Superconductivity and Its Enhancement in Polycyclic Aromatic Hydrocarbons

The discovery of superconductivity above 30 K in aromatic hydrocarbons in the past years has been an exciting event in condensed matter physics, chemistry, and materials. Superconductivity was realized by simply introducing electrons in the network made by carbon and hydrogen - the two components for organic systems. However, the accurate amount of doped electrons and their positions have not been established experimentally. No agreement has been reached for the mechanism of their superconductivity. Here, the first principles theory with the combination of electron correlations is used to address these problems. Our calculations show that both the single-ring compound (benzene) and the two-benzene rings compound (naphthalene) are superconductors with transition temperatures (Tc’s) of 6.2 K and 5.8 K respectively after upon doing of potassium. By examining the existing experimental data carefully, we find that there exists a unique superconducting phase with Tc in the range of 5 - 7 K range for all aromatic hydrocarbons. The almost constant low density of states at Fermi level with about two electrons doping accounts for this unified phase. The high density of states at Fermi level upon three-electron doping seems responsible for the high-Tc phase in these hydrocarbons with long benzene rings. Meanwhile, the electronic correlations also increase with increasing the number of benzene rings. These findings offer clues to the understanding of the superconductivity as well as to the search of new superconductors in this family.


INTRODUCTION
Organic materials with a crystal structure made primarily of a complex network based on carbon and/or hydrogen were postulated to superconduct with high transition temperature (T c ) and may even above room temperature [1]. Organic superconductors such as charge-transfer compounds are made up of electron donor molecules such as tetrathiafulvalene (TTF), bis-ethylenedithrio-TTF (BEDT-TTF, abbreviated as ET), and tetramethyltetraselenafulvalene (TMTSF) derivatives, as well as electron acceptor molecules such as tetracyanoquinodimethane (TCNQ) and fullerides [2][3][4][5], which were discovered many years ago and are being extensively studied. In 2010 and subsequent years, polycyclic aromatic hydrocarbons (PAHs) were found to exhibit superconductivity [6][7][8][9], and their discoveries have attracted great interest in these organic compounds. Experimental data showed that there usually exist multiple superconducting phases with different superconducting HPSTAR 740-2019 transition temperature T c , and the highest T c seems to increase with the number of benzene rings within the unit cell [6][7][8][9]. These compounds all have a superconducting phase with T c ∼ 5 − 7 K. In addition, spin-liquid state has been also realized in an ionic PAH [10]. Recently, the crystal structures of potassium-intercalated PAHs have been experimentally determined [11]. This information is very useful in theoretical studies on the superconducting phase and for understanding the driving force for superconductivity. However, a systematic study on the superconductivity in PAH, e.g., the relation between T c and the number of benzene rings, is lacking. The issue of what is the exact amount of the cation intercalation or doping into the PAHs to induce superconductivity remains unsettled. Benzene−, being the simplest aromatic hydrocarbon, has six carbons on a ring and six hydrogens attached, i.e., C 6 H 6 . Other PAHs are made of more benzene rings. Thus, being the basic unit of PAH, solid benzene stands out as an ideal system to start with and address relevant issues. A natural question is whether such a compound could superconduct when doped by alkali atoms. If it superconducts, will its properties mimic those of all aromatic hydrocarbons? What factor is important for higher T c ? Exploring structures of doped solid benzene and examining corresponding phase stability, followed by the study of superconductivity, are the tasks of this work.
Solid benzene, whose crystallization characteristics have been extensively investigated [12][13][14], has a rich phase diagram with evolution into several phases by changing temperature and pressure. Liquid benzene crystallizes at room temperature and about 0.07 GPa, in an orthorhombic phase with space group Pbca (phase I), containing four formula units (f.u.) per unit cell [12]. At 1.4 GPa, a transition occurs from phase I to phase II (P4 3 2 1 2), and then at 4 GPa, phase II transfers to phase III (P2 1 /c), followed by phase IV (11 GPa) and phase V at higher pressures. Solid benzene has been found to be nonmetallic and non-superconducting over a very broad pressure range. Upon heavy compression, solid benzene was predicted to undergo an insulator-to-metal transition in the pressure range of 180-200 GPa [15], suggesting that benzene is a possible superconductor under pressure. However, inherent to highpressure techniques is the difficulty for experimental diagnosis and observation. Therefore, instead, we chose potassium (K) to dope solid benzene at ambient pressure and examine the possible realization of metallization and superconductivity. If doped solid benzene superconducts at ambient pressure, it would serve as an example to uncover the underlying origin of superconductivity in PAHs.
