Description of the thermodynamic properties of Bi H 5 and Bi H 6 superconductors beyond the mean-field approximation
M. W. Jarosik, E. A. Drzazga, I. A. Domagalska, K. M. Szczęśniak, U. Stępień
aa r X i v : . [ c ond - m a t . s up r- c on ] M a r Description of the thermodynamic properties of
BiH and BiH superconductors beyond the mean-field approximation M. W. Jarosik (1) , E. A. Drzazga (1) , ∗ I. A. Domagalska (1) , K. M. Szcz¸e´sniak (2) , and U. St¸epie´n (1) (1)
Institute of Physics, Cz¸estochowa University of Technology,Ave. Armii Krajowej 19, 42-200 Cz¸estochowa, Poland and (2)
Ul. Pomorska 37/55, Zawiercie, 42-400, Poland (Dated: September 11, 2018)The ab initio calculations suggest (Y. Ma et al. ), that the high-pressure ( p = 200 GPa) supercon-ducting state in BiH and BiH compounds characterizes with a high value of the critical tempera-ture ( T C ∼
100 K). Due to the large value of the electron-phonon coupling constant ( λ ∼ . and BiH have been determinedbeyond the mean-field approximation - in the framework of the Eliashberg equations formalism.We have calculated the dependence of the order parameter (∆) and the wave function renormal-ization factor on the temperature. Then we have estimated the free energy difference between thesuperconducting state and the normal state, the thermodynamic critical field ( H C ) and the specificheat of the superconducting state ( C S ) and the normal state ( C N ). The values of the dimen-sionless ratios R ∆ = 2∆ (0) /k B T C , R C = ∆ C ( T C ) /C N ( T C ) and R H = T C C N ( T C ) /H C (0) areequal R ∆ BiH5 = 4 .
17 and R ∆ BiH6 = 4 . R C BiH5 = 2 .
54 and R C BiH6 = 2 . R H BiH5 = 0 .
146 and R H BiH6 = 0 .
146 respectively.
PACS numbers: 74.20.Fg, 74.25.Bt, 74.62.Fj
Keywords:
Hydrogen-rich materials, High-pressureeffect, Electron-phonon superconductivity, Eliashbergtheory, Thermodynamic propertiesIn 1935 Wigner and Huntington noted that hydrogenaffected by the high pressure ( ∼
25 GPa) should tran-sit into a metallic state [1]. So far, it has been possibleto experimentally demonstrate the metallization of liquidhydrogen for the pressure at 140 GPa, but the tempera-ture was up to 3000 K [2]. In 2017 the information wasspread that hydrogen metallization was observed at roomtemperature under the pressure at 495 GPa [3]. Unfor-tunately, this result has not been confirmed again.In 1968 Ashcroft noted that the metallic hydrogenshould be a high-temperature superconductor [4]. Hegave the following arguments: first of all, hydrogen hasthe lowest atomic mass of the nucleus, which implies theDebye frequency ( ω D ) of the crystal lattice of solidifiedhydrogen to be high; secondly, there are no electrons onthe internal shells in the hydrogen atom, which favors astrong electron-phonon coupling. In the following years,thanks to the rapid development of computer technology,more detailed calculations were carried out. The hydro-gen metallization pressure was estimated to be around400 GPa [4, 5]. It has been suggested that in the molec-ular hydrogen phase, within the range of pressures from400-500 GPa there should be a sharp rise in the criticaltemperature from approximately 80 K to about 350 K[6–9]. Above the pressure at 500 GPa the numerical cal-culations suggest that the metallic molecular hydrogenshould dissociate into the atomic phase [6], whereas the ∗ Electronic address: [email protected] critical temperature value should be within the limit from300 K to 470 K [10]. In the literature on the subjectthere are also works analyzing the superconducting stateof hydrogen for extremely high pressures (up to 3 . T C was foreseen at 630 Kfor p = 2 TPa.Due to the very high value of the hydrogen’s met-allization pressure, in 2004 Ashcroft suggested to tryto look for a high temperature superconducting statenot in pure hydrogen but in hydrogenated compounds[15]. Induction of high temperature superconductingstate in hydrogenated compounds should be supportedby chemical precompression caused by heavier elements.The real breakthrough occurred only in December 2014,when Drozdov, Eremets, and Troyan have experimentallydemonstrated the existence of a high-temperature super-conducting state in H S ( T C = 203 K for p = 155 GPa)[16, 17].In the presented work, we have determined the thermo-dynamic properties of the superconducting state in hy-drogenated compounds BiH and BiH . Due to the highpredicted value of the electron-phonon coupling constant( λ BiH = 1 .
