Correlated electron-hole mechanism for molecular doping in organic semiconductors

The electronic and optical properties of the paradigmatic F4TCNQ-doped pentacene in the low-doping limit are investigated by a combination of state-of-the-art many-body \emph{ab initio} methods accounting for environmental screening effects, and a carefully parametrized model Hamiltonian. We demonstrate that while the acceptor level lies very deep in the gap, the inclusion of electron-hole interactions strongly stabilizes dopant-semiconductor charge transfer states and, together with spin statistics and structural relaxation effects, rationalize the possibility for room-temperature dopant ionization. Our findings reconcile available experimental data, shedding light on the partial vs. full charge transfer scenario discussed in the literature, and question the relevance of the standard classification in shallow or deep impurity levels prevailing for inorganic semiconductors.


I. INTRODUCTION
Doping of organic semiconductors (OSC) 1,2 by introduction in the host matrix of strong electron-or hole-donating molecules has been shown to increase their electrical conductivity by orders of magnitude, leading to enhanced performances in organic light-emitting devices and photovoltaic cells. However, in contrast to inorganic semiconductors where doping 3 is understood to proceed via the formation of shallow dopant levels, the case of OSC remains The emerging picture is that of strongly interacting molecular host-dopant pairs resulting in only partially ionized dopants. On the other hand, the scanning tunneling microscopy (STM) images by Ha and Kahn 15 instead show that isolated F4TCNQ in pentacene films are fully ionized at room temperature, an observation that has been rationalized on the basis of electrostatic modeling 18 .
In this paper, we revisit the case of F4TCNQ-doped pentacene in the low-doping regime and analyze its electronic and optical properties with a combination of many-body ab initio and model Hamiltonian electronic structure calculations, both explicitly accounting for electron-hole correlations. We show that despite very deep acceptor levels in the pen-tacene gap, electron-hole interactions result in thermally accessible states with fully ionized F4TCNQ dopants. The broader picture obtained from our correlated electron-hole model describes doping as a competition between neutral and ionized dopants, passing through a narrow window of fractional CT. Our electronic structure calculations locate the pentacene- Our computational approach relies on state-of-the-art ab initio electronic structure calculations using Green's function many-body perturbation theories within the GW 19,20 and Bethe-Salpeter equation (BSE) 20,21 formalisms. Extensive benchmarks against reference quantum-chemical calculations have demonstrated the accuracy of these approaches for calculating quasiparticle energy levels [22][23][24] and optical excitation energies 25,26 , BSE properly accounting for the long-range electron-hole interactions crucial for CT excitations 27,28 . The GW formalism is embedded in a recently developed 29 hybrid quantum/classical (QM/MM) scheme where many-body effects in the QM system are combined with an accurate discrete polarizable model accounting for the dielectric screening by the MM environment, known to largely affect the energies of charged and CT excitations 30 . This approach that proved to reproduce accurately the experimental photoemission gap and bulk (periodic) GW calculations in pristine pentacene 29 is here extended to optical excited states within the BSE framework.
The paper is organized as follows. We first present in Section II the embedded ab initio many-body formalism used, and the complementary model Hamiltonian that allows us to explore finite size effects and polaronic coupling. Our results are presented in Section III, followed by a discussion in Section IV. After the conclusions and perspectives (Section V), we gather in an Appendix convergence and validation tests performed on a small but representative subsystem. For sake of consistency, we start with an hybrid functional PBEh(α=0.4), 37 namely a functional with 40% of exact exchange, selected so as to match the starting Kohn-Sham gap with the GW one (see below). BSE calculations are performed at the Tamm-Dancoff (TDA) level that produces accurate energies for CT excitations.
The MM region is described by the charge response (CR) model by Tsiper and Soos 38 in its MESCAL code implementation. 39 This approach describes the anisotropic molecular response to electric fields in terms of induced atomic charges and induced dipoles, providing an accurate description of the static dielectric tensor of molecular solids. 40