Although experimental studies on doping K into solid benzene have yet to be performed, the interaction between the benzene molecule and the K + ion has already been observed experimentally as well as studied theoretically by López et al. [16], a resource which implies the feasibility of doping potassium into solid benzene. Notably, doping potassium has been shown to be the most effective and common method to induce superconductivity in PAHs [6][7][8][9]. One reason is that K has a relatively low melting point as compared to other alkali elements, which makes the chemical reaction and crystallization from liquid states of K and benzene suitable.

Structural Optimizations and Electronic Properties Calculation
All structural optimizations and electronic properties were calculated by using the Vienna ab initio simulation package (VASP) [17,18]. In this work, the projector augmented wave (PAW) method [19] was adopted, and the exchange correlation energy was described by the Ceperley-Alder local density approximation (LDA) [20] as parameterized by Perdew and Zunger [21], which has been confirmed to be adequate by previous studies [22]. An energy cutoff of 500 eV was used for the plane wave basis sets. Thus, C-2p, H-1s, and K-3p4s were treated as valence states. The Brillouin zone (BZ) was sampled by 4 × 4 × 4 Monkhorst-Pack k-point grids during the optimization, and double k-point in the electronic structural calculations. The convergence thresholds were set as 10 −6 eV in energy and 0.005 eV/Å in force. A conjugate-gradient algorithm was used to relax the ions into their instantaneous ground state.

Electron-Phonon Interaction and Relevant Parameter Calculations
To calculate electron-phonon interaction and relevant parameters, we used the QUANTUM-ESPRESSO package (QE) [23] with a cutoff energy of 60 and 450 Ry for wave functions and charge densities, respectively. Forces and stresses for the converged structures were optimized and checked to within the error allowance of the VASP and QE codes. Troullier-Martins norm-conserving scheme [24] was used to generate the pseudopotentials for C, H, and K atoms. An 8 × 8 × 8 Monkhorst-Pack k-point grid with Gaussian smearing of 0.03 Ry was used for the electron-phonon interaction matrix element calculations at 2 × 2 × 2 q-point mesh.

Prediction of Superconductivity in Benzene and Naphthalene
As a comparison and a check on accuracy, we start our investigation by looking at solid benzene. Figure 1A shows the optimized crystal structure of solid benzene from different orthographic directions, with crystal lattice constants a = 7.041 Å, b = 8.903 Å, and c = 6.357 Å at zero temperature and ambient pressure, which are 3-6% less than experimental values at 78 K [12]. If the effect of temperature is considered, the error between theoretical prediction and experimental measurement is acceptable. Solid benzene is a wide band gap insulator as shown in Figure S1A. Our calculated band gap of 4.1 eV is in good agreement with previous results [13]. From the reported data on PAHs, their superconductivity is highly sensitive to both the doping concentration and the cation positions in the unit cell. To examine both effects, we performed optimization studies on doped solid benzene, K x C 6 H 6 , by varying potassium contents (x = 1, 2, 3) and the position of K atoms in the unit cell. The optimized crystal lattice constants and atomic coordinates are summarized in Table S1. The formation energy (E f ) is a good indication for examining the stability. We calculated the formation energy of the doped system using the following: where E K x C 6 H 6 and E C 6 H 6 is the total energy of the doped and host solid benzene, respectively. x is the number of doped K atoms. µ K is the chemical potential of the K species. µ K (bulk) can be obtained from the energy per K atom in the K metal with the bcc structure. When µ K = µ K (bulk), the doped element content is so rich that the related pure element phase can form. When E f < 0, the doped compound can stably exist. The calculated result (see Figure S2) shows that K 2 C 6 H 6 has lower formation energy than K 1 C 6 H 6 and K 3 C 6 H 6 in a certain region of chemical potential, and its associated negative sign suggests easier synthesis. Figure 1B shows the optimized K 2 C 6 H 6 crystal structure from different directions, with the following lattice constants: a = 8.605 Å, b = 10.705 Å, and c = 6.415 Å. When compared with the undoped case, we found that the lattice constants expand largely in the a and b directions. The insertion of eight K-atoms makes the volume expand by 26.9%. By K doping, a visible layered structure is formed, differing from that of the solid benzene. As shown in Figure 1B, all benzene rings are rotated and parallel to the bc-plane under the interaction of dopants, and K atoms are intercalated into two layers of benzene rings, occupying the Wyckoff 8c sites. The optimized structure of K 2 C 6 H 6 is found to tend toward the higher symmetry of Fmmm from the lower Pbca symmetry. Significantly, as shown in Figure S3, the layered structure of K 2 C 6 H 6 is similar to those of YbC 6 and CaC 6 [25]. This similarity indicates the possibility of superconductivity, because both YbC 6 and CaC 6 are known superconductors, with their T c 's reaching 6.5 K and 11.5 K, respectively [25,26].
