Dramatically Enhanced Superconductivity in Elemental Bismuth from Excitonic Fluctuation Exchange

Motivated by the remarkable discovery of superconductivity in elemental Bismuth at ambient pressure, we study its normal state in detail using a combination of tight-binding (TB) band-structure supplemented by dynamical mean-field theory (DMFT). We show that a two-fluid model composed of preformed and dynamically fluctuating excitons coupled to a tiny number of carriers provides a unified rationalization of a range of ill-understood normal state spectral and transport data. Based on these, we propose that resonant scattering involving a very low density of renormalized carriers and the excitonic liquid drives logarithmic enhancement of vertex corrections, boosting superconductivity in Bi. A confirmatory test for our proposal would be the experimental verification of an excitonic semiconductor with electronic nematicity as a ‘competing order’ on inducing a semi-metal-to semiconductor transition in Bi by an external perturbation like pressure

involving strong coupling between the charge carriers and plasmons 8 . Optical data raise additional issues in this context 9 . While a tiny Drude component is visible at low energy, finding of sizable mid-infra-red (mid-IR) absorption is inexplicable in a free-electron framework. Given the tiny Fermi energy 25  meV, a large T-dependent transfer of spectral weight over a much larger scale 300  meV also presents a challenge for the standard (uncorrelated band) view.
On the theoretical front, first-principles density-functional theory (DFT) and Slater-Koster tight-binding fits 6 conclusively show that: (i) there are multiple tiny pockets in Bi, with two being almost perfectly compensated, and the others supplying a tiny additional number of carriers. Moreover, the importance of spin-orbit coupling (SOC) is shown by the fact that the correct positions of the e(h) pockets are only found when sizable SOC, O(1.3) eV, is included in band calculations. Thus, DFT + SOC calculations indeed yield good description of ground state properties. In particular, they yield the correct shape and size of the carrier pockets at the T and L points in the Brillouin zone. However, it is also well known that such a ground state theory can fall short of describing the dynamical excitation spectrum and, consequently, the finite-temperature (T) responses 10 . This is especially true when coulomb interaction-induced bound state formation is involved: in the specific context of Bi, it is long known 9 that the dominant interaction between conduction and valence band carriers in Bi is of short-range excitonic character. This raises the possibility of exciton bound state formation, but its role in Bi has never been satisfactorily addressed. While DFT calculations work well in moderately correlated solids, they are, by construction, inadequate when such (e.g, excitonic) bound states appear due to correlations. Generically, one also expects a symmetry-adapted coupling of such interband excitons to intervalley phonons 11 in Bi: since the ratio of the phonon Debye frequency kΘ D to E F is large ( 0 5  . in Bi), sizable self-energy effects due to this additional coupling will also influence normal state responses above T c . As far as SC is concerned, the huge enhancement of the SC transition temperature, , and the large discrepancy 3 between the measured and calculated (within BCS theory) ratio of the upper critical field to T c also mandates a non-adiabatic 'strong coupling' SC beyond a traditional BCS view.
These limited observations must constrain theoretical modelling: both unconventional metallicity and SC must find explication in a picture based on (i) the special band structure of Bi and (ii) strong scattering (electron-hole and/or electron-phonon interaction) processes beyond DFT and, in particular, microscopic processes which generate a T dc 5 ρ  should also be involved in generating the SC pair glue. Our first observation is that two of the multiple electron(hole) pockets of Bi are almost perfectly compensated, leading to a situation famously encountered in transition-metal dichalcogenides (TMD), where preformed excitonic liquid (PEL) driven charge-density-wave (CDW) states are ubiquitous [12][13][14] . Band calculations reveal a total of four pockets in Bi, of which two (electron and hole) exhibit a propensity for an excitonic instability. Were there only these two pockets, one would have expected an electron-hole attraction-mediated excitonic insulator. However, the remaining pockets now supply a tiny number of additional carriers, leading to a physical picture of an incipient excitonic insulator self-doped with a tiny number of carriers. Thus, we are led to a model of a self-doped excitonic insulator (EI) in the intermediate coupling limit, where novel physics can arise from strong scattering between carriers and preformed but uncondensed excitons. In this report, we establish this using tight-binding-plus dynamical mean-field theory (TB-DMFT) calculations as done earlier [12][13][14] . Armed with very good accord with normal state transport, we elucidate a 'strong coupling' electronic mechanism, wherein strong resonant scattering between carriers and preformed excitons enhances vertex corrections, boosting the SC T c in Bi.
