Quantifying charge transfer energies at donor-acceptor interfaces in small-molecule solar cells with constrained DFTB and spectroscopic methods.

IOP PUBLISHING

JOURNAL OF PHYSICS: CONDENSED MATTER

J. Phys.: Condens. Matter 25 (2013) 473201 (21pp)

doi:10.1088/0953-8984/25/47/473201

TOPICAL REVIEW

Quantifying charge transfer energies at donor–acceptor interfaces in small-molecule solar cells with constrained DFTB and spectroscopic methods Reinhard Scholz1 , Regina Luschtinetz2 , Gotthard Seifert2 , Till JägelerHoheisel1 , Christian Körner1 , Karl Leo1 and Mathias Rapacioli3 1

Institut für Angewandte Photophysik, Technische Universität Dresden, D-01062 Dresden, Germany Institut für Physikalische Chemie und Elektrochemie, Technische Universität Dresden, D-01062 Dresden, Germany 3 Laboratoire de Chimie et Physique Quantiques—UMR5626 Université Paul Sabatier—Bâtiment 3R1b4, 118 route de Narbonne, F-31062 Toulouse Cedex 09, France E-mail: [email protected]

2

Received 16 September 2013 Published 18 October 2013 Online at stacks.iop.org/JPhysCM/25/473201 Abstract Charge transfer states around the donor–acceptor interface in an organic solar cell determine the device performance in terms of the open circuit voltage. In the present work, we propose a computational scheme based on constrained density functional tight binding theory (c-DFTB) to assess the energy of the lowest charge transfer (CT) state in such systems. A comparison of the c-DFTB scheme with Hartree–Fock based configuration interaction of singles (CIS) and with time-dependent density functional theory (TD-DFT) using the hybrid functional B3LYP reveals that CIS and c-DFTB reproduce the correct Coulomb asymptotics between cationic donor and anionic acceptor configurations, whereas TD-DFT gives a qualitatively wrong excitation energy. Together with an embedding scheme accounting for the polarizable medium, this c-DFTB scheme is applied to several donor–acceptor combinations used in molecular solar cells. The external quantum efficiency of photovoltaic cells based on zinc phthalocyanine–C60 blends reveals a CT band remaining much narrower than the density of states of acceptor HOMO and donor LUMO, an observation which can be interpreted in a natural way in terms of Marcus transfer theory. A detailed comparison with c-DFTB calculations reveals an energy difference of 0.32 eV between calculated and observed absorption from the electronic ground state into the CT state. In a blend of a functionalized thiophene and C60 , the photoluminescence spectra differ significantly from neat films, allowing again an assignment to CT states. The proposed computational scheme reproduces the observed trends of the observed open circuit voltages in photovoltaic devices relying on several donor–acceptor blends, finding an offset of 1.16 eV on average. This value is similar as in polymer–fullerene photovoltaic systems where it amounts to about 0.9 eV, indicating that

0953-8984/13/473201+21$33.00

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c 2013 IOP Publishing Ltd Printed in the UK & the USA

J. Phys.: Condens. Matter 25 (2013) 473201

Topical Review

the photophysics of CT states in molecular donor–acceptor blends and in polymer–fullerene blends are governed by the same mechanisms. S Online supplementary data available from stacks.iop.org/JPhysCM/25/473201/mmedia (Some figures may appear in colour only in the online journal)

Contents 1. Introduction 2. Constrained DFT scheme 2.1. Self-consistent charge density functional tight binding 2.2. Charge constraints 3. Polarizable medium 4. Orbital alignment, ionization potentials, and electron affinities 5. Geometry optimization of donor–acceptor pairs 5.1. Zinc phthalocyanine–C60 interface 5.2. Model geometries for donor–acceptor interfaces in solar cells 6. Calculation of CT states: CIS, time-dependent DFT, and constrained DFT 6.1. Configuration interaction of singles 6.2. Time-dependent DFT 6.3. Constrained DFT 7. CT energy in a polarizable medium 7.1. DIP–DCV3T 7.2. ZnPc–C60 7.3. Other donor–acceptor pairs 8. Spectroscopic properties of CT states 8.1. ZnPc–C60 8.2. DCV4T–C60 9. Comparison of calculated CT energies and observed open circuit voltages 10. Discussion 11. Conclusion, outlook Acknowledgments References