Once the crystal structure is determined, it is possible to obtain information about the electronic band structure. For K 2 C 6 H 6 , eight conduction bands (called CB1, CB2, ... and CB8) along a high symmetry k-direction, Figure 2A. Specifically, under either Pbca or Fmmm symmetry, the lowest unoccupied molecular orbitals (LUMO) and LUMO+1 split into eight bands from Ŵ to Z, then they merge into four degenerate bands along the Z − T − Y − S − X − U directions, and further into two highly degenerated bands along the U − R direction. Electrons transferred from the K atoms to the C 6 H 6 molecule occupy low energy orbitals and raise the Fermi level. Three bands-CB1, CB2, and CB3-are fully occupied by six of eight electrons, while the other three-CB4, CB5 and CB6-are partially filled by the remaining two electrons with Fermi level passing through. Both CB7 and CB8 are empty. Thus, K 2 C 6 H 6 is metallic but in the low-spin state. Compared with the solid benzene, we noticed that the intermolecular distance along benzene rings becomes smaller on the bc-plane of K 2 C 6 H 6 . This effect enhances the intermolecular interaction and broadens the dispersion of energy bands. Analyzing the electronic characters near the Fermi level, as shown in Figure 2B, we found that the hole-like Fermi surfaces are formed by CB4 along the Ŵ − Z k-direction, while both CB5 and CB6 contribute the electron-like Fermi surfaces around the Y k-point. Fermi surface nesting is quite clear. Calculation of the projected density of states (PDOS) shows that the Cp y and C-p z states are fully occupied and far below the Fermi level, while the C-p x state has a large PDOS around the Fermi level, as shown in Figure 2C. Most PDOS of K-(s, p x , p y , p z ) states are above the Fermi level, indicating the electrons transfer from K to C-p x orbital. The large ratio of C-p x PDOS over K PDOS follows that C-p x electronic states contribute more to the Fermi surfaces.
Next we examine whether this simplest aromatic hydrocarbon could exhibit superconductivity like other PAHs observed with superconductivity [6][7][8][9]. Assuming the pairing mechanism is still electron-phonon interaction mediated [22,27], we evaluated the Eliashberg spectral function α 2 F(ω), the electron-phonon coupling integral λ(ω), and projected phonon density of states (PhDOS) of K 2 C 6 H 6 as a function of frequency. The results are shown in Figure 3. The vibration in the low frequency range of 0-250 cm −1 comes mainly from K, C, and H atom modes as shown in Figure 3C. The appearance of phonon modes of C and H atoms in the region of low frequency implies a weak intermolecular interaction. The modes with moderate frequencies of 300-1450 cm −1 are formed by C and H vibrations (Figure 3C), which results from the intramolecular interaction. The total λ is 0.67, which is comparable to that of K 3 C 22 H 14 [22] when assuming that the superconductivity in K 3 C 22 H 14 is mainly mediated by the electron-phonon interaction. As shown in Figure 3B, we also noted that vibrational modes from C and H atoms contribute 58% and 19% to the total λ value, respectively, while the remaining 23% is provided by the dopant phonon modes. Following the modified McMillan equation of Allen and Dynes [28], with the obtained values of λ(ω) and logarithmic average of phonon frequencies ω log , and a typical value of 0.10 for the Coulomb pseudopotential µ ⋆ , we obtained T c = 6.2 K for K 2 C 6 H 6 .