We begin with a tight-binding description of the DFT band structure 6 by using the Slater-Koster (SK) fit with form factors and parameters including SOC as in earlier work (given in SI Table I 6 ). The resulting band structure in Fig. 1 excellently reproduces all the pockets seen in full DFT calculations, constituting the appropriate band structural input for the correlation calculations. Guided by the discussion above, we focus on inter-band x y (represented by the green line in Fig. 1 Fig. 1) where E p is on-site energy (−9.643 eV), V ppπ and V ppσ (2.271 eV and −0.679 eV, respectively) are third nearest neighbour hoppings and the interband matrix element ⋍ x y , as extracted earlier 6 from an SK fit. Here c k and c k † represent anihilation and creation operator and μ, ν represent the band indices for the two bands crossing E F (=0) in Fig. 1. The interaction terms are taken to be dominantly local with the understanding 9 that the coupling between valence and conduction band carriers is of short-range excitonic character, as pointed out above. Here U 1,1 = U 2,2 and U′ are the intra-and inter-band Coulomb interactions, n i,μ = ∑ σ n i,μ,σ . It is also long known that coupling of carriers to intervalley phonons is significant in Bi 11 . Since the dimensionless measure of the electron-phonon (e-p) coupling, defined as (ω D the Debye frequency, and E F = Fermi energy) is actually large in Bi 3 , one is in a non-adiabatic limit of the electron-phonon problem. This intermediate coupling regime is also interesting in that in contrast to the anti-adiabatic (g → ∞) or the adiabatic (g → 0) limits, one cannot "integrate out" these phonons to give a further e-h attraction O(g 2 /ħω D ). One must solve H including an explicit e-p term by coupling the intervalley phonons to the k-dependent hybridization as done in earlier work 13,14 . Technically, coupling of inter-band excitons to phonons is described by adding a term i are the intervalley phonon operators. We have followed our earlier procedure 13,14 to incorporate the sizable e-p coupling effects for Bi in the two-band Hubbard-like model above within the DMFT procedure (see SI(A) for details).

TB+DMFT Results and Transport
We solve the two-band model above using DMFT, with multi-orbital iterated perturbation theory (IPT) as the 'impurity' solver. Though not numerically exact, it works very well in real multi-band cases for all temperatures and band-fillings, especially in cases where there is a sizable crystal-field splitting between the bands (here, between valence and conduction band). The bands that serve as a band-structural input to the correlation calculations are obtained by diagonalizing the one-electron part of the two-band model above, and labelled henceforth as "a" and "b" band states. IPT is a fast solver and its efficacy in a wide range of real systems is known [13][14][15] . In contrast, though state-of-the-art continuous-time quantum Monte Carlo (CTQMC) solvers are much more "numerically exact", they cannot access temperatures below O(20) K at present 16 , which makes them unsuitable for investigation of the low T(<1 K) behavior in Bi. We choose U 1,1 = U 2,2 = 0.5-0.7 eV (henceforth we use the notation U = U 1,1 = U 2,2 ) and U′ = 0.1-0.2 eV as appropriate parameters, and while ab-initio estimates will yield more precise estimates, our present choice is physically motivated. Given widths of O(1.0-1.5) eV for the valence (VB) and conduction (CB) bands, we are in the intermediate coupling limit of the two-band model. This is precisely the case where DMFT works best (for specific studies in the BCS-BEC crossover, see ref. 17).
