Superconductivity of Bi-III phase of elemental Bismuth: insights from Muon-Spin Rotation and Density Functional Theory
Rustem Khasanov, Hubertus Luetkens, Elvezio Morenzoni, Gediminas Simutis, Stephan Schönecker, Andreas Östlin, Liviu Chioncel, Alex Amato
SSuperconductivity of Bi-III phase of elemental Bismuth:insights from Muon-Spin Rotation and Density Functional Theory
Rustem Khasanov, ∗ Hubertus Luetkens, Elvezio Morenzoni, GediminasSimutis, Stephan Sch¨onecker, Andreas ¨Ostlin, Liviu Chioncel, and Alex Amato Laboratory for Muon Spin Spectroscopy, Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland Applied Materials Physics, Department of Materials Science and Engineering,KTH Royal Institute of Technology, SE-10044 Stockholm, Sweden Augsburg Center for Innovative Technologies, and Center for Electronic Correlations and Magnetism,Theoretical Physics III, Institute of Physics, University of Augsburg, D-86135 Augsburg, Germany
Using muon-spin rotation the pressure-induced superconductivity in the Bi-III phase of elementalBismuth (transition temperature T c (cid:39) .
05 K) was investigated. The Ginzburg-Landau parameter κ = λ/ξ = 30(6) ( λ is the magnetic penetration depth, ξ is the coherence length) was estimatedwhich is the highest among single element superconductors. The temperature dependence of thesuperconducting energy gap [∆( T )] reconstructed from λ − ( T ) deviates from the weak-coupledBCS prediction. The coupling strength 2∆ /k B T c (cid:39) .
34 was estimated thus implying that Bi-IIIstays within the strong coupling regime. The Density Functional Theory calculations suggest thatsuperconductivity in Bi-III could be described within the Eliashberg approach with the characteristicphonon frequency ω ln (cid:39) . Following the intense research on superconductivity incuprates and Fe-based superconductors, simpler mate-rials such as elementary or binary compounds have at-tracted renewed interest. Recent results have shown thatin spite of their simple composition they represent a play-ground for the discovery of new phenomena and uncon-ventional characteristics. Binary compounds were re-ported to superconduct at surprisingly high critical tem-peratures ( T c ’s) reaching (cid:39)
40 K for MgB , as well asthe highest ever obtained T c of ∼
200 K in SH at the(extreme) pressure of p (cid:39)
200 GPa. The basic elementswith high T c ’s include Li ( T c ≈ −
20 K at 30 GPa),
Ca ( T c ≈ −
29 K at 220 GPa), Sc and Y ( T c ≈
20 Knear 100 GPa),
V ( T c ≈
17 K at 120 GPa), andS ( T c ≈
17 K at 220 GPa). Such high transition tem-peratures of basic elements indicate the importance ofpressure as a tuning parameter of superconducting prop-erties and its role in unraveling the intrinsic propertiesof the electronic system.Among single element superconductors the Bi-IIIphase of elemental Bismuth is one of the most inter-esting system to study. Following Refs. 12–19, Bismuthconverts into a phase Bi-III, exhibiting superconductiv-ity at T c (cid:39) (cid:39) . It consists of two interpene-trating structures, with ”guest“ atoms forming chainswithin cylindrical cavities in the ”host“ lattice. In the ab − plane, the unit cells of guest and host match, whilethe ratio along the c − axis lattice parameters is incom-mensurate: c host /c guest = 1 . Remarkably, thisincommensurate structure may give rise to an additionalacoustic mode, in the phonon spectrum, at very low fre-quencies arising from the sliding of one structure throughthe other, a process which has almost no energy cost.