The dominant contribution to the total λ from the vibrational modes of the C atoms in K 2 C 6 H 6 shows some similarity with other C 6 -based or low-carbon-bearing compounds. In reality, CaC 6 was discovered to exhibit superconductivity at 11.5 K [25]. Superconductivity has also been discovered in intercalation compounds of graphite with alkali metals AC 8 (A =K, Rb, or Cs) [29]. The predicted T c of 6.2 K in K 2 C 6 H 6 is comparable to the temperature range of these low-carbon-based superconductors [25,29]. Naphthalene (C 10 H 8 ) is the second simple aromatic compound. Following the benzene crystal, a careful analysis on the 2-ring case also showed that K-doped naphthalene (K x C 10 H 8 ) possesses a superconducting phase with T c ∼ 5.2 K comparable to the doped solid benzene. The crystal structures induced by doping are shown in Figure S4 and the lattice parameters are listed in Table S2. K 2 C 10 H 8 was identified to be a more stable phase from the calculated formation energy result shown in Figure S5. As observed in solid benzene, the doping of two electrons per molecule is the most stable. Different from the metallicity in K 2 C 6 H 6 , K 2 C 10 H 8 is a semiconductor with a band gap about 0.75 eV. Similar to charged chrysene (C 18 H 12 ) [30] and picene (C 22 H 14 ), however, naphthalene would become a metal if doped with two electrons with a small amount of charge fluctuations (see [24] and Figures S6-S8). For charged K 2 C 10 H 8 , the Eliashberg spectral function α 2 F(ω) and the electron-phonon coupling integral λ(ω) as a function of frequency were also investigated ( Figure S9). The values of λ = 0.64 and T c = 5.8

Variation of Superconducting Transition Temperature Among Aromatic Hydrocarbons
By the same approach mentioned above, we have also investigated the superconductivity of K-doped phenanthrene (K x C 14 H 10 ). The theoretically predicted T c for this material is marked out in Figure 4, which is in good agreement with the experimental value. The calculated superconducting parameters of K x C 14 H 10 are presented in Figure S10C and Table S3. A comparison of the predicted superconducting of benzene, naphthalene and phenanthrene with other PAHs reveals a common feature of aromatic hydrocarbons (Figure 4). Besides the reported high-T c phases with T c = 18 K for K-doped picene (K x C 22 H 14 ) [6], 15 K for K-doped coronene (K x C 24 H 12 ) [7], and 33 K for K-doped 1,2:8,9-dibenzpentacene (K x C 30 H 18 ) [9], multi-superconducting phases in long-benzenering hydrocarbons were generally observed. A low T c phase with almost the same T c range, 5 − 7 K, exists in these hydrocarbon superconductors discovered so far [6-9]-5 K for K x C 14 H 10 [8], 7 K for K x C 22 H 14 [6,7], and 7.4 K for K x C 30 H 18 [9]-though several other transitions with higher T c 's were also observed. T c of 5 − 7 K was also reported for K x C 24 H 12 [7] in addition to three other transitions. Moreover, both Rb 3 C 22 H 14 and Ca 1.5 C 22 H 14 FIGURE 4 | Superconducting transition temperatures of aromatic hydrocarbons. Variation of superconducting transition temperature T c with the number of benzene rings (n) for various aromatic hydrocarbons. The gray region is the guide for eyes. Solid red squares represent our predicted results, while the hollow red square implies the predicted result by Casula et al. [22] Other data are taken from experiments.
show the superconducting transition at temperatures around 7 K [6]. In particular, Sm-doped phenanthrene, chrysene and picene show the superconductivity around T c ∼ 5 K [31]. It is important to note that the superconducting transitions were readily observed in the temperature range of 5 − 7 K for all reported aromatic hydrocarbons with various numbers of benzene rings. These observations are in good agreement with our current prediction for benzene, naphthalene, and phenanthrene, the relatively simple compounds of this series. So a natural question is what accounts for this most common low-T c phase and why the compounds with multiple benzene rings have high-T c phases.
