Δ2n

i 2 3=2

Z Sn ω2n þ Δn

;

ð12Þ

where e is the electron charge and vF is the Fermi velocity. We find that the temperature dependence of normalized λ2L also presents a non-conventional behaviour with marked curvature with respect to the BCS curve. On the basis of thermodynamic functions determined in this paper, we estimated the dimensionless ratios: RΔ 2Δð0Þ=kB T C , RC ΔC ðT C Þ=C N ðT C Þ and RH T C C N ðT C Þ=H 2C ð0Þ. The above ratios play very important role in theory of superconductivity because they can be determined in experimental measurements and compared with theoretical predictions. We notice that in the framework of the BCS theory, the above parameters take the universal values: ½RΔ BCS ¼ 3:53, ½RC BCS ¼ 1:43 and ½RH BCS ¼ 0:168 [26,27]. In the case of antiperovskite MgCNi3, the obtained results significantly deviate from the predictions of the BCS theory. In particular RΔ ¼ 4:19, RC ¼ 2:27 and RH ¼ 0:141. Note that the value of RC is in excellent agreement with the results of previous experimental studies where ½RC exp ¼ 2:3 [13]. In the case of RΔ

λ

0

Ω

For MgCNi3, ωln ¼ 7:99 meV and the discussed ratio is equal to 0.086, while in the weak-coupling limit kB T C =ωln -0. 3. Conclusion In the present paper we explore the thermodynamic properties of the superconducting state induced in MgCNi3 below the critical temperature (TC ¼8 K). In the first step we determined the critical value of the Coulomb pseudopotential which must be fitted in the framework of the Eliashberg formalism in order to reproduce the experimental data. Here we obtained a very high value, μ⋆ C ¼ 0:29.

68


In the next step, to finding the dimensionless parameters RΔ , RH and RC the energy gap, thermodynamic critical field, and specific heat were calculated. The values obtained in the case of transitionmetal based antiperovskite MgCNi3 are clearly differ from the predictions of the BCS theory, in particular RΔ ¼ 4:19, RC ¼ 2:27 and RH ¼ 0:141. It was found that this is caused by the strongcoupling and retardation effects existing in antiperovskite MgCNi3. These results are expected to stimulate further exploration and discovery of new antiperovskite-type materials like MgCNi3. Acknowledgements A.P. Durajski would like to express his gratitude to Lena Durajska, for her assistance in the numerical calculations. All calculations are based on the Eliashberg function sent to us by Wenjian Lu to whom we are very thankful. References [1] K. Kamishima, T. Goto, H. Nakagawa, N. Miura, M. Ohashi, N. Mori, T. Sasaki, T. Kanomata, Phys. Rev. B 63 (2000) 024426. [2] K. Asano, K. Koyama, K. Takenaka, Appl. Phys. Lett. 92 (2008) 161909. [3] T. Tohei, H. Wada, T. Kanomata, J. Appl. Phys. 94 (2003) 1800. [4] K. Takenaka, K. Asano, M. Misawa, H. Takagi, J. Appl. Phys. 92 (2008) 011927. [5] P. Tong, Y.P. Sun, X.B. Zhu, W.H. Song, Phys. Rev. B 77 (2006) 224416. [6] E.O. Chi, W.S. Kim, N.H. Hur, Solid State Commun. 120 (2001) 307. [7] T. He, Q. Huang, A.P. Ramirez, Y. Wang, K.A. Regan, N. Rogado, M.A. Hayward, M.K. Haas, J.S. Slusky, K. Inumara, et al., Nature 411 (2001) 54. [8] B. Wiendlocha, J. Tobola, S. Kaprzyk, Phys. Rev. B 73 (2006) 134522. [9] B. Wiendlocha, J. Tobola, S. Kaprzyk, D. Fruchart, J. Alloy. Compd. 442 (2007) 289. [10] J.-Y. Lin, P.L. Ho, H.L. Huang, P.H. Lin, Y.-L. Zhang, R.-C. Yu, C.-Q. Jin, H.D. Yang, Phys. Rev. B 67 (2003) 052501. [11] P.M. Singer, T. Imai, T. He, M.A. Hayward, R.J. Cava, Phys. Rev. Lett. 87 (2001) 257601. [12] R. Prozorov, R.W. Giannetta, Supercond. Sci. Technol. 19 (2006) R41. [13] Z.Q. Mao, M.M. Rosario, K.D. Nelson, K. Wu, I.G. Deac, P. Schiffer, Y. Liu, T. He, K.A. Regan, R.J. Cava, Phys. Rev. B 67 (2003) 094502. [14] R. Prozorov, A. Snezhko, T. He, R.J. Cava, Phys. Rev. B 68 (2003) 180502 (R). [15] A.Y. Ignatov, S.Y. Savrasov, T.A. Tyson, Phys. Rev. B 68 (2003) 220504 (R).

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