Possible Fano resonance for high Tc multi-gap superconductivity in p-Terphenyl doped by K at the Lifshitz transition

TL;DR

This paper proposes that high-temperature superconductivity in potassium-doped p-Terphenyl arises from quantum Fano resonances near a Lifshitz transition, with numerical evidence showing a narrow energy window for maximum Tc.

Contribution

It introduces a novel mechanism involving shape resonances for high Tc in organic superconductors and provides numerical support for the role of Lifshitz transitions in enhancing Tc.

Findings

High Tc (up to 123K) occurs near a Lifshitz transition in p-Terphenyl.

Superconducting gaps exhibit Fano resonance behavior near the transition.

Predicted experimental signatures include variations in isotope coefficients and gap ratios.

Abstract

Recent experiments have reported the emergence of high temperature superconductivity with critical temperature $T_c$ between 43K and 123K in a potassium doped aromatic hydrocarbon para-Terphenyl or p-Terphenyl. This achievement provides the record for the highest Tc in an organic superconductor overcoming the previous record of Tc=38 K in Cs3C60 fulleride. Here we propose that the driving mechanism is the quantum resonance between superconducting gaps near a Lifshitz transition which belongs to the class of Fano resonances called shape resonances. For the case of p-Terphenyl our numerical solutions of the multi gap equation shows that high Tc is driven by tuning the chemical potential by K doping and it appears only in a narrow energy range near a Lifshitz transition. At the maximum critical temperature, Tc=123K, the condensate in the appearing new small Fermi surface pocket is in the BCS-BEC crossover while the Tc drops below 0.3 K where it is in the BEC regime. Finally we predict the experimental results which can support or falsify our proposed mechanism: a) the variation of the isotope coefficient as a function of the critical temperature and b) the variation of the gaps and their ratios 2Delta/Tc as a function of Tc.

Equations

$${\Delta }_{n,k_y}=-\frac{1}{2N}\sum _{n^{\prime },{\mathbf{k}}^{\prime }}{V}_{\mathbf{k},{\mathbf{k}}^{\prime }}^{n,n^{\prime }}\frac{\tanh (\frac{E_{n,k_y^{\prime }}+{ϵ}_{k_x}-\mu }{2T_c})}{E_{n,k_y^{\prime }}+{ϵ}_{k_x}-\mu }{\Delta }_{n^{\prime },k_y^{\prime }},$$

$${\Delta }_{n,k_y}=-\frac{1}{2N}\sum _{n^{\prime },k_y^{\prime },k_x^{\prime }}\frac{V_{\mathbf{k},{\mathbf{k}}^{\prime }}^{n,n^{\prime }}{\Delta }_{n^{\prime },k_y^{\prime }}}{\sqrt{(E_{n^{\prime },k_y^{\prime }}+{ϵ}_{k_x^{\prime }}-\mu {)}^2+{\Delta }_{n^{\prime },k_y^{\prime }}^2}},$$

$$V_{\mathbf{k},{\mathbf{k}}^{\prime }}^{n,n^{\prime }}={\tilde{V}}_{\mathbf{k},{\mathbf{k}}^{\prime }}^{n,n^{\prime }}\times \theta ({\omega }_0-|E_{n,k_y}+{ϵ}_{k_x}-\mu |)\theta ({\omega }_0-|E_{n^{\prime },k_y^{\prime }}+{ϵ}_{k_x^{\prime }}-\mu |).$$

$$\begin{array}{rl}{\tilde{V}}_{\mathbf{k},{\mathbf{k}}^{\prime }}^{n,n^{\prime }}=& -\frac{{\lambda }_{n,n^{\prime }}}{N_0}S\times \\ & \times {\int }_S{\psi }_{n^{\prime },-k_y^{\prime }}(y){\psi }_{n,-k_y}(y){\psi }_{n,k_y}(y){\psi }_{n^{\prime },k_y^{\prime }}(y)dxdy,\end{array}$$

$$\rho =\frac{1}{L_x{L}_y}\sum _n^{N_b}\sum _{k_x,k_y}\left[1-\frac{E_{n,k_y}+{ϵ}_{k_x}-\mu }{\sqrt{(E_{n,k_y}+{ϵ}_{k_x}-\mu {)}^2+{\Delta }_{n,k_y}^2}}\right].$$

