Electronic excitations in solution-processed oligothiophene small-molecules for organic solar cells , F. Gala , L. Mattiello, F. Brunetti, and G. Zollo
Citation: J. Chem. Phys. 144, 084310 (2016); doi: 10.1063/1.4942501 View online: http://dx.doi.org/10.1063/1.4942501 View Table of Contents: http://aip.scitation.org/toc/jcp/144/8 Published by the American Institute of Physics
THE JOURNAL OF CHEMICAL PHYSICS 144, 084310 (2016)
Electronic excitations in solution-processed oligothiophene small-molecules for organic solar cells F. Gala,1,a) L. Mattiello,1 F. Brunetti,2 and G. Zollo1
1
Dipartimento di Scienze di Base e Applicate per l’Ingegneria, Universitá di Roma “La Sapienza,” Via A. Scarpa 14–16, 00161 Rome, Italy 2 Dipartimento di Ingegneria Elettronica, Universitá di Roma “Tor Vergata,” Via del Politecnico 1, 00133 Rome, Italy
(Received 7 December 2015; accepted 3 February 2016; published online 29 February 2016) First principles calculations based on density functional theory and many body perturbation theory have been employed to study the optical absorption properties of a newly synthesized oligothiophene molecule, with a quaterthiophene central unit, that has been designed for solutionprocessed bulk-heterojunction solar cells. To this aim we have employed the GW approach to obtain quasiparticle energies as a pre-requisite to solve the Bethe-Salpeter equation for the excitonic Hamiltonian. We show that the experimental absorption spectrum can be explained only by taking into account the inter-molecular transitions among the π-stacked poly-conjugated molecules that are typically obtained in solid-state organic samples. C 2016 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4942501]
I. INTRODUCTION
Nowadays organic photovoltaic materials are considered as promising alternatives to silicon for next-generation solar cells suitable for the green economy. Such organic cells, indeed, have many appealing properties such as low fabrication cost, solution processability, transparency, and flexibility. Moreover, power conversion efficiencies (PCEs) of polymer based solar cells (PSCs) with bulk-heterojunction (BHJ) architecture have been greatly improved in the recent years.1 A BHJ material2 is a solid state mixture of two nanostructured components: the donor and the acceptor; light absorption at the donor generates strongly bound Frenkeltype excitons3,4 that, at the donor-acceptor interface, have to be separated to generate the electrical current between the electrodes. Fullerene derivates are common choices for acceptor materials in BHJs, while many efforts have been devoted to the design and synthesis of donor materials, in order to improve the organic solar cells’ performances. In addition, huge efforts have been devoted to the development of π-conjugated small molecules to be employed as donors in BHJ solar cells, being the PCEs of such solar cells greatly increased up to 9%−10%5–8 in the last years. Although the PCEs of solution-processed small molecule organic solar cells have not matched those achieved by PSCs, their advantages of manageable synthesis and stability, high purity, batch to batch control, and easier energy level design, cannot be ignored. Small π conjugated molecule with push-pull structure, i.e., made of an electron-donating unit (D) as central a)Author to whom correspondence should be addressed. Electronic mail:
[email protected] 0021-9606/2016/144(8)/084310/8/$30.00
building block and electron-accepting units (A) as end groups, has been intensively investigated5,9–11 but a full quantitative understanding of the processes involved in the formation of excitons in such molecules (that might be also employed as donors in BHJ solar cells) is still lacking. Thus, theoretical calculations of the excited properties of such small molecules can provide a better understanding of the exciton processes involved also in BHJ solar cells. Such theoretical investigations, however, require an accurate description of electron-hole (eh) interactions that, in principle, could be addressed in the context of quantum-chemical approaches (such as, for instance, the coupled-cluster method) but with prohibitive computational effort. A good compromise between accuracy and computational cost can be achieved by using the GW approximation12 and the Bethe-Salpeter equation (BSE),13,14 in the framework of many-body perturbation theory (MBPT). Such methods have been successfully employed to calculate the optical excitations in organic crystals15,16 and in small organic molecules.17–19 However a straightforward comparison between theoretical calculation and experiments is, in general, a difficult task, due to a few different effects that are neglected in our GW+BSE calculations, such as dynamical effects in exciton screening or electron-phonon coupling. In this work we study the absorption spectra of a newly assembled oligomer-like molecule for organic solar cells. The absorption spectrum calculated by GW-BSE of such molecules is compared to the experimental one showing a good agreement for stacked configurations showing that stacking properties are of great importance to explain the observed optical properties. As mentioned, the subject of our investigations is a π small molecule, the diethyl 3, 3′-(3, 3′′′-dioctyl[2, 2′:5′, 2′′:5′′, 2′′′-quaterthiophene]-5, 5′′′-diyl) (2E,2′E)-bis-
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(2-cyanoacrylate), with chemical formula C44H52N2O4S4 (BT2N in the following), to be employed as donor material in BHJ solar cells. BT2N has been obtained by the means of relatively simple and low-cost synthetic procedures and possesses good solubility in most common organic solvents. The detailed synthesis of BT2N will be described and published elsewhere. In principle, a quantitative comparison with the experimental absorption of a BT2N thin-film should be carried out with an organic molecular crystal made of BT2N units; however, due to the large physical size of such oligothiophene molecules, together with the high level of accuracy needed to obtain precise results in the framework of GW+BSE theory, we will try to compare experimental spectra with two systems; one made of a single moleculed and a second one made of two stacked molecules.
where ε (0) and ⟨x|n⟩ = φ n (r) are the eigenvalues and n eigenfunctions of the DFT Hamiltonian, respectively, and ) −1 ( ∂Σn , (2) Zn = 1 − ∂ω ω=ε (0) n where Σn (ω) = ⟨n|Σ(r1r2,ω)|n⟩ is the Fourier transform of the dynamical electron self-energy operator. Within the GW approximation,29 Σ(r1r2,ω) can be cast in terms of the ground state DFT Green’s function G(r1r2,ω) as dω ′ iω′η e G(r1r2,ω − ω ′)W (r1r2,ω ′), Σ(r1r2,ω) = i lim+ η→ 0 2π (3) with G(r1r2,ω) = lim+ η→ 0
n
and W (r1r2,ω) =
The ground state electronic and geometric properties of a single isolated BT2N molecule and of two isolated BT2N molecules have been obtained by density functional theory20,21 (DFT) with a generalized gradient approximation based on the Perdew-Burke-Ernzerhof (PBE) formula22 for the electron exchange and correlation potential Vxc [n(r)], and norm-conserving pseudopotentials have been constructed with the Troullier-Martins scheme23 in the framework of a plane-wave basis set expansion. DFT calculations have been performed using the QUANTUM-ESPRESSO package,24 with a plane-wave energy cutoff of 70 Ry for the wave functions. Structural optimization has been achieved in a cubic box of 20 × 40 × 20 Å3 using the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method25 together with the Hellmann-Feynman forces acting on the ions; an empirical dispersion forces26 is included to handle the long range interactions (π stacking) between the molecules. The chosen box size is large enough to prevent any spurious interaction arising from the effects of the periodically repeated images of the molecules. All the calculations have been performed using the Γ point for the Brillouin zone sampling, and the ionic minimization was done until the convergence threshold of 0.001 a.u. for the total force was achieved. Theoretical bandgaps have been obtained with three different levels of accuracy: in the context of ground state DFT either as the difference between the lowest unoccupied molecular orbital (LUMO) and the highest occupied molecular orbital (HOMO) (simply referred as ε gap) or as the difference between the electron affinity and the ionization potential F ε ∆SC = E N +1 − E N − (E N − E N −1) (the so called ∆-SCF gap method) where E x is the DFT ground state total energy of the system containing x electrons,27 and finally through the GW method that allows the calculation of the quasi-particle (QP) energies in the context of MBPT; the last task has been attained using the YAMBO code.28 In particular, QP energies are computed to first order as (0) (0) ε QP n = ε n + Z n ⟨n|[Σ(ε n ) − Vxc ]|n⟩,
(1)
ω−
[ε (0) n
II. THEORETICAL AND EXPERIMENTAL METHODS A. Theoretical methods
φ n (r1)φ∗n (r2)
dr3
+ iηsgn(µ − ε (0) n )]
ϵ −1(r3r2,ω) |r1 − r3|
(4)
(5)
is a dynamically screened interaction, expressed in terms of the inverse dielectric function ϵ −1 of the system and of the bare Coulomb interaction V . The inverse dielectric matrix is related to the response function χ(12) = δ ρ(1)/δVe xt (2) (where (1) is a short-hand notation for (r1,t 1)) via the relation χ(32) ϵ −1(12) = δ(12) + d(3) . (6) |r1 − r3| In the RPA,28 χ is related to the non-interacting response function χ(0)(12) = G(12)G(21+), through the Dyson-like equation 1 χ(12) = χ(0)(12) + d(34) χ(0)(13) χ(42). (7) |r3 − r4| Another common approximation for the screened interaction is the plasmon pole approximation (PPA),28,30 in which it is assumed that the imaginary part of W is characterized by a strong peak corresponding to a plasmon excitation at the plasmon frequency; both methods (the GWRealAxis and the GW-PPA schemes in the following), will be employed to evaluate the quasiparticle corrections to DFT energy gaps. The above equations, alternatively, can be expressed in reciprocal space, by exploiting the corresponding Fourier ′ ′ representation f (r, r′) = G,G′ eiG·r f GG′e−iG ·r for all the twopoint functions defined above, thus, for example, Eq. (7) can be recast as −1 4π (0) χG1,G2(ω) = δG1,G2 − χG ,G (ω) χ(0) (8) G3,G2(ω) 2 1 3 |G3| G 3
(the reader is referred to the Appendix for a detailed discussion of the parameters chosen in the numerical implementation of Eqs. (3), (4) and (8)). A box-like potential cutoff in the long-range Coulomb potential28,31 is used at this stage to simulate truly isolated molecular excited states, this fact is essential17 for reaching good convergence in the self-energy calculations.