Following Refs. 18,19 in Bi-III such mode was suggested to be responsible for a very high electron-phonon cou-pling strength and gave rise to the enhanced transitiontemperature and the high upper critical field ( H c2 ). Itis worth to note, however, that the number of physicalquantities studied so far for Bi-III phase of elemental Bis-muth were mostly limited to T c and H c2 , which maynot be enough to draw unambiguous conclusion on typeof the superconducting mechanism.In this paper, we report on the results of experimen-tal and theoretical studies of Bi-III phase of elementalBismuth. The measurements of the temperature de-pendence of the magnetic field penetration depth ( λ )were performed in the muon-spin rotation ( µ SR) exper-iments ( p (cid:39) .
72 GPa). The magnetic field distribu-tion in the sample below T c reflects the formation ofa vortex lattice thus confirming that Bi-III is a type-II superconductor. A zero-temperature value of themagnetic penetration depth λ (0) = 301(4) nm was deter-mined. With the coherence length ξ = 10(2) nm takenfrom the upper critical measurements, a Ginzburg-Landau parameter κ = 30(6) ( κ = λ/ξ ) was obtainedwhich turns out to be the highest among single elementsuperconductors. Density Functional Theory (DFT) cal-culations were used to determine the equilibrium lat-tice structure parameters of Bi-III under pressure. TheDFT results of electronic structure, phonons, and Fermisurfaces, are used to analyze different possible scenar-ios for the onset of superconductivity in Bi-III. Withinthe framework of the Eliashberg theory, for the givenspectral density of Bi-III, the most effective phonon en-ergy ω ln (cid:39) . a r X i v : . [ c ond - m a t . s up r- c on ] F e b (cid:0)(cid:2) (cid:3)(cid:2) (cid:4)(cid:2) (cid:5)(cid:2) (cid:2)(cid:6)(cid:2) (cid:2)(cid:6)(cid:7) (cid:8)(cid:6)(cid:2)(cid:2)(cid:6)(cid:2)(cid:2)(cid:6)(cid:7)(cid:8)(cid:6)(cid:2) (cid:2)(cid:6)(cid:3) (cid:2)(cid:6)(cid:4) (cid:2)(cid:6)(cid:5) (cid:8)(cid:6)(cid:2)(cid:2)(cid:6)(cid:2)(cid:2)(cid:6)(cid:7)(cid:8)(cid:6)(cid:2) (cid:0) (cid:2)(cid:3) (cid:4) (cid:5) (cid:6) (cid:4) (cid:7) (cid:8) (cid:4) (cid:9)(cid:10) (cid:11) (cid:12) (cid:2) (cid:4) (cid:13) (cid:9) (cid:14) (cid:5) (cid:2)(cid:10) (cid:0)(cid:9)(cid:10)(cid:11)(cid:12)(cid:13) (cid:5)(cid:9)(cid:14)(cid:2)(cid:6)(cid:0)(cid:7)(cid:9)(cid:14) (cid:0) (cid:2) (cid:2) (cid:15)(cid:16) (cid:17)(cid:18)(cid:19)(cid:11)(cid:20)(cid:21)(cid:22)(cid:23)(cid:24)(cid:24)(cid:24)(cid:9)(cid:12)(cid:25)(cid:9)(cid:26)(cid:2)(cid:9)(cid:11)(cid:12)(cid:10)(cid:15)(cid:13) (cid:0) (cid:15) (cid:0) (cid:16) (cid:17) (cid:0) (cid:15) (cid:18) (cid:16) (cid:9)(cid:12)(cid:25)(cid:9)(cid:8)(cid:2)(cid:9)(cid:11)(cid:12)(cid:9)(cid:12)(cid:25)(cid:9)(cid:26)(cid:2)(cid:9)(cid:11)(cid:12) (cid:3)(cid:27)(cid:3) (cid:4) (cid:0)(cid:2)(cid:27)(cid:28) (cid:17) (cid:3) (cid:29) (cid:30)(cid:3)(cid:6)(cid:26)(cid:3)(cid:10)(cid:8)(cid:13) (cid:10)(cid:31)(cid:13) (cid:9)(cid:12)(cid:25)(cid:9)(cid:8)(cid:2)(cid:9)(cid:11)(cid:12)(cid:9)(cid:12)(cid:25)(cid:9)(cid:26)(cid:2)(cid:9)(cid:11)(cid:12)(cid:9) !(cid:6)(cid:9)(cid:0)(cid:9)" (cid:2)(cid:6), (cid:17)(cid:18) (cid:2)(cid:6)(cid:26) *(cid:9)-&% (cid:2) (cid:15) (cid:0) (cid:16) (cid:7)(cid:17)(cid:7) (cid:2) (cid:3)(cid:27)(cid:3) (cid:29) (cid:10)(cid:29)(cid:13) FIG. 1: (a) Fourier transform of TF- μ SR time spectra ( μ H ap = 30 mT) reflecting the internal field distribution P ( B ) inthe Bi-III sample above ( T = 8 . T = 0 .