Full text

1 Introduction

Recently high temperature superconductivity with $T_{c}$ in the range of 43-123 K has been reported following different growth procedures in a potassium doped aromatic hydrocarbon p-Terphenyl [1, 2, 3]. The superconducting crystalline phase is expected to be $K_{3}C_{18}H_{14}$. Superconducting pairing with a large 15 meV gap opening at about 60K was confirmed on a K-doped surface of a p-Terphenyl single crystal [4]. Para-Terphenyl, a linear molecule made of a chain of 3 benzene rings, and its derivatives are aromatic biological molecules present in edible mushrooms [5]. Pharmaceutical research is in progress for their use as immunosuppressive, anti-inflammatory and anti-tumor agents, moreover it has technological applications as laser dye, sunscreen lotion and in photon detectors. If these results will be confirmed, $K_{x}$ p-Terphenyl provides today the record for the highest critical temperature in carbon based materials and larger than in many cuprates oxides and in many iron based superconductors. The search for macroscopic quantum coherence at high temperature in organics and organometallics has been a long standing search for the holy grail of room temperature superconductors. Superconductivity in graphite intercalation compounds has been an active topic for several decades, but $T_{c}$ ranges only up to 11.5 K for $CaC_{6}$ [6]. The highest critical temperature $T_{c}$=38 K in doped fullerides $A_{3}C_{60}$ $A=K,Rb,Cs$ has been found in $Cs_{3}C_{60}$ with A15 structure by application of 7 Kbar hydrostatic pressure [7]. This system provides a complex phase of condensed matter where structural polymorphism controls both magnetic and superconducting properties [8] and shows a fluctuating microscopic heterogeneity made of the coexistence of both localized Jahn Teller active and itinerant electrons [9]. Recently the material research for high temperature superconductors was oriented toward metal-intercalated aromatic hydrocarbons by the discovery of superconductivity with $T_{c}$=18K by doping potassium into picene ($C_{22}H_{14}$) [10, 11]. Picene is a hydrocarbon molecule made of five benzene rings condensed in an armchair manner. Looking for high $T_{c}$ in alkali-metals or alkali-earth-metals doped polycyclic-aromatic-hydrocarbons (PAHs), the previous record $T_{c}$=33 K was held by in $K_{x}$ 1,2:8,9-dibenzopentacene ($C_{30}H_{18}$) [12]. The recent indications for $T_{c}$=123K in p-Terphenyl open a new road map for the search of higher critical temperatures in the large family of doped metal-organic compounds, with the hope to overcome the highest superconducting critical temperature known so far, $T_{c}$=203K in pressurized $H_{3}S$  [13, 14]. There is today high interest on understanding both the quantum mechanism beyond the emergence of $T_{c}$=123K in p-Terphenyl and the relation between the nanostructure and quantum functionality in organics. This is needed to develop novel $quantum$ $plastics$ materials, taking advantage that conducting polymers can be prepared in a large variety of polymeric heterostructures at atomic limit [15, 16, 17] and possess the combination of easy processability, light weight, and durability. Using density functional theory and Eliashberg’s theory of superconductivity in the single band approximation the superconducting critical temperature has been predicted to be in the range $3<T_{c}<8$ K in $K_{3}$ picene [18] and around 6.2 K in K_{2}$$C_{6}$$H_{6} moreover, it was argued that all hydrocarbons should show $T_{c}$ in a similar temperature range of $3<T_{c}<7$K [19]. A hot topic today is the search for superconductivity in the extreme high pressure phases of benzene [20, 21, 22]. Therefore unconventional pairing mechanisms were invoked for high $T_{c}$ in doped p-Terphenyl: a) the Bose-Einstein Condensation (BEC) of preformed bipolarons [1, 2, 3]; b) the Resonating Valence Bond (RVB) theory in a scenario with two coexisting doped Mott insulators [23] c) the so called $s_{+/-}$ pairing mechanism mediated by a repulsive interaction in a scenario with two strongly correlated bands forming two similar Fermi surfaces connected by a nesting vector [24].

Here we propose the mechanism for $T_{c}$ amplification driven by Fano resonance [25, 26] in a multigap superconductor between different gaps occurring where the chemical potential is tuned at a electronic topological Lifshitz transition [27, 28]. In this regime the configuration interaction between the pairing scattering channels in the (n-1)-th bands with high Fermi energy $E_{F(n-1)}$ in the BCS approximation and the n-th pairing scattering channel in the new appearing n-th small Fermi surface with a low Fermi energy $E_{Fn})$ can give a resonance in the superconducting gaps, called shape resonance[29, 30] like in nuclear physics and molecular physics [31] or Feshbach resonance as in the jargon of ultracold gases [32]. The Fano resonance give high $T_{c}$ domes around Lifshitz transitions in the energy range of $E_{Fn}$ of the order of the energy of the pairing cut-off and the n-th condensate is in the BCS-BEC crossover [33, 34].