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Using the above obtained dynamically screened interaction, the macroscopic complex dielectric function that includes the excitonic effects is obtained by solving BSE in the eh space made of eh pairs |eh⟩ and antipairs |he⟩; its solution, in fact, can be mapped onto an eigenvalue problem for a two particle excitonic Hamiltonian13,14 of the form H res H coupl H exc = , (9) −(H coupl)∗ −(H res)∗ where the resonant term H res = (Ee − Eh )δ e,e′δ h,h′ + ⟨eh|K |e ′h ′⟩ is Hermitian, the coupling part H coupl = ⟨eh|K |h ′e ′⟩ is symmetric, and K = W − 2V is the excitonic kernel (the reader is referred to Ref. 14 for a detailed explanation of the block terms in H exc). The macroscopic dielectric function ϵ M (ω) is calculated from the eigenvalues Eλ and the eigenstates |λ⟩ of H exc as ϵ M (ω) = 1 − limV(q) lim+ q→ 0
× Sλ−1λ ′
η→ 0
⟨n1|e−iq·r|n2⟩
λ λ ′ n 1n 2
Anλ1n2 ω − Eλ + iη
( f n3 − f n4)⟨n4|eiq·r |n3⟩(Anλ3n4)∗, ′
′
(10)
n 3n 4 λ ′ with Ann ′ = ⟨nn |λ⟩, and Sλ λ ′ is an overlap matrix of the generally non-orthogonal eigenstates of H exc. λ From the excitonic eigenfunctions Ann ′ the electron-hole wave function is obtained as λ ∗ (11) Φ λ (re , rh ) = Ann ′φ n (re )φ n ′(rh ).
(TBABF4) as supporting electrolyte at a scan rate of 100 mVs−1. The concentration of all the substrates was 1 mM. In these conditions the standard redox potential of the ferrocenium/ferrocene system is E◦(Fc+/Fc) = +0.46 V vs SCE. Next, a 18 mg/ml solution in chloroform of BT2N has been spin coated in controlled nitrogen atmosphere at 1000 RPM for 60 s on glass substrates previously cleaned in ultrasonic bath with acetone and ethanol (10 min each step). Subsequently, 15 min of thermal annealing at 50 ◦C has been performed to remove any residual solvent without degrading the film performances. The UV/Vis absorption spectrum has been obtained from Shimadzu UV-2550 spectrophotometer.
III. RESULTS AND DISCUSSION A. Single BT2N molecule
Different isomers of the BT2N molecule have been studied, and the corresponding DFT ground state structures have been reported in Fig. 1, together with their corresponding relative energies with respect to the most stable one (corresponding to the configuration shown in Fig. 1(a)); such configuration shows an almost planar geometry as well as a definite π conjugation as can be inferred from the HOMO and the LUMO orbitals (Figs. 2(a) and 2(b), respectively). The corresponding energy levels and the DFT bandgap have
nn ′
Finally, we obtain the polarizability of the system (isolated or stacked molecules) using the Clausius-Mossotti relations19 α∥ =
3Ω ϵ ∥ − 1 4π ϵ ∥ + 2
(12)
along the leading dimension of BT2N molecule (the transverse polarizability α⊥ is negligible with respect α ∥ for all the cases here studied) and a comparison between the GW+BSE prediction and the experimental absorption spectra is obtained from the imaginary part of α ∥ . Dynamical effects in the excitonic Hamiltonian have been neglected, since it has been shown18 that they affect optical gaps of oligothiophenes by a small uniform reduction of 0.1 eV in the corresponding optical gaps; electron-phonon effects are completely neglected in our calculations, thus no vibronic replicas will appear in the calculated theoretical spectra. B. Experimental methods
Voltammetric experiments were performed using an AMEL System 5000 apparatus. Measurements were carried out, under nitrogen atmosphere, in a conventional three electrode cell with a saturated calomel electrode with multiple junctions as the reference electrode, a Pt wire as the counter electrode and a glassy carbon electrode (GCE) as the working electrode. Prior to experiments, the glassy carbon electrode was polished with alumina paste 0.05 µm. All experiments were carried out at 25 ◦C, in anhydrous dichloromethane (DCM) with 0.1M tetrabutylammonium tetrafluoroborate
FIG. 1. Different ground state configurations studied for the single BT2N molecule, together with the relative energies with respect to the most stable one.
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FIG. 2. Isosurface plots of ∥ψ(r)∥ 2 for the calculated DFT HOMO and LUMO orbitals (in (a) and (b), respectively), together with their normalized x z average values as a function of y (c). Isosurfaces have been set to ∼8% (HOMO) and ∼12% (LUMO) of the maximum value of ∥ψ(r)∥ 2, in order to evidence their π and π ∗ characters of the HOMO and LUMO levels.
been reported in Table I, together with the corresponding experimental values measured by cyclic voltammetry (CV) for a BT2N solution in dichloromethane. The BT2N chemical structure suggests a typical acceptordonor-acceptor (A-D-A) configuration; such an A-D-A character emerges from the orbital localization, even though it is not marked probably because of the reduced length size of the molecule; indeed, looking at HOMO and the LUMO probability densities along the molecule axis (the y axis), that have been obtained by integrating the probability densities in the xz plane, it can be seen that the HOMO is only partially localized on the electron-donating unit (i.e., the two central thiophene units between y1 and y2 in Fig. 2(c)), the probability to find an electron there being about PyHOMO ∼ 0.53 implying 1−y2 that the HOMO state is almost equally spread between the electron accepting and donating parts of the molecule; the same quantity for the LUMO is PyLUMO ∼ 0.31 meaning that 1−y2 this orbital is preferentially located at the acceptor ends of the molecule.