B. Model Hamiltonian for molecular doping
Our accurate many-body ab initio analysis is complemented with a generalized Mulliken model for intermolecular CT that allows us to describe systems of larger size and introduce structural relaxation (polaronic) effects.
where ε CT im are CT states energies, t CT im is the DOP-OSC charge transfer integral and t h ij is the hole transfer integral between OSC sites. Triplet excitations are described by the same Hamiltonian restricted to the subspace of CT states |i m with triplet spin pairing.
Hamiltonian 1 describes Coulombically-bonded and possibly delocalized electron-hole pairs and can be considered an extension of similar models successfully applied to the description of intra- 43,44 and intermolecular 45 CT.
Hamiltonian 1 applies to doping in molecular or polymer OSC and is here accurately parametrized from first principles for F4TCNQ-doped pentacene. Diabatic CT states energies can be written as: 30 Molecular orbital energies are calculated from gas-phase evGW calculations that yield E PEN HOMO−1 = 7.84 eV, E PEN HOMO = 6.36 eV and E DOP LUMO = 3.76 eV. The polarization energy P ± im accounts for electrostatic and screening effects in the solid-state. P ± i0 is evaluated for each CT state with CR calculations, whose results are extrapolated in the infinite crystal limit. Polarization energies of CT states with the hole in the pentacene HOMO−1 are set to Hole transfer couplings t h are computed at the DFT level (PBE0 functional, 6-31G(d) basis) with the projective method. 46 The same approach proved to be strongly dependent on the functional for pentacene-F4TCNQ CT couplings t CT , for which we instead applied  the CT couplings and the energies of localized (diabatic) CT states annotated on the lattice of doped pentacene. Figure 3 shows the electron-hole distance dependence of the exciton binding energy of diabatic CT states from CR calculations, which closely follows a screened Coulomb potential even at relatively short distance.
Hamiltonian 1 can be extended to account for intramolecular structural relaxation upon charging within the framework of the Mulliken-Holstein model. We hence introduce one effective mode per molecule with coordinate q i , here treated within the adiabatic (Born-Oppenheimer) approximation, linearly modulating the energies of frontier molecular orbitals.
The electronic Hamiltonian in the presence of Holstein coupling is formally equivalent to being N is the number of molecular sites. The total energy includes the harmonic elastic contribution from molecular deformation, where λ + and λ − are the polaron binding energies for hole and electrons on OSC and DOP sites, respectively. These quantities have been calculated for PEN (λ + = 52 meV) and F4TCNQ (λ − = 140 meV) at the DFT level (PBE0 functional, 6-311G(d) basis) using differences of total energy obtained at the molecular geometries fully relaxed in the neutral and charged state (∆SCF scheme).

III. RESULTS
A. Embedded GW and BSE calculations of F4TCNQ-doped pentacene The model system investigated here considers F4TCNQ substitutional defects in the pentacene crystal lattice 15 -see Figure 1. Within our hybrid formalism, we describe a supramolecular complex (CPX) formed by one F4TCNQ molecule surrounded by its first shell of six pentacene neighbors (1+6 CPX henceforth) at the GW /BSE level. This CPX is then embedded into the pentacene crystal described within the charge response (CR) model 38,39 , which provides an accurate description of the anisotropic static dielectric response of molecular crystals 39 . F4TCNQ adopts the same position and orientation as that of the replaced pentacene molecule and its geometry has been optimized in vacuum at the CCSD level. The pentacene structure is taken from X-ray diffraction data for the vapor-grown polymorph 49 . We stress that such an approach goes significantly beyond previous DFT electronic structure calculations 4 , in terms of method accuracy, QM system size and account of an atomistic polarizable embedding. where shallow impurity levels are located within a few dozen of meVs from the band edges.
As shown in Figure 4(b), the GW HOMO-LUMO gap of the CPX is indeed found to be 0.67 eV, dramatically larger than room temperature thermal energy, clearly evidencing that the standard theory accepted for inorganic semiconductors does not apply to OSC.
The analysis of the frontier orbital isocontours in Figure 4 We now turn to optical excitations as obtained within the BSE formalism. The (screened) electron-hole interaction dramatically lowers the optical gap as compared to the HOMO-LUMO gap, as evidenced by comparing Figure 4(b) and 4(c). The resulting CPX optical absorption spectrum is shown in Figure 5(a). The most salient feature is that the lowest singlet excitation energy (S 1 ) is found to be extremely low in energy, namely 34 meV above the ground state. The analysis of the corresponding BSE electron-hole two-body eigenstate, represented in Figure 5 The analysis of the contributing levels reveals that these excitations correspond mainly to CT transitions from the manifold of pentacene HOMO−1 orbitals to the F4TCNQ LUMO.
We will come back to this point in Section IV.