25 K) the superconducting transition temperature. The strongerbroadening and the asymmetric field distribution in the superconducting state is caused by the formation of the vortex lattice.The solid lines are fits (see the Supplemental Part, Ref. 25). (b) The temperature dependence of the superfluid density ρ ( T ) /ρ (0) = λ − ( T ) /λ − (0) of Bi-III sample obtained in TF- μ SR experiments at the applied field μ H ap = 10 mT (bluetriangles) and 30 mT (red circles), respectively. The solid line is the fit of Eq. 1 with the gap described by Eq. 2. (c) Thetemperature dependence of the superconducting gap reconstructed from the superfluid density data. The solid line is the fit ofEq. 2 to the Δ( T ) data. The dashed line is the weak-coupled BCS temperature behavior, Ref. 26. The + and × symbols referto Δ( T ) / Δ points for the superconducting Sn and Pb . Bi . obtained in tunneling experiments (after Townsend and Sutton,Ref. 27 and Adler and Chen, Ref. 28). ( (cid:2) . (cid:2)
12 mm height) and placedinside a double-wall piston-cylinder pressure cell madeof MP35/NiCrAl alloy. The construction of the pressurecell is similar to the one described in Refs. 29,30.The ac susceptibility measurements at p (cid:2) .
72 GPareveal the presence of a sharp transition with T c (cid:2) .
05 K(see the Supplemental Part, Ref. 25). No indication ofthe transition at T ∼ . e.g. , in Ref. 17 for pressures up to 2 . μ SR experiments were car-ried out at the μ E1 beam line by using the dedi-cated GPD (General Purpose Decay) spectrometer (PaulScherrer Institute, Switzerland). The details of TF- μ SRunder pressure experiments are provided in the Supple-mental Part, Ref. 25, and in the Ref. 29. Figure 1 (a)shows the Fourier transform of TF- μ SR time spectra (thepressure cell background subtracted, see the Supplemen-tal Part, Ref. 25) reflecting the internal field distribution P ( B ) in the Bi-III sample. The asymmetric P ( B ) dis-tribution at T (cid:2) .
25 K possesses the basic features ex-pected for an ordered vortex lattice, namely: the cutoff atlow fields, the peak shifted below μ H ap and the long tailtowards the high field direction (see e.g. Refs. 31,32 andreferences therein). Our experiments confirm, therefore,that the superconductivity of Bi-III is of a type-II.
The temperature dependence of the superfluid density[ ρ s ( T ) /ρ s ( T = 0) = λ − ( T ) /λ − ( T = 0), Fig. 1 (b)]was further obtained from the second central mo-ment of the magnetic field distribution ( (cid:4) Δ B (cid:5) ). The superconducting part of the second moment, whichfor extreme type-II superconductors becomes a di-rect measure of the magnetic penetration depth λ ,was calculated via: (cid:4) Δ B (cid:5) sc = (cid:4) Δ B (cid:5) − σ /γ μ =0 . /λ . Here σ nm is the contribution of nu-clear magnetic moments measured at T > T c and Φ =2 .
068 10 − Wb is the magnetic flux quantum. From themeasured γ μ (cid:4) Δ B (cid:5) / sc (0 .
25 K ,
10 mT) = 1 . μ s − and γ μ (cid:4) Δ B (cid:5) / sc (0 .