Structure
GW-RealAxis
GW-PPA
∆-SCF
BT2Nx1 BT2Nx2
3.84 3.52
3.93 3.60
3.89 3.33
As expected, the HOMO-LUMO gap is largely underestimated (being nearly of 1.39 eV) at the DFT-PBE level with respect to the experimental value obtained by CV (see Table I). The correct theoretical bandgap obtained from ∆-SCF and the quasi-particle bandgaps, calculated by GW (with either GW-RealAxis or GW-PPA approximation), are in nice accordance between them (see Table II) indicating that the computational scheme adopted for GW is correct. The obtained results, however, are much larger than the experimental bandgap that is a clear sign of the leading role of excitons in the BT2N molecule that will affect markedly the optical absorbance phenomena too. The quasi-particle energy levels obtained for both the GW-PPA and the GW-RealAxis are compared to the DFT-PBE eigenvalues and are shown in Fig. 3(c) in the energy range [−1.5 eV, 2 eV] from the HOMO level considered as a reference value at 0 eV. Subsequently we have calculated the optical absorption spectrum by solving the BSE starting from both GW-PPA and GW-RealAxis energy levels; our results are shown in Fig. 3(b) together with the experimental absorption spectrum obtained from a BT2N thin-film. Although the distance between the experimental curve and the calculated ones is greatly reduced, some important differences emerge between the theory and the experiment: the first one is that GW+BSE are clearly redshifted with respect to the experimental absorption spectrum by nearly ∆λ ∼ 50 nm and ∆λ ∼ 100 nm for, respectively, GW-PPA and GW-RealAxis approximations while a better agreement is found for optical bandgaps (calculated from the onset of the absorption curves35), located at 704 nm, 715 nm, and 769 nm for the experimental, GW-PPA, and GW-RealAxis curves, respectively. A second major difference is that the GW+BSE spectra do not show the second peak, located near 600 nm, that is present in the experimental spectrum. The third difference is the absence of the vibronic peak in the GW+BSE calculated spectrum that, on the contrary, is present in the experimental absorption spectrum and is located at nearly 525 nm. This last discrepancy, however, is not surprising, since electron-phonon effects have been completely neglected in the present model.
TABLE I. DFT-PBE HOMO-LUMO energies, together with the experimental bandgap measured with cyclic voltammetry for the single BT2N isolated molecule and for two π-stacked BT2N molecules (in parentheses). ε HOMO (eV)
ε LUMO (eV)
ε gap (eV)
Name
DFT-PBE
Expt.
DFT-PBE
Expt.
DFT-PBE
Expt.
BT2N
−4.96 (−4.56)
−5.49
−3.57 (−3.27)
−3.47
1.39 (1.30)
2.02
Structure
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FIG. 3. Normalized absorption spectra for BT2N molecule, for light polarized parallel to the main axis of the molecule (a). In (b) benchmark of the main peak position as a function of the electron states included in the construction of the excitonic Hamiltonian H exc defined in (9), while in (c) QP corrections to DFT-PBE electron eigenvalues in an energy window ranging from −1.5 eV below to +2.0 eV above the e HOMO level. For the sake of clarity absorption curves have been obtained by imposing an artificial Lorentzian broadening of 100 meV and they are normalized to the maximum value of the curve obtained by GW-RealAxis+BSE.
Because the calculated values of the QP electronic gap (3.93 eV and 3.84 eV for GW-PPA and GW-RealAxis, respectively) are much larger than the optical bandgap obtained from the optical spectrum, it is clear that the calculated optical spectrum reflects the existence of strong excitonic transitions, with high excitonic binding energy (1.92 eV and 1.96 eV for GW-PPA+BSE and GW-RealAxis+BSE, respectively) that is found to be entirely associated to single particle π → π ∗ transition from the HOMO to the LUMO levels. Although there is a similarity among the calculated and the experimental spectra, they are still quite different, with two of the three main differences described above being included in principles in the level of theory here adopted. As a consequence we have examined the adsorption spectrum obtained with two molecules that are stacked, as one should expect in a real sample. B. Two BT2N molecules
Indeed, the disagreement between the experimental and the theoretical absorption properties of the BT2N molecule previously highlighted can be understood by considering a system that is closer to the thin-film solid state phase where molecular packing can play a key role in the optical properties measured. On the contrary, atomistic models of isolated molecules are suitable to study the optical properties of such molecules in the gas phase, rather than solid state one. We emphasize that a true solid state model with a nearly periodic arrangement of the molecules is still nowadays not accessible at this level of theory for computational reasons. Regular packing of the molecules is often based on a herringbone arrangement (where molecules are organized in layers with an edge-to face arrangement), or on a cofacial π-stacked interaction of adjacent molecules.32 We
have chosen the latter model since this configuration has been experimentally observed in blend films made of simple oligomer-like small molecules8,33 similar to BT2N, showing a π-π stacking with the distance depending on the number of central thiophene units of the molecules and ranging from 3.56 Å (7 units) to 3.74 Å (6 units). Thus we have studied the DFT ground state of two BT2N molecules in a typical π stacked configuration; after structural relaxation the electronic structure of the system shows a DFT bandgap of 1.3 eV, slightly smaller than the one obtained for the isolated molecule; this fact is due to the splitting into two states of the HOMO π state (one for each molecule) as reported in Fig. 5(c). Another remarkable difference with respect to the single isolated BT2N molecule case is that the HOMO and the LUMO do not lie anymore on the same molecule (see Figs. 4(b) and 4(c)) suggesting that the HOMO-LUMO optical transition may be accompanied by an electron charge transfer (CT) between the two molecules that might change the transition energy with respect to the isolated molecule. GW corrections to DFT-PBE energies, with both GWRealAxis and GW-PPA schemes, result in a gap opening (ε QP gap = 3.6 eV for GW-RealAxis and 3.5 eV for GW-PPA) that is, again, close to the ∆-SCF value of 3.3 eV (see Table II). The calculated energy gap values again still overestimate markedly the experimental bandgap measured by CV thus confirming the leading role of the excitonic phenomena that must be considered in order to explain the experimental results. The optical absorption spectrum is calculated for the π stacked configuration by solving again the BSE starting from both GW-PPA and GW-RealAxis energy levels. Our results are shown in Fig. 5(a) where it can be seen that, now, the obtained theoretical curves are in close agreement with the experimental absorption spectrum, especially concerning the GW-RealAxis approximation that is the highest level of
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FIG. 4. Isosurface plots of ∥ψ(r)∥ 2 for the DFT calculated HOMO-1, HOMO, LUMO, LUMO+1, and LUMO+2 orbitals in (a), (b), (c), (d), and (e), respectively; isosurfaces have been set to ∼5% of the corresponding maximum values in order to evidence their symmetry properties.
FIG. 5. Normalized absorption spectra for two π conjugated BT2N molecules, for light polarized parallel to the main axis of the molecule (a), while in (b) QP corrections to DFT-PBE electron eigenvalues in an energy window ranging from −1.5 eV below to +2.0 eV above the e HOMO level. As for the single molecule case absorption curves have been obtained by imposing an artificial Lorentzian broadening of 100 meV and they are normalized to the maximum value of the curve obtained by GW-PPA+BSE.
theory here adopted; indeed the absorption peak predicted in this case is located at 558 nm that is just 4 nm away from the experimental peak position measured at the wavelength of 554 nm. A better understanding of the excitonic nature of the optical transitions can be obtained by looking at the eigenvalues and eigenvectors of the excitonic Hamiltonian defined in Eq. (11) and obtained with the GW-RealAxis scheme; the results of such inspection are shown in Fig. 6, where the Lorentzian broadening parameter η of Eq. (10) is reduced so that the contribution to the system polarizability of the single eigenstates Eλ of the excitonic Hamiltonian can be isolated and studied. From Fig. 6 it appears clearly that
the main peak in the normalized experimental absorption BT2N thin-film spectrum is due to the formation of two nearly degenerate excitons ((b) and (c) peaks in Fig. 6) with binding energies of 1.34 eV and 1.29 eV, respectively. A careful inspection of the H exc eigenstates evidences that the eh transition corresponding to the (b) peak is the HOMO → LUMO+2 transition (see Table III) between the HOMO and the LUMO+2 that are mainly located at two different BT2N molecules thus involving a marked CT between the stacked molecules. The other contribution to the main absorption peak is the eh eigenstate at (c) that arises from three transitions the HOMO-1 → LUMO, the HOMO-1 → LUMO+1, and the HOMO → LUMO+1. Of
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FIG. 6. Im[α ∥ ] obtained from Eq. (10) with a Lorentzian broadening of 10 meV compared with the normalized absorption spectrum. In the inset the electron-hole transitions that weigh more than 10% in the excitonic states (a)–(c) are schematically represented.