quantification of the energy alignment between the primary photoexcitation and the resulting charge pair would prove beneficial. During the last decades, organic photovoltaics have benefited from several innovative concepts including donor–acceptor (DA) blends and tandem cells, making it possible to reach a technologically relevant power conversion efficiency of 12% [1]. In a first step, the light is absorbed at a donor or acceptor site, from where the exciton diffuses towards the heterointerface between both compounds. At this interface with a staggered (type-II) alignment of the frontier orbitals, the optical excitation is split into a charge transfer (CT) complex with the electron localized on an acceptor molecule and the hole on a donor site. These CT complexes form a reservoir of localized charge pairs which may be ionized into free electrons and holes, diffusing eventually towards the electrodes of the solar cell. The precise mechanism of the separation of molecular excitations and CT states into free charge carriers is still under intense debate. When the photon energy is much larger than the lowest CT state, the excess energy seems to support the dissociation of a ‘hot’ exciton into free charges in specific systems [2], whereas other donor–acceptor combinations do not show a significant dependence of the internal quantum efficiency on photon energy [3]. Some spectroscopic studies indicate that the detailed potential landscape around a donor–acceptor interface may be influenced by electrostatic effects arising from the difference of the dielectric constants and the resulting polarization shifts [4, 5]. Entropic contributions lower the free energy barrier for charge separation, reducing in turn the recombination via CT states [6]. In any case, in typical donor–acceptor blends the CT state represents the lowest excited state, so that its formation, dissociation, and recombination competes with the collection of photogenerated charge carriers. In polymer–fullerene bulk heterojunctions, weak absorption below the gaps of donor and acceptor was successfully assigned to CT states localized at the interface, revealing a linear relationship between the CT absorption energy and the open circuit voltage [7]. Similarly, donor–acceptor blends can show an electroluminescence band below the energies of the electroluminescence in the pure materials, correlating again with the open circuit voltage [8]. The assumption of detailed balance provides a relationship between the quantum efficiencies of a photovoltaic cell and its electroluminescence spectra, involving the CT energy as the low-temperature limit of the open circuit voltage [9, 10]. Concerning donor–acceptor blends of small molecules, both absorption and photoluminescence from CT states have only been obtained for mixtures of zinc phthalocyanine (ZnPc) and

2 4 4 4 4 5 8 8 8 10 10 10 11 12 13 14 14 14 14 16 16 18 19 19 19

1. Introduction Both in biological light-harvesting systems and in molecular solar cells, an incoming photon is transformed into a molecular or intermolecular excitation, resulting eventually in a charge separated state. These charge pairs form a reservoir for subsequent photochemical reactions in a living cell or for the collection of photocurrent at the contacts of the photovoltaic device. Even though the initial charge separation is of primary interest for the functionality of natural and artificial photoreceptors, the respective states may be difficult to detect because they show only weak spectroscopic signatures. Hence, theoretical schemes allowing 2


Topical Review

C60 [11]. Therefore, donor–acceptor crystals have emerged as alternative reference systems making it possible to obtain some insight into the basic properties of the low-lying CT states [12–14]. Even though the efficiency of organic photovoltaics depends on the energy alignment of molecular singlet excitations, triplet configurations, and CT states, reliable calculations of the latter are rather scarce. Hartree–Fock based linear response calculations like configuration of singles (CIS) overestimate molecular excitations and CT energies by a large amount [15], whereas more advanced methods like coupled cluster theory (CC2) prove more reliable but become computationally much more demanding [16]. Both schemes share the advantage that due to the inclusion of exact non-local exchange, the asymptotic −1/r dependence of the Coulomb interaction between a pair of oppositely charged donor and acceptor molecules is correctly included. In the most widely used scheme for excited state calculations, time-dependent density functional theory (TD-DFT), the use of local exchange–correlation functionals inhibits the correct asymptotics of the anion–cation interaction [17]. Several remedies for this systematic failure have been proposed, including in particular long-range corrections of the exchange–correlation functional [17–19]. Constraints in DFT represent an elegant technique to circumvent the spurious errors introduced by the wrong asymptotics of the exchange–correlation functional and problems related to the self-interaction in DFT [20]. In applications to charged donor–acceptor pairs, the additional constraints fix the net charges on each constituent, so that the DFT scheme corresponds to ground state calculations of the ionized donor and acceptor molecules D+ and A− together with their mutual Coulomb interaction, guaranteeing the correct Coulomb asymptotics proportional to −1/r [21]. Hence, in constrained DFT, the excitation energy of a D+ A− pair with respect to the neutral ground state D0 A0 relies mainly on ground state calculations of molecules in different charge states. The ionization potentials and electron affinities are known to be rather robust against the use of different exchange–correlation functionals [22, 23], providing reliable results even in quite small variational basis sets [24, 25]. In the present work, we calculate CT excitations in donor–acceptor pairs with a constrained DFT method [26, 27]. By localizing opposite charges on the cationic donor and anionic acceptor molecules with a scheme based on Lagrange multipliers, the resulting electronic configuration gives the correct Coulomb asymptotics proportional to −1/r together with a realistic interaction between the delocalized charge densities for shorter donor–acceptor distances. Our implementation relies on a density functional tight binding (DFTB) scheme in a minimal orbital basis [28] where the self-consistent charge distribution is included via a second order expression in the Mulliken charges [29]. The charge constraints for this DFTB scheme (c-DFTB) have been implemented starting from the deMonNano code [30]. The choice of this DFTB approach is motivated by the objective of investigating very large systems in future work, including