B. Model validation and size effects
Even though based on an accurate ab initio many-body framework which is at the forefront of what can be achieved today in terms of system size and complexity, the present GW /BSE calculations on the 1+6 CPX miss to incorporate potentially important effects associated with the delocalization of the transferred hole over pentacene molecules beyond the first shell of neighbors and local structural relaxation or polaronic effects. To extend the reach of our analysis to larger system sizes, we resort to the generalized Mulliken model for intermolecular CT presented in Section II B.
The diagonalization of Hamiltonian 1 with parameters specific to F4TCNQ-doped pentacene, as described in Section II B, yields ground and excited states for systems large enough The absorption spectrum computed with Hamiltonian 1 for the 1+6 CPX is compared to BSE results in Figure 5(a). Again, the low-energy region is characterized by three CT transitions to electronic excited states where the hole is equally shared on the HOMOs of pairs of symmetry-equivalent pentacene molecules. In the 1.2-1.8 eV energy span, we predict three other absorption peaks corresponding to CT excitations where the hole lies in the pentacene HOMO−1 orbitals. The agreement with BSE is excellent for both the relative energy and intensity of the electronic transitions and the shape of the excited-state electron-hole maps, see Figure 5(b-c). This provides an important validation step for our model.
Finite-size effects are also addressed in Figure 5 We hence extended the model to account for one effective Holstein mode per molecule linearly coupled to the site charge as described in Section II B. The optimization of the ground state energy with respect to the set of intra-molecular coordinates leads to a qualitatively different symmetry-broken ground state of full-CT character, shown in the inset of Figure  6(c). Indeed, because of the two low-lying degenerate CT states at the undistorted molecular geometries, a structural (Jahn-Teller like) instability develops in the system, leading to the collapse of the hole on one of the nearest pentacene neighbors -see Figure 6 Extensive spectroscopic (photoemission, optical and vibrational) investigations by Koch and co-workers allowed identifying two different scenarios for doped molecular crystals, characterized by strong orbital hybridization and partial CT, and conjugated polymers, mostly exhibiting full ionization of dopant impurities. 2,4,8,14 Our accurate theoretical analysis, tightly connected to experimental evidence, concludes that F4TCNQ-doped pentacene represents an exception to this empirical rule, at least in the low-doping regime targeted by our calculations. We further note that at least one case of partial CT has also been reported in conjugated polymers. 13 We emphasize that the absence of intragap features in the photoemission spectra of doped pentacene is not inconsistent with full dopant ionization and with the presence of pentacene integrals and the coupling to vibrations are found to be key for dopant ionization. All these quantities can be strongly dependent on the material and on the morphology, calling for a detailed analysis of structure-property relationships. The proposed approach provides a robust framework to such a scope.
On more general grounds, this study puts the accent on the central role of electronhole interaction and polaronic effects in favoring dopant ionization, two effects that are missed in the well-established theories for electrical doping in inorganic semiconductors, whose applicability to the case of organic systems is here called into question. In contrast to the picture prevailing for doped inorganic semiconductors, the explanation for the high conductivity in heavily doped organics should thus be sought beyond independent-electron theories.
Appendix A: Test calculations with the embedded many-body approach

Stability with respect to starting Kohn-Sham eigenstates and basis set
To perform extensive benchmark calculations at a reasonable cost, we study a relatively small pentacene-F4TCNQ complex (CPX) including the dopant and two of its pentacene neighbors, named 1+2 CPX and shown in Figure 7. The actual choice of such a geometry is motivated by the calculations on the larger 1+6 CPX discussed in Section III, presenting its lowest-energy excitation S1 localized on the two molecules of the 1+2 CPX (see Figure 7).
We first address the stability of the evGW and Bethe-Salpeter results, namely the fact that the calculated gap and excitation energies hardly depend on the Kohn-Sham eigenstates chosen as input for the GW calculations. As shown in several publications, 35,36 the Kohn-Sham HOMO-LUMO gap of donor-acceptor complexes varies dramatically with the amount of exact exchange. This is confirmed in the present case (see Table I We show in Table II that the 6-311G(d) atomic basis set provides energy differences, namely HOMO-LUMO gap and excitation energies, well converged as compared to the ones obtained with the much larger correlation-consistent cc-pVTZ basis set.

Full BSE versus Tamm-Dancoff approximation and BSE Hamiltonian size
We show in Table III that