In Fig. 2, we show the TB+DMFT local density-of-states (LDOS) as U, U′ are cranked up. While correlations gradually close the band gap for the VB ('a'-orbital) as expected 18 , they reduce the LDOS at E F for the CB ('b' orbital): these contrasting behaviors are direct consequences of distinct effects of local electronic correlations on band-insulating and metallic subsets of the non-interacting band structure. Around  U W b , the b-fermion band-width, we observe eventual opening of a Mott-like gap in the LDOS, seen by the fact that Im Σ b (ω) develops a pole structure at E F in this case. Correspondingly, the a-fermion states retain metallic character: this would be an interesting manifestation of orbital-selective Mott physics in semi-metals and, were such a strong coupling regime to be realized, would open up the possibility to development of novel instabilities with concomitant competing orders 19, 20 as, for instance, in Fe-arsenides. However, this is not the regime applicable to Bi, and so we concentrate on the smaller U regime.
Choosing U = 0.5 eV, U′ = 0.15 eV, we next compute the dc resistivity using the Kubo formalism in DMFT. This task is facilitated by the finding 21 that irreducible vertex corrections apprearing in the Bethe-Salpeter equations for conductivities are negligible and can be ignored to a very good approximation. Interestingly, as shown in Fig. 3, we find that a linear-in-T behavior of ρ dc (T) at 'high' T ≥ 40-50 K smoothly crosses over to a Fermi-liquid-like T 2 behavior up to about 10 K and, remarkably, exhibits a further low-T crossover to a 'good' metal with . This is in very good accord with experimental trends, and mandates deeper microscopic rationalization. To cement the link between transport and excitonic liquid fluctuations, Fig. 3 shows the excitonic average, computed as Δ exc = (−1/π)∫dω Im G 12 (ω) (G 12 (ω) is itself computed from two-band DMFT(IPT) as previously done for TMDs 13,14 ). We find a clear correlation between Δ exc (T) and the T-dependence of ρ dc (T ) over a wide T range. This provides strong theoretical evidence linking the dc resistivity to microscopic processes involving scattering of the tiny number of carriers off well-formed and quasi-local excitonic correlations: at high-T, the latter are incoherent, leading to a quasi-linear-in-T resistivity, while increasing one-fermion coherence via suppression of incoherent excitonic fluctuations provides a rationalization for onset of enhanced metallic coherence, and the near T 2 behavior at intermediate T.
ScienTific REPORTS | 7: 10993 | DOI:10.1038/s41598-017-11269-y Remarkably, however, we also obtain the  ρ T T ( ) dc 5 dependence at very low T, the latter correlated with a reduced excitonic fluctuation at low T. Thus, we identify a new element: the T-dependence of sizable and dynamical 'preformed' excitonic correlations governs the T-dependent resistivity in a wide T window. Our proposal is distinct from: (i) the plasmaron view 8 , where weak-coupling RPA-like analysis is employed to get  T dc 5 ρ from long-range interactions, and (ii) pure e-p coupling models, which could, in principle, also yield similar behavior when T D  Θ , something that may also obtain in Bi. We emphasize that e-p coupling is included on an equal footing with local excitonic correlations in our approach. We propose that a way to distinguish between these distinct scenarios could be T-dependent tunnelling measurements at small-to-intermediate T: as a function of T, the conductance g(V) = dI/dV would, in a picture involving coupling of carriers to any bosonic mode(s), show finite-voltage (energy) peak-dip-hump features. The energies and spectral weights of such features could be compared with estimates from different bosonic channels, allowing a determination of the most important fermion-boson scattering channel. However, since there will always be a symmetry-dictated coupling of interband excitons to intervalley phonons, one also generally expects two bosonic modes at different energies to show up in g(V): their relative weights provide an estimate of the relative importance of carrier-exciton vis-a-vis carrier-phonon coupling.  Further support for our view arises when we compare the DMFT optical conductivity, σ xx (ω) in Fig. 4 as a function of T with published data 9 . Since we have kept only the two lowest bands crossing E F from the SK fit, we do not expect accord at higher energies, but can readily make a comparison for the relevant energy range (few hundred milli-eV) of interest. Specifically, up to about 100 meV, our result matches quite well with data, including (i) the plasmon edge, (ii) the detailed optical lineshape as a function of energy up to about 100 meV, (iii) sizable optical spectral weight transfer upon raising T. The plasmon-like features are clearly visible on both regular and log-log plots as a clear absorption onset at ≃15 meV at low T = 10 K. Interestingly, it is also preceded by a 'prepeak' structure, centered at ≃10 meV, in good accord with observations 9 . This prepeak feature is also washed out with increasing T, in accord with data. Clear spectral weight transfer up to ≃200 meV is also found: given the tiny E F in Bi, this is quite a large energy scale, attesting to considerable dynamical correlations. Finally, we find an isosbestic point around 5.0 meV as a function of T, which is another characteristic signature of dynamical electronic correlations that could be tested in extant work.