scheme is still affected by large errors concerning also the position of the main peak that is away from the experimental position by 50-100 meV. This discrepancy is completely recovered once the π conjugated geometry of BT2N molecules is considered as a possible organization pattern of oligothiophene molecules in the solid state. Indeed the absorption spectrum of two π conjugated BT2N molecules shows a nice agreement between the experiment and the prediction obtained with the highest level of accuracy employed in this article. The analysis of the eh transitions that contribute to the main absorption peak evidences the important role played by the inter-molecular optical transition in explaining the observed absorption behavior. Therefore, the importance of the π conjugation between the molecules is put in evidence in our study showing that the absorption properties of such molecules are tightly related to the way they are arranged in the matrix of organic solar cell device. ACKNOWLEDGMENTS
TABLE III. Electron-hole pairs that contributes to excitonic states more than 10%. Peak
Energy (eV)
eh pairs
Weight
(a)
2.02
HOMO-1 → LUMO+1 HOMO → LUMO+1 HOMO-1 → LUMO
0.61 0.25 0.12
(b)
2.18
HOMO → LUMO+2
0.87
(c)
2.23
HOMO-1 → LUMO HOMO-1 → LUMO+1 HOMO → LUMO+1
0.30 0.30 0.25
these three transitions, the first and the third ones involve molecular levels lying on the same molecules, as also reported in Table III, while the second one shows a CT behavior (see Fig. 4). In Fig. 6 a third peak at 2.02 eV is present, which again represents an exciton state with binding energy of 1.50 eV of lower optical strength with respect the (b) and (c) states; this feature is also present in the experimental curve, showing again that a strong hybridization between excitonic states on different molecules is the key for a better understanding of the optical properties of BT2N thin-films and small conjugated molecules in general. IV. CONCLUSIONS
In summary we have calculated in the context of the MBPT, using the GW approximation and the BSE for the excitonic Hamiltonian, the absorption spectra of two different systems made either of one isolated BT2N molecule or two π conjugated BT2N molecules. To this aim, we have tested two different levels of approximations, namely, the GW-PPA and the GW-RealAxis. The experimental absorption spectrum has been compared to the theoretical predictions evidencing, first of all, the excitonic nature of the measured absorption onset and main peak. However the predicted absorption spectrum of a single isolated BT2N molecule obtained with the adopted
Computational resources have been provided by CRESCO/ENEAGRID High Performance Computing infrastructure and its staff.34 CRESCO/ENEAGRID High Performance Computing infrastructure is funded by ENEA, the Italian National Agency for New Technologies, Energy and Sustainable Economic Development and by Italian and European research programmes, see http://www.cresco.enea.it/english for information. One of the authors (F. Gala) wish to express his gratitude to Dr. D. Varsano (and Giufá bookshop as well) for many helpful discussions concerning MBPT, in general, and the YAMBO code, in particular. APPENDIX: NUMERICAL CONVERGENCE OF THE CALCULATIONS
The basic component of the perturbative expansion in MBPT is the reference non-interacting system, which is represented by the solutions of the DFT equations; thus the accuracy of the obtained DFT eigenvalues have been checked by a comparison with the corresponding values calculated with an higher wave function energy cutoff (150 Ry), with the highest discrepancy found of the order of 3 · 10−2 eV. In the calculation of the quasiparticle bandgap ε QP gap, local field effects arising from charge redistribution induced by light absorption have been checked explicitly; Fig. 7(a) reports the obtained values for ε QP gap of the BT2N molecule as a function of the number of G-vectors (i.e., the matrix size) to be included in the Fourier transform of the response function χ in Eq. (8), showing that at least ∼900 G-vectors are needed to achieve convergence. Since the ε QP gap results may also depend on how much fine is the grid over which the integral in Eq. (3) is calculated, two different integration steps have been used (∆ω = 73.21 meV and 36.46 meV), but no appreciable differences emerged. Kohn-Sham energy levels up to 10.35 eV above the HOMO energy have been employed in the computation of the GW quasiparticle bandgap, corresponding 750 KS bands for the isolated BT2N molecule, while 1400 KS bands (corresponding
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FIG. 7. QP bandgap of an isolated BT2N molecule as a function of the matrix size of the response function χ for two different grid meshes in (a); while in QP (b) ε gap as a function of the number of bands included in the definition of the non-interacting Green function is shown.
to 13.83 eV from the HOMO level) have been employed in the case of two BT2N molecules. Such values ensure good convergence for quasiparticle bandgaps, as shown in Fig. 7(b) with errors of the order of 0.05 eV in the corresponding energy gaps. For the BSE, eh transitions up to ∼16 eV have been included in excitonic Hamiltonian (9), corresponding to 200 electronic states (100 occupied and 100 empty states), and ensuring no appreciable variations in the position of the leading absorbance peak, as shown in Fig. 3(b) in the case of the isolated BT2N molecule. 1L.
Dou, J. You, Z. Hong, Z. Xu, G. Li, R. Street, and Y. Yang, Adv. Mater. 25, 6642 (2013). 2A. Heeger, Adv. Mater. 26, 10 (2014). 3C. W. Tang, Appl. Phys. Lett. 48(2), 183 (1986). 4J.-L. Brédas, J. Norton, J. Cornil, and V. Coropceanu, Acc. Chem. Res. 42(11), 1691 (2009). 5Y. Liu, C.-C. Chen, Z. Hong, J. Gao, Y. Yang, H. Zhou, L. D. G. Li, and Y. Yang, Sci. Rep. 3, 3356 (2013). 6B. Kan, Q. Zhang, M. Li, X. Wan, W. Ni, G. Long, Y. Wang, X. Yang, H. Feng, and Y. Chen, J. Am. Chem. Soc. 136, 15529 (2014).