e.g. ionized molecules embedded into a larger assembly of neutral molecules. A polarizable medium embedding the donor and acceptor molecules has a twofold influence. First, the field energy around a charged molecule is reduced by the inverse of the dielectric constant of the medium, resulting in large polarization shifts of the donor ionization potential and the acceptor electron affinity, and second, the mutual Coulomb interaction between the two charged species is screened by the same dielectric constant [31]. Hence, once the distance-dependent CT potential surface has been calculated in vacuo, the influence of a polarizable medium embedding the charged donor and acceptor can be accounted for by scaling the Coulomb interaction with the inverse of the dielectric constant of the medium, 1/. The implementation of the c-DFTB method applied in the remaining parts of this work is summarized in section 2, and section 3 addresses electrostatic corrections arising for molecular ions embedded into a polarizable medium. Section 4 is devoted to a careful distinction between the Kohn–Sham energies of the frontier orbitals obtained with DFT and values of ionization potential and electron affinity determined with the same method. Only when accounting for the electrostatic corrections discussed in section 3, can the calculated ionization potential and electron affinity be directly compared to measured orbital energies. In section 5, we investigate suitable model geometries of donor–acceptor pairs with a density functional scheme augmented by dispersion interactions. Section 6 compares different methods for the calculation of CT excitation energies from a donor to an acceptor molecule in vacuum, revealing that the c-DFTB method proposed in section 2 provides a physically correct picture. In the model geometries described in section 5, the c-DFTB method gives an estimate of the unscreened Coulomb interaction between a cationic donor and an anionic acceptor molecule in direct contact, whereas in section 7, we apply the embedding into a polarizable medium. Section 8 presents absorption and photoluminescence (PL) spectra assigned to intermolecular CT excitations and compares them with computed CT energies. For the particular donor–acceptor combination ZnPc–C60 , we demonstrate that the observed linewidth of CT absorption and PL is not related to the width of the density of states assigned to the HOMO of the donor and the LUMO of the acceptor. Instead, it arises in a natural way from a configuration coordinate diagram containing the deformation between the geometry of a donor–acceptor pair in its neutral D0 A0 ground state and the relaxed D+ A− CT geometry. PL spectra obtained on DCV4T–C60 blends differ from the respective neat film, giving clear indications of radiative recombination via CT states. Section 9 is devoted to a comparison of calculated CT excitation energies with observed open circuit voltages, reproducing the essential experimental trends. Section 10 analyses the precision of the constrained DFT scheme and the embedding into a polarizable medium, and section 11 summarizes the achievements of the present work. 3


Topical Review

2. Constrained DFT scheme

of the molecular orbitals. These constraints can easily be implemented into the Lagrangian of the system, X L = ESCC-DFTB ({8i }) + 3ij h8i |8j i − δij

2.1. Self-consistent charge density functional tight binding

i,j

The self-consistent charge density functional tight binding (SCC-DFTB) scheme and its applications have been described elsewhere in some detail [28, 29, 32], so that only the key ingredients will be mentioned. The approach relies on the formulation of a tight binding model Hamiltonian within a minimal set of basis orbitals ϕkµ on atom k covering only the valence shell, so that each molecular orbital 8i can be expanded as X cikµ ϕkµ . (1) 8i =

! +V

rep

Vkl +

i

k>l

+

occ X

X

0kl qk ql ,

ni

X

D

D

A

A

ni h8i |P |8i i − N

! +V

A

X

ni h8i |P |8i i − N

(5)

i

where PD and PA are projectors of the electronic density on either donor or acceptor, N D and N A stand for the number of electrons localized on the respective fragment, V D and V A are Lagrange multipliers responsible for the charge localization, and 3ij are Lagrange multipliers ensuring orthonormality. This scheme can easily be generalized to more than two fragments, but this possibility will not be pursued in the following. In keeping with the Mulliken charge localization scheme used in DFTB, the projection onto the fragments will be performed with the same method [26, 27].