25 K ,
30 mT) = 1 . μ s − thevalue of the magnetic penetration depth was found tobe λ (0 .
25 K) = 301(4) nm. With the coherence length, ξ = 10(2) nm, estimated from the value of the upper crit-ical field H c2 ( T = 0) (cid:2) . − . this translatesinto a Ginzburg-Landau parameter κ = λ/ξ (cid:2) κ valuereported to date for single element superconducting ma-terials. This indicates that Bi-III is a representative ofextreme type-II superconductors with κ (cid:6) s − wavesymmetry and using the following functional form: ρ ( T ) ρ (0) = 1 + 2 (cid:2) ∞ Δ( T ) (cid:3) ∂f∂E (cid:4) E dEdϕ (cid:5) E − Δ( T ) , (1)where f = [1 + exp( E/k B T )] − is the Fermi function.The temperature dependence of the superconducting gapΔ( T ) was obtained by solving the nonlinear Eq. 1 forevery temperature point. The resulting Δ( T ) / Δ val-ues for Bi-III are presented in Figure 1 (c). Here Δ is
FIG. 1: (a) Fourier transform of TF- µ SR time spectra ( µ H ap = 30 mT) reflecting the internal field distribution P ( B ) inthe Bi-III sample above ( T = 8 . T = 0 .
25 K) the superconducting transition temperature. The strongerbroadening and the asymmetric field distribution in the superconducting state is caused by the formation of the vortex lattice.The solid lines are fits (see the Supplemental Part, Ref. 25). (b) The temperature dependence of the superfluid density ρ ( T ) /ρ (0) = λ − ( T ) /λ − (0) of Bi-III sample obtained in TF- µ SR experiments at the applied field µ H ap = 10 mT (bluetriangles) and 30 mT (red circles), respectively. The solid line is the fit of Eq. 1 with the gap described by Eq. 2. (c) Thetemperature dependence of the superconducting gap reconstructed from the superfluid density data. The solid line is the fit ofEq. 2 to the ∆( T ) data. The dashed line is the weak-coupled BCS temperature behavior, Ref. 26. The + and × symbols referto ∆( T ) / ∆ points for the superconducting Sn and Pb . Bi . obtained in tunneling experiments (after Townsend and Sutton,Ref. 27 and Adler and Chen, Ref. 28). ( (cid:39) . (cid:39)
12 mm height) and placedinside a double-wall piston-cylinder pressure cell madeof MP35/NiCrAl alloy. The construction of the pressurecell is similar to the one described in Refs. 29,30.The ac susceptibility measurements at p (cid:39) .
72 GPareveal the presence of a sharp transition with T c (cid:39) .
05 K(see the Supplemental Part, Ref. 25). No indication ofthe transition at T ∼ . e.g. , in Ref. 17 for pressures up to 2 . µ SR experiments were car-ried out at the µ E1 beam line by using the dedi-cated GPD (General Purpose Decay) spectrometer (PaulScherrer Institute, Switzerland). The details of TF- µ SRunder pressure experiments are provided in the Supple-mental Part, Ref. 25, and in the Ref. 29. Figure 1 (a)shows the Fourier transform of TF- µ SR time spectra (thepressure cell background subtracted, see the Supplemen-tal Part, Ref. 25) reflecting the internal field distribution P ( B ) in the Bi-III sample. The asymmetric P ( B ) dis-tribution at T (cid:39) .
25 K possesses the basic features ex-pected for an ordered vortex lattice, namely: the cutoff atlow fields, the peak shifted below µ H ap and the long tailtowards the high field direction (see e.g. Refs. 31,32 andreferences therein). Our experiments confirm, therefore,that the superconductivity of Bi-III is of a type-II.