236 and λ BiH = 1 .
259 [18]) we performed thecalculations for both cases as a part of Eliashberg formal-ism [19]. A detailed description of Eliashberg equationsand the computational methods used has been presentedin the works [20–26]. The spectral function ( α F (Ω))used in numerical calculations has been determined byY. Ma et al. with an use of the density functional the-ory [18]. Additionally, 1100 Matsubara frequencies weretaken into account, and the value of Coulomb pseudo-potential ( µ ⋆ ) equals 0 .
1. was adopted. The stabilityof the Eliashberg equations’ solutions was obtained for atemperature higher or equal to T = 20 K.In Fig. 1 (a) we have plotted a temperature dependence T=50K T=70K T=90K T=101K T=102.75 K T=T C N S / N N BiH (b) [meV] N S / N N T=50K T=70K T=90K T=98K T=99.75K T=T C BiH (c) BiH BiH T [K] o r de r pa r a m e t e r [ m e V ] (a) FIG. 1: (a) The order parameter as a function of the temper-ature obtained in the framework of the Eliashberg equationson the imaginary axis (lines). Symbols represent the valuesof the order parameter obtained on the basis of the equation(1). (b)-(c) Renormalized density of states for the selectedvalues of temperature. of the order parameter. It was obtained as part of theimaginary axis formalism and using the equation:∆( T ) = Re[∆( ω = ∆( T ))] , (1)where function of the order parameter on the real axis(∆ ( ω )) was obtained using the method of the analyticalextension, which has been described in detail in the paper[27]. We have obtained the following critical temperaturevalues: T C BiH5 = 103 K and T C BiH6 = 100 K. It meansthat BiH and BiH compounds should belong to thefamily of high-temperature superconductors.Let us consider the order parameter ∆( T ) = ∆ (0)estimated on the basis of the equation (1). Hence thedimensionless ratio R ∆ = 2∆ (0) /k B T C equals 4 .
17 and4 .
2, for BiH and BiH . respectively. Compared with thepredictions of the BCS theory, these are very high val-ues, because R ∆ BCS = 3 .
53 [28, 29]. Please note thatthe above result comes from the existence of the sig-nificant retardation and strong-coupling effects. Theycan be characterized by the parameter r = k B T C /ω ln ,where the quantity ω ln = exp h λ R Ω max d Ω α F (Ω)Ω ln (Ω) i denotes the logarithmic phonon frequency. For BiH andBiH compounds it was obtained r = 0 .
11. In the BCSlimit we have r = 0 [30].Basing on the function of the order parameter on thereal axis, one can determine the renormalized electronicdensity of states in the superconducting phase: N S ( ω ) N N ( ω ) = Re " ( ω − i Γ) p ( ω − i Γ) − (∆( ω )) , (2)where we have assumed that the pair breaking parame-ter equals Γ = 0 .
15 meV. We have plotted the obtained results in the drawings Fig. 1 (b) and (c). Note thatthe characteristic maxima of the determined curves areformed in points ω = ± ∆. On their basis, one can ex-perimentally analyze the evolution of the energy gap de-pending on the temperature.The exact form of the order parameter on the real axisis presented in the figures Fig. 2 (a)-(f). When analyzingthe case T = T , it can be seen that in the range of thelower frequencies ( ω < ω d = 85 meV) non-zero is only thereal part of the order parameter. This means no damp-ing processes that are modeled by a function Im [∆ ( ω )].It is also worth noting that as the temperature rises, ω d decreases significantly. Additionally, for illustrative pur-poses in drawings (g) and (h) we have plotted the valuesof the order parameter on the complex plane. In bothanalyzed cases, the values of ∆( ω ) on the complex planeform characteristic spirals, whose radii decrease with in-creasing temperature.The maximum value of the wave function renormal-ization factor on the imaginary axis relatively preciselydetermines the ratio of the effective mass of the electron( m ⋆e ) to the electron band mass ( m e ). However, the ex-act value of m ⋆e /m e should be determined on the basis ofthe following formula: m ⋆e m e = Z ( ω = 0) . (3)Graphs of the ratio of the electron effective mass toelectron band mass in the temperature range from 0 to T C , were collected in the drawing Fig. 3. It can be seenthat in both cases the effective mass takes values slightlyhigher than two. This result proves a strong renormal-ization of the electron band mass by the electron-phononinteraction. Additionally, the ratio m ⋆e /m e is slightly de-pendent on the temperature, and its value at the criticaltemperature can be calculated with good accuracy us-ing the formula: [ m ⋆e /m e ] T = T C = 1 + λ [30]. Let usnote that for T = T C the numerical calculations give[ m ⋆e /m e ] BiH = 2 . m ⋆e /m e ] BiH = 2 .