The matrix elements of the Hamiltonian H 0 are derived from DFT calculations for a reference electron density corresponding to the superposition of two neutral atoms [28] plus correction terms of second order in the atomic Mulliken charges qk and ql on the respective atomic sites, making it possible to obtain a self-consistent charge distribution [29]. Three centre integrals are neglected, and short range repulsive interactions involving the core region are treated as a superposition of pair potentials which can be derived from DFT calculations for atom pairs. The resulting expression for the total energy reads X

X i

kµ

ESCC-DFTB =

D

3. Polarizable medium Measured values of ionization potential (IP) and electron affinity (EA) of a molecule are affected by an embedding medium because the electrostatic field energy around each molecular ion scales with the inverse of the dielectric constant ε. Using for simplicity a spherical model for each ion with an ionic radius rion defining its size, this electrostatic field energy reads [31]

0 cikµ cilν Hkµ,lν

kµ,lν

(2)

k>l rep

Vfield (rion , ε) =

where Vkl is the repulsive potential between atoms k and l, ni the occupation number of the molecular orbital i, cikµ its expansion coefficient in terms of the basis orbital µ on atom k, H 0 the Kohn–Sham Hamiltonian at the reference density of two superimposed neutral atoms, and 0kl describes the distance-dependent interaction between the Mulliken charges on atoms k and l. The secular equation X (Hkµ,lν − εi Skµ,lν )cilν = 0 (3)

e2 . 8π ε0 εrion

(6)

Hence, when comparing the field energy of a molecular ion embedded in a polarizable medium with the respective energy in vacuum, the screening by the dielectric constant reduces the field energy, 1Vfield (rion ) = Vfield (rion , ε) − Vfield (rion , 1) < 0,

(7)

resulting in much smaller IP and much larger EA values inside the dielectric surroundings [31]. In the following, the ionic radii are calculated by equating the volume per molecule in their crystalline phase with a sphere. For simplicity, we assume equal ionic radii of the cationic and anionic configurations, so that the respective polarization energies coincide,

lν

with the overlap matrix elements Skµ,lν is solved for H = H 0 + H 1 with X 1 Hkµ,lν = 21 Skµ,lν (4) (0km + 0lm ) qm . m

P+ = P− = 1Vfield (rion ).

2.2. Charge constraints

(8)

Inside the medium, the polarization correction according to equations (6)–(8) shifts ionization potential and electron affinity towards each other,

In the present context of charged pairs of donor and acceptor molecules, the correct localization of the positive or negative net charges plays a key role. Hence, in the spirit of the constrained DFT scheme developed by Wu and van Voorhis [20, 21], the molecular orbitals {8i } which may be delocalized over the cationic donor and the anionic acceptor site are subject to a constraint of a total charge qD = e and qA = −e in addition to a constraint reflecting orthonormality

IP0 = IP + P+ EA0 = EA − P−

(9)

so that the apparent gap shrinks accordingly. As a quantity which is not affected by the polarization energies according 4


Topical Review

to equations (8) and (9), we introduce the Mulliken electronegativity χ defined as the arithmetic average of IP and EA: IP + EA IP0 + EA0 = . (10) 2 2 Hence, even though experimental spectra are shifted by the polarizable medium, an equivalent quantity can be deduced from measured orbital energies, χ=

EHOMO + ELUMO (11) 2 obtained e.g. from cyclic voltammetry. The Mulliken electronegativity can be used to order the model compounds according to their tendency to transfer electrons between them: in each pair of molecules, the compound with the lower χ serves as the donor, and the one with the larger χ as the acceptor. A c-DFTB calculation of the CT excitation energy as a function of the distance r between donor and acceptor provides the unscreened Coulomb interaction between a cationic donor molecule and an anionic acceptor, χ =−