The configuration interaction between pairing channels is determined by the symmetry and interference between the wavefunctions of electron pairs at the coexisting multiple Fermi surfaces therefore the nanoscale material architecture plays a central role. In ref [25, 26] particular material architectures made by heterostructures at atomic limit which could give high-$T_{c}$ superconductors have been disclosed. Panel (A) in Fig. 1 shows one of these heterostructures at atomic limit: a superlattice of stripes. In this case the material is made of nanoscale modules, metallic stripes of atomic thickness, as for example metallic graphene or phosphorene nanoribbons assembled in a superlattice of stripes separated by potential barriers from neighbor stripes in the same plane or in the neighbor plane. The size of the nanoscale units or modules and the superlattice period $\lambda_{p}$ is required to be in the nanoscale. In fact the Fano resonance to get the highest $T_{c}$ need to tune the wavelength $\lambda_{Fn}$ of electrons at the Fermi level in the n-th band in the range of $\lambda_{p}$. The crystalline structure of undoped p-Terphenyl [35, 36] with a monoclinic space group P21/a is shown in Panel (B) of Fig. 1 where gray dots are carbon atoms and red dots are hydrogen atoms, can be described as a packing of parallel p-Terphenyl nanoribbons or stripes of about 1.4 nm width running in the $\bf{b}$-axis direction indicated by the black arrows. The figure shows the similarity of the arrangements of the p-Terphenyl nanoscale stripes with the schematic drawing of the superlattice of stripes shown in panel (A) of Fig. 1 from ref [26]. The projections in the $\bf{ac}$, $\bf{bc}$ and $\bf{ba}$ crystal planes are shown in Fig. 2. Two parallel stripes are clearly seen in the $\bf{bc}$ projection. The $\bf{ba}$ projection in Fig. 2 shows a variable torsion angle (up to 14 degrees) of the middle ring relative with the two lateral ones. While the $H-H$ bond is 0.32 nm between the lateral rings of the neighbor molecules very short $H-H$ intermolecular bonds (dashed lines connecting neighbor molecules in the stripe in the $\bf{b}$ direction) are established for the central benzene ring, as short as 0.22 nm, like the shortest values observed in the high pressure phases of benzene [20, 21]. The stripes are not connected with the neighbor stripes also in the a-axis direction indicating the quasi one dimensional electronic structure of the stripes.

The electronic structure of p-Terphenyl [37, 38, 24] confirms the presence of one dimensional electronic states in the conduction band. A portion of the electronic structure of the conduction band [37] is shown in the upper part of Fig. 3. The band structure, upper panel of Fig. 3, displays in a clear way a strong anisotropic character in the wave-vector space. Indeed, in the irreducible representation of the band structure reported in the upper part of Fig. 3 [37, 38, 24] shows two directions in the Brillouin zone for which the band dispersions are very narrow. A narrow bandwidth of the order of hundred meV, indicates that large potential barriers have to be penetrated by the electrons to tunnel from one p-Terphenyl stripe to its neighbor stripe. Interestingly and relevantly for this work, all the bands of the band-bundle due to the different overlapping orbitals of the p-Terphenyl, have this property and are very narrow along these two independent directions, indicating that the potential barrier formed by the crystal structure and by the characteristics of the electronic bonds is felt by all the conduction and valence electrons. In the b-axis direction in the real space, orthogonal to the upper surface of the Brillouin zone the band dispersion is much broader, of the order of 700 meV, pointing toward a quasi-free conduction of the electrons with a moderate effective mass along the b direction. The energy scale of the pairing,145 meV, is taken from the relevant phonons detected at 1171 cm-1, due to a mode of the C-C bond, and/or the mode at 1222 cm-1 of the C-H mode, detected by Raman spectroscopy in metallic $K_{3}$ p-terphenyl [1]. We have simulated the band dispersion and the DOS with a periodic potential barrier with 1.4 nm periodicity which reproduces the narrow band dispersion in the transversal direction of the stripes shown in the lower panel of Fig. 3. We have obtained an electronic structure which grabs the key feature of the evolution of the electronic structure where the chemical potential is tuned at a band edge in a superlattice of stripes as shown in the upper panel of Fig. 4 showing the drawing of the patent for material design of heterostructures at atomic limit [26].