33, which is ina good agreement with the analytical result.Basing on the solutions of the Eliashberg equations onthe imaginary axis (∆ n and Z n ) we have calculated thefree energy difference between the superconducting stateand the normal state:∆ Fρ (0) = − πk B T M X n =1 (cid:16)p ω n + ∆ n − | ω n | (cid:17) (4) × ( Z Sn − Z Nn | ω n | p ω n + ∆ n ) , where ρ (0) denotes the value of the electron density ofstates at the Fermi level, ω n is the fermion Matsubarafrequency, and symbols S and N refer respectively to thesuperconducting and normal (metallic) state. On thisbasis, the thermodynamic critical field can be estimated: H C p ρ (0) = p − π [∆ F/ρ (0)] (5) -50 0 50-50050-50 0 50-500500 150 300-50050 0 150 300-50050 0 150 300-500500 150 300-50050 0 150 300-50050 0 150 300-50050
T=70K T=80K T=90K T C =100K Re( ) [meV]
BiH (h) T=100K T C =103K I m [] [ m e V ] T=70K T=80K T=90K I m [] [ m e V ] BiH (g) Re[ ] meV Im[ ] meV m e V Re( ) [meV]
20 K(a)
Re( ) [meV]
60 K(b)
Re( ) [meV]
100 K(c) m e V Re( ) [meV]
20 K(d)
Re( ) [meV]
60 K(e)
Re( ) [meV]
95 K(f)
FIG. 2: (a)-(f) The real and imaginary parts of the order parameter on the real axis for the selected temperature values. (g)and (h) The values of the order parameter on the complex plane. The curves correspond to the frequency range from 0 to ω D .
20 40 60 80 1002.102.152.202.252.302.352.40
BiH BiH m * e / m e T [K]
FIG. 3: The ratio of the electron effective mass to the electronband mass determined for BiH and BiH compounds as apart of the Eliashberg formalism of the imaginary axis (lines)and with the aid of formula (3) (stars). and the difference in specific heat between the supercon-ducting state and the normal state (∆ C = C S − C N ):∆ C ( T ) k B ρ (0) = − β d [∆ F/ρ (0)] d ( k B T ) , (6)wherein the specific heat of the normal state can be de-termined on the basis of the formula: C N ( T ) /k B ρ (0) = γk B T , where γ = π (1 + λ ) denotes the Sommerfeldconstant.Obtained results were collected in the drawing Fig.4. It can be seen that the calculated thermodynamic F / (( )) / [ m e V ] H C / (( )) / [ m e V ] T [K]
BiH BiH (a) (b) T [K] C S /k B (0) [meV] - (BiH ) C S /k B (0) [meV] - (BiH ) C N /k B (0) FIG. 4: (a) The free energy as a function of the tempera-ture (lower panel) and the thermodynamic critical field (up-per panel). (b) The specific heat of the superconducting stateand the normal state as a function of the temperature. functions assume similar values for BiH and BiH , com-pounds, which results from similar values of the electron-phonon coupling constants. Additionally, it should beemphasized that the values of the thermodynamic crit-ical field and specific heat cannot be correctly calcu-lated within the framework of the mean-field BCS the-ory. To find out about this, let us estimate the di-mensionless parameters R C = ∆ C ( T C ) /C N ( T C ) and R H = T C C N ( T C ) /H C (0). For BiH compound we havereceived R C = 2 .
54 i R H = 0 . com-pound we have R C = 2 .
58 and R H = 0 . .
43 and 0 .