The temperature dependence of the superfluid density[ ρ s ( T ) /ρ s ( T = 0) = λ − ( T ) /λ − ( T = 0), Fig. 1 (b)]was further obtained from the second central mo-ment of the magnetic field distribution ( (cid:104) ∆ B (cid:105) ). The superconducting part of the second moment, whichfor extreme type-II superconductors becomes a di-rect measure of the magnetic penetration depth λ ,was calculated via: (cid:104) ∆ B (cid:105) sc = (cid:104) ∆ B (cid:105) − σ /γ µ =0 . /λ . Here σ nm is the contribution of nu-clear magnetic moments measured at T > T c and Φ =2 .
068 10 − Wb is the magnetic flux quantum. From themeasured γ µ (cid:104) ∆ B (cid:105) / sc (0 .
25 K ,
10 mT) = 1 . µ s − and γ µ (cid:104) ∆ B (cid:105) / sc (0 .
25 K ,
30 mT) = 1 . µ s − thevalue of the magnetic penetration depth was found tobe λ (0 .
25 K) = 301(4) nm. With the coherence length, ξ = 10(2) nm, estimated from the value of the upper crit-ical field H c2 ( T = 0) (cid:39) . − . this translatesinto a Ginzburg-Landau parameter κ = λ/ξ (cid:39) κ valuereported to date for single element superconducting ma-terials. This indicates that Bi-III is a representative ofextreme type-II superconductors with κ (cid:29) s − wavesymmetry and using the following functional form: ρ ( T ) ρ (0) = 1 + 2 (cid:90) ∞ ∆( T ) (cid:18) ∂f∂E (cid:19) E dEdϕ (cid:112) E − ∆( T ) , (1)where f = [1 + exp( E/k B T )] − is the Fermi function.The temperature dependence of the superconducting gap∆( T ) was obtained by solving the nonlinear Eq. 1 forevery temperature point. The resulting ∆( T ) / ∆ val-ues for Bi-III are presented in Figure 1 (c). Here ∆ is Γ X M Γ Z R02468101214 a ngu l a r fr e qu e n c y ω ( m e V ) A|0(a) 5 10 15 20DOS (1/meV) (b)
FIG. 2: (a) Phonon dispersions relations and phonon DOS of the Bi-III approximant at 2.7 GPa. The logarithmic phononfrequency ω ln ≈ m/s. High-symmetry points of the Brillouin zone corresponding to the space group P /ncc in units of2 π ( a , a , c ). In calculations the approximantion of the incommensurate lattice structure of Bi-III by 32 atoms was used. the zero-temperature value of the superconducting en-ergy gap. Note that due to saturation of the superfluiddensity data [ ρ ( T ) /ρ (0) (cid:39) T /T c (cid:46) .
4, Fig. 1 (b)]the low-temperature ∆( T ) values can not be obtainedwith reliable accuracy. Figure 1 (c) implies that upondecreasing temperature, the gap in Bi-III grows fasterthan expected for the weak-coupled BCS ∆( T ). Forcomparison, the results for the superconducting gap inSn, Ref. 27, and Pb . Bi . , Ref. 28, which were foundto be characterized by non-BCS ∆( T ), are also shownin Fig. 1 (c). Remarkably, the temperature dependenceof the superconducting gap in Sn, Pb . Bi . and Bi-IIIhave a very similar functional form.The temperature dependence of the gap presented inFig. 1 (c) was approximated by the equation: ∆( T ) = ∆ tanh (cid:34) πk B ∆ (cid:115) c (cid:18) T c T − (cid:19)(cid:35) (2)with c and ∆ as fit parameters. Following Ref. 36 thisgeneral equation describes ∆( T )’s for superconductorswith various coupling strengths (2∆ /k B T c ) and orderparameter symmetries. The fit of Eq. 2 to the datawith c = 2 . . T ) to the superfluid density data. Usingthe value of 2∆ /k B T c (cid:39) .
34 in Carbotte’s empiricalrelation: k B T c = 3 . (cid:34) . (cid:18) k B T c ω ln (cid:19) ln ω ln k B T c (cid:35) , (3)the logarithmically averaged phonon frequency ω ln (cid:39) .