Once doped by K, the chemical potential will be raised in the conduction band and depending of the doping (and/or misfit strain, pressure, orientational disorder, magnetic field …) at the n-th band edge giving rise to a complex network of Fermi surfaces, with electron and hole-like small pockets of Fermi surfaces and Fermi arcs. The key point of this work is to predict the superconducting properties of the K-doped p-Terphenyl with numerical calculations through a simplified model of its nanoscale structure getting the key electronic structure near a band edge. The bands and wave-functions are created by using a proper superlattice of stripes, which allow the solution of the Bogoliubov gap equations [39, 40] without standard BCS approximations. giving high-$T_{c}$ superconducting state with multigaps and multi-condensates at different BCS/BEC pairing regimes. The pairing is thought to be driven by an attractive interaction within each Fermi surface, but the non-diagonal interaction between condensates can be either repulsive or attractive. The multicomponent character of the pairing and the geometry of the system will determine shape resonances in the superconducting gaps and $T_{c}$, with peculiar features predicted for the isotope effect and the gap to $T_{c}$ ratios, which can be tested in future experiments, together with the expected anisotropic transport of electrons in doped (or out of equilibrium) single crystals. The presence of high temperature superconducting domes where the chemical potential crosses the Lifshitz transitions has now well confirmed by experiments in iron based superconductors. Here we present the possible scenario of high temperature superconductivity in p-Terphenyl where the chemical potential is driven by potassium doping at a Lifshitz transition. Our model predict that at the Lifshitz transition the Fermi surface is made of multiple components: circular Fermi surface pockets and Fermi arcs as shown in Fig.4. The superconducting properties have been calculated using the Bianconi-Perali-Valletta (BPV) theory which has been used to predict the mechanism driving the emergence of high $T_{c}$ in cuprates [41, 42, 43, 44], where the 1996 proposed scenario of the coexistence of Fermi arcs and Fermi pockets was confirmed in 2009  [45]. It has been able to predict the anomalous deviation of the isotope coefficient from the BCS predicted value [46, 47]. The BPV theory has been applied to diborides [48, 49, 50], and iron based superconductors [51, 52, 53] where it has been confirmed by recent experiments [54, 55, 56, 57, 58]. In these experiments it has been reported compelling evidence that the high $T_{c}$ domes occur in the proximity of the Lifshitz transitions for the appearing of a new Fermi surface pocket. Recently similar scenarios have been proposed for pressurized sulfur hydrides [59] and in superconducting nanofilms [60]

Let us consider a system made of multiple bands with index $n$. The energy separation between the chemical potential and the bottom of the n-th band defines the Fermi energy of the n-th band. This formulation was proposed for systems with a band crossing the chemical potential having a steep free electron like dispersion in the $x$ direction and a flat band-like dispersion in the $y$ direction. The wavefunctions of electrons at the Fermi level are calculated using a lattice with quasi-one dimensional lattice potential modulation where the chemical potential is tuned near a Lifshitz transition, like in magnesium diborides, A15, cuprates, iron based superconductors and we propose here for doped p-terphenyl.

The superconducting critical temperature $T_{c}$ in the BPV theory is obtained by numerical solution of the gaps equation [39, 40] considering the simplest case of a two dimensional system [43] of stripes, but the extension to three dimensional system [50] is straightforward. The $T_{c}$ is determined by solving the following selfconsistent system of equations,

[TABLE]

where the $E_{n,k_{y}}+\epsilon_{k_{x}}$ is the energy dispersion and $\mu$ the chemical potential.

We consider a superconductor with multiple gaps $\Delta_{n,k_{y}}$ in multiple bands $n$ with flat band-like dispersion in the y direction and steep free-electron-like dispersion in the x direction for a simple model of a two dimensional metal with a one-dimensional superlattice modulation in the y-direction. The self consistent equation for the gaps at $T=0$ where each gap depends on the other gaps is given by

[TABLE]

where N is the total number of wave-vectors in the discrete summation and $V^{n,n^{\prime}}_{\mathbf{k},\mathbf{k}^{\prime}}$ is the effective pairing interaction taken in the separable and energy cutoff approximation,

[TABLE]

Here we take account of the interference effects between the single-particle wave functions of the pairing electrons in the different bands, where $n$ and $n^{\prime}$ are the band indexes, $k_{y}(k_{y}^{\prime})$ is the superlattice wave-vector and $k_{x}(k_{x}^{\prime})$ is the component of the wave-vector in the free-electron-like direction of the initial (final) state in the pairing process.