Figure 1. Frontier orbitals of zinc phthalocyanine (ZnPc, left) and C60 (right) together with their Kohn–Sham energies (centre, black bars), calculated for each molecule separately at the B3LYP/6-31G(d) level. In each case, the large difference between either ionization potential (IP, green) and Kohn–Sham energy of the HOMO or between electron affinity (EA, blue) and LUMO energy results from the wrong asymptotics of the exchange–correlation potential.

c-DFTB VCoulomb (D+ , A− , r) = Eexc (D+ , A− , r) c-DFTB − Eexc (D+ , A− , ∞). (12)

states (DOS) proportional to exp[−(E − E0 )2 /(2σ 2 )] can be determined from the energy where the tangent at the point of vanishing curvature crosses zero, so that according to this definition, each edge is shifted by 2σ into the gap with respect to the maximum of the DOS. For pure materials, the resulting values have been used for the interpretation of the transport gap [34], and in photovoltaic blends the gap between the donor HOMO and the acceptor LUMO defined in a similar way was denominated as the photovoltaic gap [35]. In each case, this definition suffers from a rather large width of each DOS, typically in the range of σ ≈ 0.2 eV for the HOMO and of σ ≈ 0.4 eV for the LUMO. Hence, the gap deduced from the edges differs by about 1.2 eV from the gap determined as the difference of the peak energies. The spectroscopic data have been interpreted with DFT calculations of the Kohn–Sham energies of the frontier orbitals. Based on the popular hybrid functional B3LYP [36, 37], the calculated LUMO energy correlates particularly well with the lower edge of the LUMO DOS as determined from IPES [34]. In figure 1, we visualize B3LYP frontier orbitals of zinc phthalocyanine (ZnPc) and C60 , calculated separately for each molecule with the G AUSSIAN 03 program package [38]. The Kohn–Sham energies of the frontier orbitals in figure 1 align indeed in a type-II (staggered) fashion, as required for efficient charge separation between a ZnPc donor and a C60 acceptor. As will become clear from the subsequent discussion, the apparent CT gap of 1.76 eV between the HOMO of ZnPc and the LUMO of C60 can only give a first indication that the CT excitation may be below the neutral excitations of each of the molecules, but it should not be misunderstood as a quantitative measure for the energy required for the excitation of a pair of oppositely charged molecules out of their neutral ground state.

Due to the finite extension of each ionized molecule, the respective charge densities are delocalized over the entire molecular volumes, resulting in strong deviations from the interaction between point charges. An embedding of the ion pair into the polarizable medium has to account for the screening of the Coulomb interaction by the inverse of the dielectric constant and for the polarization corrections, so that the excitation energy from the electronic ground state to the CT state reads screened c-DFTB Eexc (D+ , A− , r) = Eexc (D+ , A− , ∞) + (P+ + P− )

1 + VCoulomb (D+ , A− , r), (13) ε where the first term contains the difference between ionization potential of donor and electron affinity of acceptor in c-DFTB (D+ , A− , ∞) = IP − EA , the second vacuum, Eexc D A the polarization energy, and the last the screened Coulomb interaction between the oppositely charged molecules. Except for the more precise shape of the unscreened Coulomb interaction between extended ionized molecules, equation (13) defines the basis for the interpretation of electrochemical measurements [31], closely related to previous expressions for the interaction between solvated ions [33].

4. Orbital alignment, ionization potentials, and electron affinities In spectroscopic studies of organic materials, it is common to deduce the energy of the HOMO from ultraviolet photoemission spectroscopy (UPS) and the position of the LUMO from inverse photoemission spectroscopy (IPES). In each case, the edge of the measured Gaussian density of 5


Topical Review

Table 1. Comparison of measured energies of the frontier orbitals with DFT calculations at the B3LYP/6-31G(d) level. The eight compounds are ordered according to their measured electron affinity χ defined via equation (11). The experimental values for the dicyanovinyl-thiophenes are compiled from electrochemical measurements [41], and the tabulated energies for other compounds rely on UPS and IPES data [43, 35, 44]. The calculations include Kohn–Sham energies, adiabatic ionization potential (IP), and adiabatic electron affinity (EA) defined via the energy difference of the ionized molecules and the neutral species, in the optimized geometry of the neutral ground state. Electrostatic corrections are calculated via equations (7) and (8), and modified ionization potentials IP0 and electron affinities EA0 in the polarizable medium according to equation (9). Measured HOMO (eV) 6T DIP ZnPc DCV6T DCV5T DCV4T DCV3T C60