Zhang, B. Kan, F. Liu, G. Long, X. Wan, X. Chen, Y. Zuo, W. Ni, H. Zhang, M. Li et al., Nat. Photonics 9, 35 (2015). 8B. Kan, M. Li, Q. Zhang, F. Liu, X. Wan, Y. Wang, W. Ni, G. Long, X. Yang, H. Feng et al., J. Am. Chem. Soc. 137, 3886 (2015). 9H. Bai, Y. Wang, P. Cheng, Y. Li, D. Zhu, and X. Zhan, ACS Appl. Mater. Interfaces 6(11), 8426 (2014). 10Y. ad G. C. Welch, W. L. Leong, C. Takacs, and G. Bazan, Nat. Mater. 11, 44 (2012). 11D. Patra, T.-Y. Huang, C.-C. Chiang, R. Maturana, C.-W. Pao, K.-C. Ho, K.-H. Wei, and C.-W. Chu, ACS Appl. Mater. Interfaces 5, 9494 (2013). 12L. Hedin, Phys. Rev. 139, A796 (1965). 13M. Rohlfing and S. G. Louie, Phys. Rev. B 62(8), 4927 (2000). 14G. Onida, L. Reining, and A. Rubio, Rev. Mod. Phys. 74, 601 (2002). 15P. Cudazzo, M. Gatti, and A. Rubio, Phys. Rev. B 86, 195307 (2012). 16S. Sharifzadeh, A. Biller, L. Kronik, and J. B. Neaton, Phys. Rev. B 85, 125307 (2012). 17M. Palummo, C. Hogan, F. Sottile, P. Bagalá, and A. Rubio, J. Chem. Phys. 131, 084102 (2009). 18B. Baumeier, D. Andrienko, Y. Ma, and M. Rohlfing, J. Chem. Theory Comput. 8, 997 (2012). 19C. Hogan, M. Palummo, J. Gierschner, and A. Rubio, J. Chem. Phys. 138, 024312 (2013). 20P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964). 21W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965). 22J. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77(4), 3865 (1996). 23N. Troullier and J. Martins, Phys. Rev. B 43(14), 1993 (1991). 24P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni, I. Dabo et al., J. Phys.: Condens. Matter 21(19), 395502 (2009). 25R. Fletcher, Comput. J. 13(6), 317 (1970). 26S. Grimme, J. Comput. Chem. 27(15), 1787 (2006). 27R. M. Martin, Electronic Structure: Basic Theory and Practical Methods (Cambridge University Press, 2008). 28A. Marini, C. Hogan, M. Grüning, and D. Varsano, Comput. Phys. Commun. 180, 1392 (2009). 29L. Hedin and S. Lundqvist, Solid State Physics: Advances in Research and Application (Academic Press, New York, San Francisco, London, 1969). 30F. Aryasetiawan and O. Gunnarsson, Rep. Prog. Phys. 61, 237 (1998). 31C. Rozzi, D. Varsano, A. Marini, E. Gross, and A. Rubio, Phys. Rev. B 73, 205119 (2006). 32K. Gierschner, M. Ehni, H.-J. Egelhaaf, B. M. Medina, D. Belhonne, H. Benmansour, and G. C. Bazan, J. Chem. Phys. 123, 144914 (2005). 33R. Fitnzer, E. Reinold, A. Mishra, E. Mena-Osteritz, H. Ziehlke, C. K´’orner, K. Leo, M. Riedde, M. Weil, O. Tsaryova et al., Adv. Funct. Mater. 21, 897 (2011). 34G. Ponti, F. Palombi, D. Abate, F. Ambrosino, G. Aprea, T. Bastianelli, F. Beone, R. Bertini, G. Bracco, M. Caporicci et al., IEEE High Perform. Comput. Simul. 6903807, 1030 (2014). 35Optical bandgaps have been evaluated from the condition of being the intensity of the normalized absorption curve greater than 10% of its maximum value.
Citation: J. Chem. Phys. 144, 084310 (2016); doi: 10.1063/1.4942501 View online: http://dx.doi.org/10.1063/1.4942501 View Table of Contents: http://aip.scitation.org/toc/jcp/144/8 Published by the American Institute of Physics
THE JOURNAL OF CHEMICAL PHYSICS 144, 084310 (2016)
Electronic excitations in solution-processed oligothiophene small-molecules for organic solar cells F. Gala,1,a) L. Mattiello,1 F. Brunetti,2 and G. Zollo1
1
Dipartimento di Scienze di Base e Applicate per l’Ingegneria, Universitá di Roma “La Sapienza,” Via A. Scarpa 14–16, 00161 Rome, Italy 2 Dipartimento di Ingegneria Elettronica, Universitá di Roma “Tor Vergata,” Via del Politecnico 1, 00133 Rome, Italy
(Received 7 December 2015; accepted 3 February 2016; published online 29 February 2016) First principles calculations based on density functional theory and many body perturbation theory have been employed to study the optical absorption properties of a newly synthesized oligothiophene molecule, with a quaterthiophene central unit, that has been designed for solutionprocessed bulk-heterojunction solar cells. To this aim we have employed the GW approach to obtain quasiparticle energies as a pre-requisite to solve the Bethe-Salpeter equation for the excitonic Hamiltonian. We show that the experimental absorption spectrum can be explained only by taking into account the inter-molecular transitions among the π-stacked poly-conjugated molecules that are typically obtained in solid-state organic samples. C 2016 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4942501]
I. INTRODUCTION
Nowadays organic photovoltaic materials are considered as promising alternatives to silicon for next-generation solar cells suitable for the green economy. Such organic cells, indeed, have many appealing properties such as low fabrication cost, solution processability, transparency, and flexibility. Moreover, power conversion efficiencies (PCEs) of polymer based solar cells (PSCs) with bulk-heterojunction (BHJ) architecture have been greatly improved in the recent years.1 A BHJ material2 is a solid state mixture of two nanostructured components: the donor and the acceptor; light absorption at the donor generates strongly bound Frenkeltype excitons3,4 that, at the donor-acceptor interface, have to be separated to generate the electrical current between the electrodes. Fullerene derivates are common choices for acceptor materials in BHJs, while many efforts have been devoted to the design and synthesis of donor materials, in order to improve the organic solar cells’ performances. In addition, huge efforts have been devoted to the development of π-conjugated small molecules to be employed as donors in BHJ solar cells, being the PCEs of such solar cells greatly increased up to 9%−10%5–8 in the last years. Although the PCEs of solution-processed small molecule organic solar cells have not matched those achieved by PSCs, their advantages of manageable synthesis and stability, high purity, batch to batch control, and easier energy level design, cannot be ignored. Small π conjugated molecule with push-pull structure, i.e., made of an electron-donating unit (D) as central a)Author to whom correspondence should be addressed. Electronic mail:
[email protected] 0021-9606/2016/144(8)/084310/8/$30.00
building block and electron-accepting units (A) as end groups, has been intensively investigated5,9–11 but a full quantitative understanding of the processes involved in the formation of excitons in such molecules (that might be also employed as donors in BHJ solar cells) is still lacking. Thus, theoretical calculations of the excited properties of such small molecules can provide a better understanding of the exciton processes involved also in BHJ solar cells. Such theoretical investigations, however, require an accurate description of electron-hole (eh) interactions that, in principle, could be addressed in the context of quantum-chemical approaches (such as, for instance, the coupled-cluster method) but with prohibitive computational effort. A good compromise between accuracy and computational cost can be achieved by using the GW approximation12 and the Bethe-Salpeter equation (BSE),13,14 in the framework of many-body perturbation theory (MBPT). Such methods have been successfully employed to calculate the optical excitations in organic crystals15,16 and in small organic molecules.17–19 However a straightforward comparison between theoretical calculation and experiments is, in general, a difficult task, due to a few different effects that are neglected in our GW+BSE calculations, such as dynamical effects in exciton screening or electron-phonon coupling. In this work we study the absorption spectra of a newly assembled oligomer-like molecule for organic solar cells. The absorption spectrum calculated by GW-BSE of such molecules is compared to the experimental one showing a good agreement for stacked configurations showing that stacking properties are of great importance to explain the observed optical properties. As mentioned, the subject of our investigations is a π small molecule, the diethyl 3, 3′-(3, 3′′′-dioctyl[2, 2′:5′, 2′′:5′′, 2′′′-quaterthiophene]-5, 5′′′-diyl) (2E,2′E)-bis-
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(2-cyanoacrylate), with chemical formula C44H52N2O4S4 (BT2N in the following), to be employed as donor material in BHJ solar cells. BT2N has been obtained by the means of relatively simple and low-cost synthetic procedures and possesses good solubility in most common organic solvents. The detailed synthesis of BT2N will be described and published elsewhere. In principle, a quantitative comparison with the experimental absorption of a BT2N thin-film should be carried out with an organic molecular crystal made of BT2N units; however, due to the large physical size of such oligothiophene molecules, together with the high level of accuracy needed to obtain precise results in the framework of GW+BSE theory, we will try to compare experimental spectra with two systems; one made of a single moleculed and a second one made of two stacked molecules.