[TABLE]

$N_{0}$ is the DOS at the Fermi energy $E_{F}$ without the lattice modulation, $\lambda_{n,n^{\prime}}$ is the dimensionless coupling parameter, $S=L_{x}L_{y}$ is the surface of the plane and $\psi_{n,k_{y}}(y)$ are the eigenfunctions in the 1D superlattice. We have used weak coupling intraband coupling constants 0.14 for the (n-1)-th bands, 0.2 for the appearing n-th band and a small 0.1 interband attractive or repulsive exchange interband pairing constant and the distribution of calculated intraband pairing terms (n,n) (n-1,n-1) together with the (n,n-1) interband terms is shown in Fig. 5.

The system of equations for the gaps need to be solved iteratively. The anisotropic gaps depend on the band index and on the superlattice wave-vector $k_{y}$. According with Leggett[61] the ground-state BCS wave function corresponds to an ensemble of overlapping Cooper pairs at weak coupling (BCS regime) and evolves to molecular (non-overlapping) local pairs with bosonic character in the BEC regime. This approach remains valid also if a particular band is in the BCS-BEC crossover and beyond Migdal approximation because all other bands are in the BCS regime and in the Migdal approximation. In this crossover regime the chemical potential $\mu$ results strongly renormalized with respect to the Fermi energy $E_{F}n$ of the non interacting system. In the case of a Lifshitz transition, as described in this paper, nearly all electrons in the new appearing Fermi surface condense forming a condensate having a BCS-BEC crossover character.

Therefore at any chosen value of the charge density $\rho$ for a number of the occupied bands $N_{b}$, the chemical potential in the superconducting phase has to be renormalized by the gap opening solving the following equation:

[TABLE]

Three distinct regimes of multi-gap superconductivity are obtained as a function of the chemical potential tuned around the 2D-1D Lifshitz transition, as reported in Fig. 6. At the n-th band bottom, when the new Fermi surface pocket starts to appear, there is a coexistence of a BCS-like condensate of the pairs of the (n-1)-th band together with a BEC-like condensate of the pairs of the n-th band, which because of the very low density condense completely in a bosonic liquid. In this regime the critical temperature is extremely low and small variations of the parameters can lead to large variations in the gaps and in the $T_{c}$, determining a large peaked value of the isotope coefficient. The peak in the isotope coefficient reported in panel (B) is a finger print of our BPV theory and it signals the strong interplay between the Lifshitz transition at the band bottom and the onset of shape resonant superconductivity. Increasing the chemical potential trough the second Lifshitz transition, the resonant regime of maximum $T_{c}$ and gaps is obtained. This resonant regime is characterized by an interesting coexistence of BCS-like pair condensate of the (n-1)-th band and BCS-BEC crossover-like pair condensate of the n-th band. In the resonance regime the two gaps differ by a sizable 2.5 factor, the isotope effect gets its smallest values and $T_{c}$ can reach the high value of 123 K with coupling strengths in the different channels not exceeding 0.3, values still typical of metals, as Nb. The large energy scale of the phonon (145 meV) determines not only a large prefactor for the critical temperature, but also induces a large width of the resonant regimes, making relatively simple to find the right doping levels giving the highest critical temperatures. Finally, for larger chemical potentials, a third regime of conventional two-band superconductivity is reached, with coexistence of two-particles condensates having both BCS-like character, confirmed also by values of the isotope coefficient approaching 0.5 and small values of the gaps typical of weakly-coupled superconductors. The predicted zero temperature superconducting gaps to $T_{c}$ ratios, panel (A), and the Isotope coefficient as function on the critical temperature, panel (B) are reported in Fig. 7 so that the present theoretical work can be confirmed by direct experiments. This theory does not include superconducting fluctuations. However it should be noticed that in a multigap / multiband system another fundamental phenomenon helps in stabilizing high temperature superconductivity: the screening of the superconducting fluctuations. In fact the multi-band BCS-BEC crossover in a two-band superconductor (one condensate in the BCS regime, the other in the BCS-BEC regime) can determine the optimal condition to allow the screening of the detrimental superconducting fluctuations due to the large stiffness of the BCS-like condensate in the deep band. Finally an arrested or frustrated phase separation is expected to occur at a topological Lifshitz transition [62, 63] as it was observed in cuprates, and diborides [64, 65], diborides [66].

Acknowledgements.

This work has been supported by superstripes-onlus. Discussions with Gianni Profeta, Augusto Marcelli, Nicola Poccia, Andrea Ienco, Corrado Di Nicola, Fabio Marchetti, Claudio Pettinari are acknowledged.