−5.3a −5.7a −5.80c −5.43d −5.62d −5.85d −6.09e −7.0a

LUMO (eV) −1.1b −2.0a −2.82c −3.87d −3.73d −3.87d −3.90e −3.25a

B3LYP χ (eV) 3.2 3.85 4.31 4.65 4.68 4.86 5.00 5.13

HOMO (eV)

LUMO (eV)

−4.80 −5.13 −4.94 −5.48 −5.64 −5.85 −6.19 −5.99

−2.21 −2.59 −2.75 −3.33 −3.41 −3.51 −3.70 −3.18

In medium

IP (eV)

EA (eV)

χ (eV)

IP0 (eV)

EA0 (eV)

5.85 6.32 6.07 6.40 6.61 6.90 7.33 7.16

1.14 1.40 1.62 2.47 2.48 2.49 2.57 2.20

3.50 3.86 3.85 4.44 4.55 4.70 4.95 4.68

4.88 5.34 5.18 5.29 5.45 5.68 6.03 6.16

2.11 2.38 2.51 3.58 3.64 3.71 3.87 3.20

a

UPS and IPES for thin films [35]. Measured IPES–UPS gap of 4.2 eV [44] added to measured UPS value [35]. c UPS and IPES for thin films [43]. d Cyclic voltammetry in C H Cl [41]. 2 2 4 e Cyclic voltammetry in CH Cl [41]. 2 2 b

For each molecule, ionization potential and electron affinity can be calculated from the difference between the total energies of an ionized and a neutral molecule. Minor deformations between the relaxed geometries of the different charge states are ignored in the following, so that our B3LYP/6-31G(d) values correspond to the vertical IP and EA in the geometry of the neutral molecules. In order to relate IP and EA to the frontier orbitals, we visualize them as negative energies, corresponding to the opposite sign of the standard convention as used in the tables. Figure 1 reveals that the so-defined ionization potentials differ from the Kohn–Sham energies of the HOMOs, and also the Kohn–Sham energies of the LUMOs do not reproduce the electron affinities. As discussed elsewhere in more detail, this deficiency of DFT arises from the lack of an integer discontinuity of local exchange–correlation functionals in the gap region, so that HOMO and LUMO are shifted towards each other by similar amounts [39]. In the B3LYP functional, the reduction of exact non-local exchange to only 20% inhibits the fulfilment of Koopmans’ theorem which would be strictly valid for Hartree–Fock, connected with a match of the HOMO energy and the IP [40]. Although the LUMO energy in B3LYP does not very well reproduce the EA, it gives at least a rough estimate, whereas a pure Hartree–Fock treatment would fail completely due to the unphysical nature of the unoccupied virtual orbitals. Hence, the picture of the staggered orbital alignment suggested by figure 1 remains qualitatively correct, but a quantification of the respective intermolecular charge transfer excitation energy would require a detailed investigation of the Coulomb interaction as a function of distance between the ionized donor and acceptor molecules together with an embedding scheme accounting for the polarizable medium. According to figure 1, CT from a ZnPc donor to a C60 acceptor at infinite distance in vacuum corresponds to an excitation energy of 3.87 eV

defined by the difference between the ionization potential of the donor and the electron affinity of the acceptor. The embedding of the donor–acceptor pair into a polarizable medium discussed in section 3 lowers this CT excitation energy substantially. This brings the very high asymptotic CT energy of 3.87 eV calculated with B3LYP roughly into resonance with the neutral excitations of either donor or acceptor, so that the screened Coulomb interaction between adjacent cationic donor and anionic acceptor molecules can eventually result in CT states below the neutral excitations of both kinds of molecules. In the following, the alignment of the frontier orbitals will be investigated for several molecules used in photovoltaic devices. Solar cells comprising donor–acceptor blends containing functionalized thiophenes and C60 have resulted in rather high power conversion efficiencies, including in particular thiophene molecules with dicyanovinyl end groups [41]. In our calculations, we have optimized the DCVnT molecules in planar geometries resembling known crystal phases. Previous B3LYP investigations of the same compounds in vacuum have indicated deviations from planarity of less than 10◦ , and our values for the orbital energies, IP and EA, differ from these calculations by rather small deviations of the order of 0.1 eV [42]. For the DCVnT compounds, reliable IPs and EAs measured on thin films could not be found. Therefore, the respective positions of the frontier orbitals compiled in table 1 and figure 2 (peak values) rely on electrochemical measurements and give direct access to IP and EA of the DCVnT molecules embedded in highly polarizable solvents with dielectric constant as given in table 2 [41]. An interpretation of these values with B3LYP/6-31G(d) orbital energies shows a remarkably good agreement, arising however from a nearly perfect cancellation of quite large systematic errors: on the one hand a systematic deviation 6