where ε (0) and ⟨x|n⟩ = φ n (r) are the eigenvalues and n eigenfunctions of the DFT Hamiltonian, respectively, and ) −1 ( ∂Σn , (2) Zn = 1 − ∂ω ω=ε (0) n where Σn (ω) = ⟨n|Σ(r1r2,ω)|n⟩ is the Fourier transform of the dynamical electron self-energy operator. Within the GW approximation,29 Σ(r1r2,ω) can be cast in terms of the ground state DFT Green’s function G(r1r2,ω) as dω ′ iω′η e G(r1r2,ω − ω ′)W (r1r2,ω ′), Σ(r1r2,ω) = i lim+ η→ 0 2π (3) with G(r1r2,ω) = lim+ η→ 0
n
and W (r1r2,ω) =
The ground state electronic and geometric properties of a single isolated BT2N molecule and of two isolated BT2N molecules have been obtained by density functional theory20,21 (DFT) with a generalized gradient approximation based on the Perdew-Burke-Ernzerhof (PBE) formula22 for the electron exchange and correlation potential Vxc [n(r)], and norm-conserving pseudopotentials have been constructed with the Troullier-Martins scheme23 in the framework of a plane-wave basis set expansion. DFT calculations have been performed using the QUANTUM-ESPRESSO package,24 with a plane-wave energy cutoff of 70 Ry for the wave functions. Structural optimization has been achieved in a cubic box of 20 × 40 × 20 Å3 using the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method25 together with the Hellmann-Feynman forces acting on the ions; an empirical dispersion forces26 is included to handle the long range interactions (π stacking) between the molecules. The chosen box size is large enough to prevent any spurious interaction arising from the effects of the periodically repeated images of the molecules. All the calculations have been performed using the Γ point for the Brillouin zone sampling, and the ionic minimization was done until the convergence threshold of 0.001 a.u. for the total force was achieved. Theoretical bandgaps have been obtained with three different levels of accuracy: in the context of ground state DFT either as the difference between the lowest unoccupied molecular orbital (LUMO) and the highest occupied molecular orbital (HOMO) (simply referred as ε gap) or as the difference between the electron affinity and the ionization potential F ε ∆SC = E N +1 − E N − (E N − E N −1) (the so called ∆-SCF gap method) where E x is the DFT ground state total energy of the system containing x electrons,27 and finally through the GW method that allows the calculation of the quasi-particle (QP) energies in the context of MBPT; the last task has been attained using the YAMBO code.28 In particular, QP energies are computed to first order as (0) (0) ε QP n = ε n + Z n ⟨n|[Σ(ε n ) − Vxc ]|n⟩,
(1)
ω−
[ε (0) n
II. THEORETICAL AND EXPERIMENTAL METHODS A. Theoretical methods
φ n (r1)φ∗n (r2)
dr3
+ iηsgn(µ − ε (0) n )]
ϵ −1(r3r2,ω) |r1 − r3|
(4)
(5)
is a dynamically screened interaction, expressed in terms of the inverse dielectric function ϵ −1 of the system and of the bare Coulomb interaction V . The inverse dielectric matrix is related to the response function χ(12) = δ ρ(1)/δVe xt (2) (where (1) is a short-hand notation for (r1,t 1)) via the relation χ(32) ϵ −1(12) = δ(12) + d(3) . (6) |r1 − r3| In the RPA,28 χ is related to the non-interacting response function χ(0)(12) = G(12)G(21+), through the Dyson-like equation 1 χ(12) = χ(0)(12) + d(34) χ(0)(13) χ(42). (7) |r3 − r4| Another common approximation for the screened interaction is the plasmon pole approximation (PPA),28,30 in which it is assumed that the imaginary part of W is characterized by a strong peak corresponding to a plasmon excitation at the plasmon frequency; both methods (the GWRealAxis and the GW-PPA schemes in the following), will be employed to evaluate the quasiparticle corrections to DFT energy gaps. The above equations, alternatively, can be expressed in reciprocal space, by exploiting the corresponding Fourier ′ ′ representation f (r, r′) = G,G′ eiG·r f GG′e−iG ·r for all the twopoint functions defined above, thus, for example, Eq. (7) can be recast as −1 4π (0) χG1,G2(ω) = δG1,G2 − χG ,G (ω) χ(0) (8) G3,G2(ω) 2 1 3 |G3| G 3
(the reader is referred to the Appendix for a detailed discussion of the parameters chosen in the numerical implementation of Eqs. (3), (4) and (8)). A box-like potential cutoff in the long-range Coulomb potential28,31 is used at this stage to simulate truly isolated molecular excited states, this fact is essential17 for reaching good convergence in the self-energy calculations.
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Using the above obtained dynamically screened interaction, the macroscopic complex dielectric function that includes the excitonic effects is obtained by solving BSE in the eh space made of eh pairs |eh⟩ and antipairs |he⟩; its solution, in fact, can be mapped onto an eigenvalue problem for a two particle excitonic Hamiltonian13,14 of the form H res H coupl H exc = , (9) −(H coupl)∗ −(H res)∗ where the resonant term H res = (Ee − Eh )δ e,e′δ h,h′ + ⟨eh|K |e ′h ′⟩ is Hermitian, the coupling part H coupl = ⟨eh|K |h ′e ′⟩ is symmetric, and K = W − 2V is the excitonic kernel (the reader is referred to Ref. 14 for a detailed explanation of the block terms in H exc). The macroscopic dielectric function ϵ M (ω) is calculated from the eigenvalues Eλ and the eigenstates |λ⟩ of H exc as ϵ M (ω) = 1 − limV(q) lim+ q→ 0
× Sλ−1λ ′
η→ 0
⟨n1|e−iq·r|n2⟩
λ λ ′ n 1n 2
Anλ1n2 ω − Eλ + iη
( f n3 − f n4)⟨n4|eiq·r |n3⟩(Anλ3n4)∗, ′
′
(10)
n 3n 4 λ ′ with Ann ′ = ⟨nn |λ⟩, and Sλ λ ′ is an overlap matrix of the generally non-orthogonal eigenstates of H exc. λ From the excitonic eigenfunctions Ann ′ the electron-hole wave function is obtained as λ ∗ (11) Φ λ (re , rh ) = Ann ′φ n (re )φ n ′(rh ).
(TBABF4) as supporting electrolyte at a scan rate of 100 mVs−1. The concentration of all the substrates was 1 mM. In these conditions the standard redox potential of the ferrocenium/ferrocene system is E◦(Fc+/Fc) = +0.46 V vs SCE. Next, a 18 mg/ml solution in chloroform of BT2N has been spin coated in controlled nitrogen atmosphere at 1000 RPM for 60 s on glass substrates previously cleaned in ultrasonic bath with acetone and ethanol (10 min each step). Subsequently, 15 min of thermal annealing at 50 ◦C has been performed to remove any residual solvent without degrading the film performances. The UV/Vis absorption spectrum has been obtained from Shimadzu UV-2550 spectrophotometer.
III. RESULTS AND DISCUSSION A. Single BT2N molecule
Different isomers of the BT2N molecule have been studied, and the corresponding DFT ground state structures have been reported in Fig. 1, together with their corresponding relative energies with respect to the most stable one (corresponding to the configuration shown in Fig. 1(a)); such configuration shows an almost planar geometry as well as a definite π conjugation as can be inferred from the HOMO and the LUMO orbitals (Figs. 2(a) and 2(b), respectively). The corresponding energy levels and the DFT bandgap have
nn ′
Finally, we obtain the polarizability of the system (isolated or stacked molecules) using the Clausius-Mossotti relations19 α∥ =
3Ω ϵ ∥ − 1 4π ϵ ∥ + 2
(12)
along the leading dimension of BT2N molecule (the transverse polarizability α⊥ is negligible with respect α ∥ for all the cases here studied) and a comparison between the GW+BSE prediction and the experimental absorption spectra is obtained from the imaginary part of α ∥ . Dynamical effects in the excitonic Hamiltonian have been neglected, since it has been shown18 that they affect optical gaps of oligothiophenes by a small uniform reduction of 0.1 eV in the corresponding optical gaps; electron-phonon effects are completely neglected in our calculations, thus no vibronic replicas will appear in the calculated theoretical spectra. B. Experimental methods
Voltammetric experiments were performed using an AMEL System 5000 apparatus. Measurements were carried out, under nitrogen atmosphere, in a conventional three electrode cell with a saturated calomel electrode with multiple junctions as the reference electrode, a Pt wire as the counter electrode and a glassy carbon electrode (GCE) as the working electrode. Prior to experiments, the glassy carbon electrode was polished with alumina paste 0.05 µm. All experiments were carried out at 25 ◦C, in anhydrous dichloromethane (DCM) with 0.1M tetrabutylammonium tetrafluoroborate
FIG. 1. Different ground state configurations studied for the single BT2N molecule, together with the relative energies with respect to the most stable one.