51 meV was calculated. In fact, ω ln corresponds to thedynamics of the superconducting state and represents themost effective phonon energy for a given T c . Followingphenomenological arguments of Carbotte, for Bi-IIIone expects ω ln ∝ k B T c (cid:39) . in the caseof single element superconductors, and MgB andPbBi alloys for binary ones. The deviation from weak-coupled BCS ∆( T ) is quite often observed in cuprateand Fe-based high-temperature superconductors. The-oretically, a non-BCS behavior of ∆( T ) is expectedfor superconductors with high values of the electron-phonon coupling constant, within the Eliashberg-Nambuformalism, and is explained by damping of quasi-particle excitations caused by a strong electron-phononinteraction. In the following part of the paper the results of Den-sity Functional Theory calculations are discussed. Thestructural analysis and the electronic band structure arepresented in the Supplementary Part, Ref. 25. The calcu-lations reveal that at ambient pressure, where Bi adoptsa rhombohedral structure (Bi-I phase), its density ofstates (DOS) shows bands which are clearly separatedinto bonding/antibonding s − and p − states. The Fermilevel, E F , is situated in a gap between the bonding andantibonding p -states thus giving rise to the semimetal-lic behavior of Bi-I. As pressure increases, the p -statesstart to overlap, leading to a metallic DOS at E F . Fromthe analysis of electronic bands and DOS (see the Supple-mentary Part, Ref. 25) the host and guest sites in Bi-IIIphase were found to have similar magnitudes, in agree-ment with Ref. 48. The electronic bands of p -characterwould provide electrons for the pairing.Figure 2 (a) shows the phonon dispersion and thephonon DOS obtained with the approximation of the in-commensurate lattice structure of Bi-III by 32 atoms. Adifferent structure of 42 atoms was considered in Ref. 18and it was found to demonstrate the presence of a lowfrequency phason mode. This mode was suggested tobe responsible for a very high electron-phonon couplingstrength and gave rise to the enhanced transition temper-ature and the high upper critical field. Note, how-ever, that in the actual calculations with 32 atoms nophason modes were found to appear in the phonon spec-tra. One would also mention that in accordance witharguments of the Eliashberg theory, neither the very lownor the very high frequencies are important, while fre-quencies around the middle of the spectrum, ≈ T c . Note that this value stays close to ω ln (cid:39) .
51 meV ob-tained from measured 2∆ /kT c (cid:39) .
34 by means of Eq. 3.The Eliashberg theory provides no arguments on limiting T c for a given pairing mechanism (phonon, phason or anyother boson exchange). Considering the electron-phononinteraction, the limitation on the magnitude of T c for Bi-III is most probably related to the lattice (in)stability fora given value of the applied pressure.An alternative superconducting pairing mechanism,based on correlations in the Fermi liquid rather thanelectron-phonon coupling, was proposed by Kohn andLuttinger. This pairing mechanism is associated withthe effective interaction, between quasiparticles occur-ring as a result of polarization of the fermionic back-ground, which is proportional to the static susceptibility(Lindhard function). The appearance of a divergence inthe static susceptibility χ ( q ) is determined by the exis-tence of nesting vectors q nest arising if the Fermi sur-face possesses parallel fragments such that pairs of elec-tronic states can be connected by the same wave vec-tor q nest . In this case the possibility of Cooper pairingis determined by the characteristics of the energy spec-trum (band structure) in the vicinity of the Fermi leveland by the effective interaction. In Fig. 2 (b) the re-sults of the Fermi surface (FS) calculations are shown.Based on the FS shape one can not exclude that such asimple mechanism of Fermi surface instabilities underliesthe pairing formation in Bi-III, in particular in conjunc-tion with the Kohn-Luttinger mechanism. Moreover, due to proximity to a maximum in the DOS, the super-conducting transition temperature (predicted within theframework Kohn-Luttinger mechanics) may increase fur-ther as a consequence of the combination with van Hovesingularities. To conclude, muon-spin rotation experiments per-formed under pressure p (cid:39) .
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