Topical Review

Table 2. Molecular radii, static dielectric constant, and polarization energies for an embedded molecular ion.

6T DIP ZnPc DCV6T DCV5T DCV4T DCV3T C60

˚ Radius (A)

ε (1)

|P+ | = |P− | (eV)

5.02a 4.97a 5.19a 5.77e 5.52e 5.23a 4.92e 5.54a

3.1b 3.1c 2.8d 8.50f 8.50f 8.50f 8.93g 4.4h

0.97 0.98 0.89 1.10 1.15 1.21 1.30 1.00

a

Ionic radii from the respective crystal phases [41, 46–49]. b Assumed to correspond to DIP. c Static interpolation from optical data for crystalline phase [51]. d Static interpolation from optical data for crystalline phase [52]. e Ionic radii obtained by assuming the same mass density as in crystalline DCV4T. f Embedding medium C H Cl for cyclic voltamme2 2 4 try [41, 50]. g Embedding medium CH Cl for cyclic voltamme2 2 try [41, 50]. h Capacitance measurement [53].

Figure 2. Ionization potential, electron affinity, and Mulliken electronegativity of different model compounds: B3LYP (red, left bar for each compound), observed (grey and black, central bar), and DFTB (blue, right bar). In each case, the shaded bar extends from the ionization potential IP0 in the polarizable medium to the electron affinity EA0 , and the horizontal line indicates the Mulliken electronegativity χ, compare tables 1–3 including the polarization corrections according to equation (9). ×: peak positions deduced from UPS and IPES measured on molecular films, +: peak positions from cyclic voltammetry data in highly polarizable solvents, see table 1.

spectra and B3LYP values of IP and EA [54, 55]. The rising deviations for compounds containing fused aromatic rings may indicate a systematic deficiency of the B3LYP hybrid functional when applied to such molecules. Moreover, the IP and EA from films may be influenced by different orientations of the molecules [56], and changes of the polarization energy between molecules embedded into the bulk or at the surface of thin films [44]. For 6T, the origin of the rather large deviation between the calculated EA0 and the value deduced from UPS and IPES remains unclear. The visualization of the measured and calculated positions of IP0 , EA0 and χ in figure 2 indicates that the B3LYP calculations reproduce the general trends quite well, with a few particularly large deviations as mentioned above. Except for C60 , the ordering of the measured values of the Mulliken electronegativities is correctly reproduced. For the eight model compounds, the IPs and EAs obtained in optimized SCC-DFTB geometries are compiled in table 3 and figure 2. For the DCVnT compounds, the embedding into a polarizable medium brings the calculated IP and EA into quantitative agreement with the values deduced from electrochemistry, with residual deviations similar to the B3LYP values. The calculated IP for 6T seems reasonable, but similarly to the B3LYP calculations reported in table 1, the measured EA cannot be reproduced. For DIP, both the calculated IP and EA are systematically too high. The calculated IP of ZnPc shows a fair agreement with measured photoemission spectra, and for C60 , the computed IP is superior to the B3LYP value in table 1. For the donor–acceptor pairs DCV3T–C60 and DIP-ZnPc, the experimental ordering of the Mulliken electronegativity is inverted. In summary, for six out of the eight compounds studied, the precision of the SCC-DFTB approach is competitive with B3LYP, but