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J. Chem. Phys. 144, 084310 (2016) TABLE II. Bandgaps (in eV) for the BT2N structures studied obtained with three different computational schemes.
FIG. 2. Isosurface plots of ∥ψ(r)∥ 2 for the calculated DFT HOMO and LUMO orbitals (in (a) and (b), respectively), together with their normalized x z average values as a function of y (c). Isosurfaces have been set to ∼8% (HOMO) and ∼12% (LUMO) of the maximum value of ∥ψ(r)∥ 2, in order to evidence their π and π ∗ characters of the HOMO and LUMO levels.
been reported in Table I, together with the corresponding experimental values measured by cyclic voltammetry (CV) for a BT2N solution in dichloromethane. The BT2N chemical structure suggests a typical acceptordonor-acceptor (A-D-A) configuration; such an A-D-A character emerges from the orbital localization, even though it is not marked probably because of the reduced length size of the molecule; indeed, looking at HOMO and the LUMO probability densities along the molecule axis (the y axis), that have been obtained by integrating the probability densities in the xz plane, it can be seen that the HOMO is only partially localized on the electron-donating unit (i.e., the two central thiophene units between y1 and y2 in Fig. 2(c)), the probability to find an electron there being about PyHOMO ∼ 0.53 implying 1−y2 that the HOMO state is almost equally spread between the electron accepting and donating parts of the molecule; the same quantity for the LUMO is PyLUMO ∼ 0.31 meaning that 1−y2 this orbital is preferentially located at the acceptor ends of the molecule.
Structure
GW-RealAxis
GW-PPA
∆-SCF
BT2Nx1 BT2Nx2
3.84 3.52
3.93 3.60
3.89 3.33
As expected, the HOMO-LUMO gap is largely underestimated (being nearly of 1.39 eV) at the DFT-PBE level with respect to the experimental value obtained by CV (see Table I). The correct theoretical bandgap obtained from ∆-SCF and the quasi-particle bandgaps, calculated by GW (with either GW-RealAxis or GW-PPA approximation), are in nice accordance between them (see Table II) indicating that the computational scheme adopted for GW is correct. The obtained results, however, are much larger than the experimental bandgap that is a clear sign of the leading role of excitons in the BT2N molecule that will affect markedly the optical absorbance phenomena too. The quasi-particle energy levels obtained for both the GW-PPA and the GW-RealAxis are compared to the DFT-PBE eigenvalues and are shown in Fig. 3(c) in the energy range [−1.5 eV, 2 eV] from the HOMO level considered as a reference value at 0 eV. Subsequently we have calculated the optical absorption spectrum by solving the BSE starting from both GW-PPA and GW-RealAxis energy levels; our results are shown in Fig. 3(b) together with the experimental absorption spectrum obtained from a BT2N thin-film. Although the distance between the experimental curve and the calculated ones is greatly reduced, some important differences emerge between the theory and the experiment: the first one is that GW+BSE are clearly redshifted with respect to the experimental absorption spectrum by nearly ∆λ ∼ 50 nm and ∆λ ∼ 100 nm for, respectively, GW-PPA and GW-RealAxis approximations while a better agreement is found for optical bandgaps (calculated from the onset of the absorption curves35), located at 704 nm, 715 nm, and 769 nm for the experimental, GW-PPA, and GW-RealAxis curves, respectively. A second major difference is that the GW+BSE spectra do not show the second peak, located near 600 nm, that is present in the experimental spectrum. The third difference is the absence of the vibronic peak in the GW+BSE calculated spectrum that, on the contrary, is present in the experimental absorption spectrum and is located at nearly 525 nm. This last discrepancy, however, is not surprising, since electron-phonon effects have been completely neglected in the present model.
TABLE I. DFT-PBE HOMO-LUMO energies, together with the experimental bandgap measured with cyclic voltammetry for the single BT2N isolated molecule and for two π-stacked BT2N molecules (in parentheses). ε HOMO (eV)
ε LUMO (eV)
ε gap (eV)
Name
DFT-PBE
Expt.
DFT-PBE
Expt.
DFT-PBE
Expt.
BT2N
−4.96 (−4.56)
−5.49
−3.57 (−3.27)
−3.47
1.39 (1.30)
2.02
Structure
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FIG. 3. Normalized absorption spectra for BT2N molecule, for light polarized parallel to the main axis of the molecule (a). In (b) benchmark of the main peak position as a function of the electron states included in the construction of the excitonic Hamiltonian H exc defined in (9), while in (c) QP corrections to DFT-PBE electron eigenvalues in an energy window ranging from −1.5 eV below to +2.0 eV above the e HOMO level. For the sake of clarity absorption curves have been obtained by imposing an artificial Lorentzian broadening of 100 meV and they are normalized to the maximum value of the curve obtained by GW-RealAxis+BSE.
Because the calculated values of the QP electronic gap (3.93 eV and 3.84 eV for GW-PPA and GW-RealAxis, respectively) are much larger than the optical bandgap obtained from the optical spectrum, it is clear that the calculated optical spectrum reflects the existence of strong excitonic transitions, with high excitonic binding energy (1.92 eV and 1.96 eV for GW-PPA+BSE and GW-RealAxis+BSE, respectively) that is found to be entirely associated to single particle π → π ∗ transition from the HOMO to the LUMO levels. Although there is a similarity among the calculated and the experimental spectra, they are still quite different, with two of the three main differences described above being included in principles in the level of theory here adopted. As a consequence we have examined the adsorption spectrum obtained with two molecules that are stacked, as one should expect in a real sample. B. Two BT2N molecules
Indeed, the disagreement between the experimental and the theoretical absorption properties of the BT2N molecule previously highlighted can be understood by considering a system that is closer to the thin-film solid state phase where molecular packing can play a key role in the optical properties measured. On the contrary, atomistic models of isolated molecules are suitable to study the optical properties of such molecules in the gas phase, rather than solid state one. We emphasize that a true solid state model with a nearly periodic arrangement of the molecules is still nowadays not accessible at this level of theory for computational reasons. Regular packing of the molecules is often based on a herringbone arrangement (where molecules are organized in layers with an edge-to face arrangement), or on a cofacial π-stacked interaction of adjacent molecules.32 We
have chosen the latter model since this configuration has been experimentally observed in blend films made of simple oligomer-like small molecules8,33 similar to BT2N, showing a π-π stacking with the distance depending on the number of central thiophene units of the molecules and ranging from 3.56 Å (7 units) to 3.74 Å (6 units). Thus we have studied the DFT ground state of two BT2N molecules in a typical π stacked configuration; after structural relaxation the electronic structure of the system shows a DFT bandgap of 1.3 eV, slightly smaller than the one obtained for the isolated molecule; this fact is due to the splitting into two states of the HOMO π state (one for each molecule) as reported in Fig. 5(c). Another remarkable difference with respect to the single isolated BT2N molecule case is that the HOMO and the LUMO do not lie anymore on the same molecule (see Figs. 4(b) and 4(c)) suggesting that the HOMO-LUMO optical transition may be accompanied by an electron charge transfer (CT) between the two molecules that might change the transition energy with respect to the isolated molecule. GW corrections to DFT-PBE energies, with both GWRealAxis and GW-PPA schemes, result in a gap opening (ε QP gap = 3.6 eV for GW-RealAxis and 3.5 eV for GW-PPA) that is, again, close to the ∆-SCF value of 3.3 eV (see Table II). The calculated energy gap values again still overestimate markedly the experimental bandgap measured by CV thus confirming the leading role of the excitonic phenomena that must be considered in order to explain the experimental results. The optical absorption spectrum is calculated for the π stacked configuration by solving again the BSE starting from both GW-PPA and GW-RealAxis energy levels. Our results are shown in Fig. 5(a) where it can be seen that, now, the obtained theoretical curves are in close agreement with the experimental absorption spectrum, especially concerning the GW-RealAxis approximation that is the highest level of
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FIG. 4. Isosurface plots of ∥ψ(r)∥ 2 for the DFT calculated HOMO-1, HOMO, LUMO, LUMO+1, and LUMO+2 orbitals in (a), (b), (c), (d), and (e), respectively; isosurfaces have been set to ∼5% of the corresponding maximum values in order to evidence their symmetry properties.