Taken together, our results strongly support the idea that a common underlying scattering process involving tiny number of carriers and uncondensed fluctuating interband excitons may be operative in Bi. That the dominant interaction in Bi has a short-range excitonic character is well known 9 . Our results show that it seems to be a sufficient minimal input to understand transport in Bi, and ties in our view with plasmarons 8 . Generally, in a correlated metal, appearance of mid-infra-red features (e.g, DMFT for one-band Hubbard type models) involve interband transitions between low-energy itinerant and high-energy localized Hubbard-band states 10 . In our case, they arise from transitions involving carrier states scattering off the incoherent and interband excitons involving precisely the renormalized VB and CB in the two-band Hubbard-like model, since σ xx (ω) in DMFT is just a direct convolution of the one-particle DMFT spectral functions. In Bi, this feature occurs at a very low energy because of the tiny E F . The interband electronic excitations involving carriers coupled to such exciton-like entities have been christened plasmaron in prior work. We find that this characterization is not in conflict with our excitonic fluctuation picture, since we have now shown that it can be well described by analyzing effects of predominantly local and sizable excitonic liquid-like correlations in a quasi-realistic model for Bi. Finally, since excitonic correlations can trigger changes in lattice parameters, whether development of excitonic (liquid) correlations can rationalize the small but unexplained dependence of the rhombohedral parameter (u) 4 on T is interesting to inquire, but out of scope of the present work.
Buoyed by the agreement with the phenomenology so far, we venture to propose a specific model for superconductivity in Bi. To do so, we need to generate an effective pairing interaction arising from excitonic and/or phonon fluctuation exchange. Given the efficacy of the excitonic liquid view above for the normal state, and that the exciton formation energy scale ( ex exc  Ω ∆ ) is quite large, we are clearly in the non-adiabatic limit. This precludes a BCS-like instability to SC in Bi. In the normal state, the interband excitons involving the VB and CB form an isospin T = 1/2 degree of freedom, described by , where ρ = ρ(E F ) is the LDOS at E F and  χ J t (0) 12 2 12 , with χ 12 (0) the excitonic susceptibility at ω = E F (=0): correlated FL behavior obtains below T K ex , thanks to an excitonic Kondo screening implicit in a situation where a tiny number of carriers are now coupled to fluctuating isospins T. It is precisely this resonant scattering that governs transport in the normal state in good accord with transport data as found above.
As T is lowered, intersite, residual interactions develop in full analogy with the way inter-site spin correlations, mediated by metallic carriers, arise in the Kondo lattice. They correspond to exchange coupling between isospins . TB+DMFT Optical Conductivity in the normal state for Bi. A tiny Drude-like contribution is followed by inter-band absorption: this feature onsets at ≃15 meV and originates from strong resonant scattering between the tiny number of carriers and interband excitons in DMFT. The pre-peak in the absorption, centered at ≃10 meV, also originates from the same mechanism. The inset reveals the same features in more detail on a log-log plot. These findings are in very good accord with data 9 .