For the number of benzene rings from one to seven, we have compared the growth situations of all aromatic hydrocarbons based on the calculated formation energies shown in Figures S11, S12. For K-doped phenanthrene shown in Figure S11C, similar to benzene and naphthalene, K 2 C 14 H 10 is more stable in the chemical potential region satisfying E f < 0. Hence, we suggest that the superconducting phase of T c ∼ 5 K observed by experiment [8] is K 2 C 14 H 10 instead of K 3 C 14 H 10 . For K-doped chrysene (K x C 18 H 12 ), a fourbenzene-ring system, our previous study [30] has indicated that K 2 C 18 H 12 is the most stable case in the chemical potential region satisfying E f < 0 (Also see Figure S11D). As a result, when intercalating two K atoms, the doped system is the most stable for benzene, naphthalene, phenanthrene, and chrysene. Here, we call them short-benzene-ring systems. Differing from these short-benzene-ring systems, satisfying E f < 0, as shown in Figure S12A, K 2 C 22 H 14 firstly obtains stable phase with the increase of K chemical potential, and then, under the K-rich limited condition, the high-concentration doping phase of K 3 C 22 H 14 can also be obtained. The same  situations also occur in K-doped [6]phenacene (K x C 26 H 16 ) and K x C 30 H 18 (See Figure S12). Thus we put picene, [6]phenacene, and 1,2;8,9-dibenzopentacene in the same class as longbenzene-ring systems. From these calculated results, for shortbenzene-ring systems, the doping of two electrons is the most stable, while for long-benzene-ring systems, both the dopings of two and three electrons are, respectively, the more stable phase in the different regions of K chemical potential. Thus, the doping phases with different K contents can be obtained in long-benzene-ring systems. However, in all aromatic hydrocarbons, stability with two-electron doping is a common feature. The unified phases with two-electron dopant for all the studied compounds in this series are consistent with recent experiments [11]. The experimental results of band gaps are taken from previous reports [7,[42][43][44][45][46].
With these analyses above, we are confident to conclude that the presence of the superconducting phase with T c around 5 − 7 K is a common feature of all aromatic hydrocarbons. In most cases, the low-T c phase was easily fabricated and more stable than the higher T c phase [6,7,9]. Considering the fact that the cation concentration has never been experimentally identified by using the reliable neutron scattering or single crystal Xray diffraction techniques, all the reported doping levels are an ideal chemical estimation based on the synthesis procedure. Our phase stability examination for K-doped aromatic hydrocarbons pins down the doping concentration of the exact two electrons for the stable superconducting phase of these hydrocarbons with T c of 5 − 7 K. The superconducting phase with high-T c is apparently the characteristic feature of long-benzene-ring systems, tending toward high-concentration doping such as three or more electrons.
In addition, we have investigated the density of states at Fermi level (N(E F )) to analyze the origin of this unified superconducting phase, since the stable phase of low-concentration doping can be obtained in all aromatic hydrocarbons. As shown in Figure 5, the low-concentration K 2 -doping results in small and comparable values on the N(E F ) for all aromatic hydrocarbons. Moreover, the obtained ω log values of 199.8 K for K 2 C 6 H 6 , 210.4 K for charged K 2 C 10 H 8 , and 193.7 K for K 2 C 14 H 10 are also comparable in quantity with 207 K of K 3 C 22 H 14 as shown in [19]. These results indicate that almost the same strength of electronphonon coupling exists in all the K 2 -doping cases. Based on the superconductivity mediated by the electron-phonon interaction, we suggest that the unified superconducting phase of 5 − 7 K mainly comes from the low and the almost-same N(E F ) value induced by two-electron doping. The electron-phonon interaction is enough to describe the superconductivity of this unified superconducting phase. With the help of the analysis of stability and DOS at the Fermi level above, we predict that the T c of K-doped chrysene is in the range of 5 − 7 K as shown in Figure 4. Although a complete understanding of the mechanism of superconductivity in aromatic hydrocarbons is yet to be achieved, the common feature of the 5 − 7 K superconducting phase for all reported aromatic hydrocarbons provides important clues for understanding the superconductivity itself as well as for the search of new superconductors in this series.