FIG. 5. Normalized absorption spectra for two π conjugated BT2N molecules, for light polarized parallel to the main axis of the molecule (a), while in (b) QP corrections to DFT-PBE electron eigenvalues in an energy window ranging from −1.5 eV below to +2.0 eV above the e HOMO level. As for the single molecule case absorption curves have been obtained by imposing an artificial Lorentzian broadening of 100 meV and they are normalized to the maximum value of the curve obtained by GW-PPA+BSE.
theory here adopted; indeed the absorption peak predicted in this case is located at 558 nm that is just 4 nm away from the experimental peak position measured at the wavelength of 554 nm. A better understanding of the excitonic nature of the optical transitions can be obtained by looking at the eigenvalues and eigenvectors of the excitonic Hamiltonian defined in Eq. (11) and obtained with the GW-RealAxis scheme; the results of such inspection are shown in Fig. 6, where the Lorentzian broadening parameter η of Eq. (10) is reduced so that the contribution to the system polarizability of the single eigenstates Eλ of the excitonic Hamiltonian can be isolated and studied. From Fig. 6 it appears clearly that
the main peak in the normalized experimental absorption BT2N thin-film spectrum is due to the formation of two nearly degenerate excitons ((b) and (c) peaks in Fig. 6) with binding energies of 1.34 eV and 1.29 eV, respectively. A careful inspection of the H exc eigenstates evidences that the eh transition corresponding to the (b) peak is the HOMO → LUMO+2 transition (see Table III) between the HOMO and the LUMO+2 that are mainly located at two different BT2N molecules thus involving a marked CT between the stacked molecules. The other contribution to the main absorption peak is the eh eigenstate at (c) that arises from three transitions the HOMO-1 → LUMO, the HOMO-1 → LUMO+1, and the HOMO → LUMO+1. Of
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FIG. 6. Im[α ∥ ] obtained from Eq. (10) with a Lorentzian broadening of 10 meV compared with the normalized absorption spectrum. In the inset the electron-hole transitions that weigh more than 10% in the excitonic states (a)–(c) are schematically represented.
scheme is still affected by large errors concerning also the position of the main peak that is away from the experimental position by 50-100 meV. This discrepancy is completely recovered once the π conjugated geometry of BT2N molecules is considered as a possible organization pattern of oligothiophene molecules in the solid state. Indeed the absorption spectrum of two π conjugated BT2N molecules shows a nice agreement between the experiment and the prediction obtained with the highest level of accuracy employed in this article. The analysis of the eh transitions that contribute to the main absorption peak evidences the important role played by the inter-molecular optical transition in explaining the observed absorption behavior. Therefore, the importance of the π conjugation between the molecules is put in evidence in our study showing that the absorption properties of such molecules are tightly related to the way they are arranged in the matrix of organic solar cell device. ACKNOWLEDGMENTS
TABLE III. Electron-hole pairs that contributes to excitonic states more than 10%. Peak
Energy (eV)
eh pairs
Weight
(a)
2.02
HOMO-1 → LUMO+1 HOMO → LUMO+1 HOMO-1 → LUMO
0.61 0.25 0.12
(b)
2.18
HOMO → LUMO+2
0.87
(c)
2.23
HOMO-1 → LUMO HOMO-1 → LUMO+1 HOMO → LUMO+1
0.30 0.30 0.25
these three transitions, the first and the third ones involve molecular levels lying on the same molecules, as also reported in Table III, while the second one shows a CT behavior (see Fig. 4). In Fig. 6 a third peak at 2.02 eV is present, which again represents an exciton state with binding energy of 1.50 eV of lower optical strength with respect the (b) and (c) states; this feature is also present in the experimental curve, showing again that a strong hybridization between excitonic states on different molecules is the key for a better understanding of the optical properties of BT2N thin-films and small conjugated molecules in general. IV. CONCLUSIONS
In summary we have calculated in the context of the MBPT, using the GW approximation and the BSE for the excitonic Hamiltonian, the absorption spectra of two different systems made either of one isolated BT2N molecule or two π conjugated BT2N molecules. To this aim, we have tested two different levels of approximations, namely, the GW-PPA and the GW-RealAxis. The experimental absorption spectrum has been compared to the theoretical predictions evidencing, first of all, the excitonic nature of the measured absorption onset and main peak. However the predicted absorption spectrum of a single isolated BT2N molecule obtained with the adopted
Computational resources have been provided by CRESCO/ENEAGRID High Performance Computing infrastructure and its staff.34 CRESCO/ENEAGRID High Performance Computing infrastructure is funded by ENEA, the Italian National Agency for New Technologies, Energy and Sustainable Economic Development and by Italian and European research programmes, see http://www.cresco.enea.it/english for information. One of the authors (F. Gala) wish to express his gratitude to Dr. D. Varsano (and Giufá bookshop as well) for many helpful discussions concerning MBPT, in general, and the YAMBO code, in particular. APPENDIX: NUMERICAL CONVERGENCE OF THE CALCULATIONS
The basic component of the perturbative expansion in MBPT is the reference non-interacting system, which is represented by the solutions of the DFT equations; thus the accuracy of the obtained DFT eigenvalues have been checked by a comparison with the corresponding values calculated with an higher wave function energy cutoff (150 Ry), with the highest discrepancy found of the order of 3 · 10−2 eV. In the calculation of the quasiparticle bandgap ε QP gap, local field effects arising from charge redistribution induced by light absorption have been checked explicitly; Fig. 7(a) reports the obtained values for ε QP gap of the BT2N molecule as a function of the number of G-vectors (i.e., the matrix size) to be included in the Fourier transform of the response function χ in Eq. (8), showing that at least ∼900 G-vectors are needed to achieve convergence. Since the ε QP gap results may also depend on how much fine is the grid over which the integral in Eq. (3) is calculated, two different integration steps have been used (∆ω = 73.21 meV and 36.46 meV), but no appreciable differences emerged. Kohn-Sham energy levels up to 10.35 eV above the HOMO energy have been employed in the computation of the GW quasiparticle bandgap, corresponding 750 KS bands for the isolated BT2N molecule, while 1400 KS bands (corresponding
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FIG. 7. QP bandgap of an isolated BT2N molecule as a function of the matrix size of the response function χ for two different grid meshes in (a); while in QP (b) ε gap as a function of the number of bands included in the definition of the non-interacting Green function is shown.
to 13.83 eV from the HOMO level) have been employed in the case of two BT2N molecules. Such values ensure good convergence for quasiparticle bandgaps, as shown in Fig. 7(b) with errors of the order of 0.05 eV in the corresponding energy gaps. For the BSE, eh transitions up to ∼16 eV have been included in excitonic Hamiltonian (9), corresponding to 200 electronic states (100 occupied and 100 empty states), and ensuring no appreciable variations in the position of the leading absorbance peak, as shown in Fig. 3(b) in the case of the isolated BT2N molecule. 1L.
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