For long-benzene-ring systems, multi-doping concentrations are feasible. At a high-concentration doping level, as shown in Figure 6, K 3 -doping clearly leads to larger values of N(E F ) compared with K 2 -doping. Within the framework of the BCS theory, K 3 -doping can drive a stronger electron-phonon interaction to exhibiting higher-T c superconductivity. Hence, for each long-benzene-ring system, this high-T c phase is related to the high N(E F ) value induced by the three-electron doping. Regarding the enhancement of superconductivity among aromatic hydrocarbons, this is also related to the increase of the N(E F ) with the number of benzene rings as shown in Figure 6B. Based on the discussion above combining the stability with the values of N(E F ), we predicted that the T c of K 2 -doped PAHs is in the range of 5 − 7 K, while the T c of K 3

Electronic Correlations
In this section we address mechanism for high-T c phases. Although electron-phonon interaction is still possible to account for higher transition temperatures with certain crystal structure, other possibilities exist. Like other superconductors such as cuprates, further investigations indicate that the electronelectron correlations seem to play an important role on superconductivity in these systems. In fact, this was similar to charge-transfer organic superconductors [32][33][34][35][36][37] and fullerides [3,38], where the importance of electron correlations on superconductivity was emphasized. Magnetic measurements [8] on phenanthrene suggest the existence of local spin moments and indicate similar interplay between magnetism and superconductivity observed in fullerides [3].
To gain some insights on the exchange-correlation effects of electron-electron (e-e), we have carried out the series of calculations using the hybrid functionals method (HSE) [39,40]. The exchange-correlation energy E HSE xc was written as where µ is a screened Coulomb potential which splits the electron-electron exchange interaction into short-range (SR) and long-range (LR) components, the mixing coefficient ν represents the amount of exact/DFT exchange, indicating the extent of Fock exchange and reflecting the intensity of electronic correlation. Within the framework of the HSE method, the correlation interaction is still presented by the PBE functional form [41], the exchange interaction is separated into two parts-SR and LR. The LR exchange item is also described by PBE functional form, while the SR exchange item further splits into exact exchange and PBE forms. Both µ and ν are adjustable parameters; in this calculation, we adjusted them to investigate the extension of e-e interactions in PAH compounds. For PAH crystal, the band gap is one of the electronic properties reflected by the exchange-correlation interactions. Usually, the LDA functional underestimates the value of band gap due to the incompletely counting of exchange-correlation effects. As mentioned above, the HSE functional can correct the exchange-correlation interaction in PAH compounds as possible as well. The parameter ν just describes the magnitude of the exchange-correlation effects by fitting the band gap between theoretical values and experimental ones. Table 1 lists out the parameters of µ and ν fitting experimental band gaps for the typical systems such as solid phenanthrene (3rings-A), chrysene [4rings-A(B)], picene (5rings-A), 1,2:5,6dibenzanthracene (5rings-B), [6]phenacene (6rings-A), 1,2:7,8dibenzanthracene (6rings-B), [7]phenacene (7rings-A), and 1,2:8,9-dibenzopentacene (7rings-B), respectively. Results show that the ratio ν of exact/DFT exchange becomes larger with the increase of the number of rings whether µ is 0.1 or 0.2, indicating the enhancement of electronic exchangecorrelation interactions with the increase of the number of benzene rings.
Another quantity to calculate is the effective Coulomb interaction, e.g., the Hubbard U. Combining the estimated Hubbard U with the calculated bandwidth of bands crossing the Fermi level (see the computational details in the Supplementary Material), we indeed found that the electron-electron correlation becomes important as the number of benzene rings increases, as shown in Figure 7. Multi-ring hydrocarbons are in favor of strong correlations and thus tend to have higher T c s. It is therefore believed that both the electron-phonon interaction and electron-electron correlations work together to enhance T c of the PAHs with increasing the number of benzene rings.

CONCLUSIONS
In summary, carrying out first-principles calculations, we have investigated the crystal structures and possible superconductivity in potassium-doped solid benzene and naphthalene. Furthermore, we have analyzed the stability from formation energy, electronic structures near the Fermi level, and electronic correlations of K-doped PAHs. It was found that: (1) Solid benzene can be superconducting when doping with two K atoms; (2) the low T c phase of 5 − 7 K is a common feature in all aromatic hydrocarbon superconductorsfor this unique superconducting phase, the doping level is around two electrons in both short-benzene-ring and longbenzene-ring systems, and the electron-phonon interaction is enough to account; and (3) for higher T c phases in longbenzene-ring systems, the doping level tends toward three electrons. Both the electron-phonon interaction and electronelectron correlations work together to enhance the T c in long-benzene-ring systems.