between calculated orbital energies and either IP or EA in DFT, on the other hand a complete neglect of the polarizable medium. The calculated adiabatic IP and EA for the DCV3T to DCV6T molecules in table 1 differ strongly from voltammetry, but accounting for the polarizable medium via equation (8) with the ionic radii in table 2 gives modified values in excellent agreement with the experimental data: except for EA0 of DCV6T, all deviations remain below 0.2 eV. This indicates that both a careful interpretation of DFT and a suitable model for the polarizable medium are required for a meaningful comparison between experimental results and model calculations, whereas the seemingly good agreement between voltammetry and B3LYP Kohn–Sham orbital energies in table 1 is merely fortuitous. Discrepancies between the IPs of DCVnT deduced from cyclic voltammetry and UPS observed earlier [42, 45] find a natural explanation in the dependence of the polarization energy P+ on the dielectric constant of the embedding medium. Table 1 and figure 2 include HOMO and LUMO peak energies deduced from ultraviolet photoemission spectra (UPS) and inverse photoemission spectra (IPES) [35, 43] measured on thin films of sexithienyl (6T), diindenoperylene (DIP), ZnPc and C60 . Due to the smaller dielectric constants of the molecular films, the influence of the polarizable medium remains somewhat smaller, so that the gap energies shrink by a smaller amount with respect to the dissolved DCVnT molecules. The B3LYP estimates for IP or EA plus corrections arising from the polarizable medium remain somewhat less convincing: 6T, DIP and C60 show deviations of about 0.4 eV from measured IP values, and ZnPc about 0.6 eV. In the case of C60 , a similar energy offset occurs between gas phase 7


Topical Review

Table 3. Experimental energies for the frontier orbitals and Mulliken electronegativity of several model compounds as in table 1, and calculated ionization potential, electron affinity and Mulliken electronegativity obtained with the SCC-DFTB method using the local density approximation (LDA). All energies refer to geometries of neutral molecules optimized with the same method. The electrostatic corrections of the embedding medium are calculated via equations (7) and (8), and modified ionization potentials IP0 and electron affinities EA0 in the polarizable medium according to equation (9) with the ionic radii tabulated in table 2. Measured HOMO (eV) 6T DIP ZnPc DCV6T DCV5T DCV4T DCV3T C60

−5.3a −5.7a −5.80c −5.43d −5.62d −5.85d −6.09e −7.0a

LUMO (eV) −1.1b −2.0a −2.82c −3.87d −3.73d −3.87d −3.90e −3.25a

SCC-DFTB

In medium

χ (eV)

IP (eV)

EA (eV)

χ (eV)

IP0 (eV)

EA0 (eV)

3.2 3.85 4.54 4.65 4.68 4.86 5.00 5.13

6.43 7.10 6.48 6.67 6.88 7.14 7.50 7.61

2.01 2.17 1.98 2.91 2.85 2.78 2.68 2.32

4.22 4.64 4.23 4.79 4.87 4.96 5.09 4.97

5.46 6.12 5.59 5.56 5.73 5.93 6.20 6.61

2.98 3.15 2.87 4.01 4.00 3.99 3.98 3.32

a

UPS and IPES for thin films [35]. Measured IPES–UPS gap of 4.2 eV [44] added to measured UPS value [35]. c UPS and IPES for thin films [43]. d Cyclic voltammetry in CH Cl [41]. 2 2 e Cyclic voltammetry in C H Cl [41]. 2 2 4 b

Table 4. Optimized geometries of C60 in contact to the ZnPc plane: binding energies EDFTB obtained from DFTB without dispersion interactions, Edisp from DFTB with dispersion [57], and the respective distances dDFTB and ddisp between the closest C atom and the ZnPc plane defined via the nitrogen atoms, together with the distances rDFTB and rdisp between the centre of the fullerene and the phthalocyanine plane. The respective geometries are visualized in figure 3.

Lh Lc Lb1 Lb2 La

˚ dDFTB (A)

˚ rDFTB (A)

3.15 2.87 2.96 2.96 2.90

6.42 6.24 6.46 6.46 6.48

EDFTB (eV) −0.07 −0.19 −0.19 −0.17 −0.20

˚ ddisp (A)

˚ rdisp (A)

2.96 2.60 2.71 2.72 2.65

6.23 6.23 6.20 6.21 6.21

Edisp (eV) −1.02 −1.15 −1.17 −1.14 −1.16

Lc where the fullerene is laterally displaced towards one of the nitrogen atoms encircling Zn. The binding energies EDFTB in local density approximation without dispersion and the dispersion-corrected values Edisp are reported in table 4 together with the optimized centre-to-centre distances r between the two molecules, and d the distances between the average plane of the nitrogen atoms in ZnPc and the closest carbon atom of the fullerene. Without dispersion interaction, the central Zn atom is elongated considerably out of the phthalocyanine plane towards the fullerene, with an average ˚ over all five configurations, see figure 3. Including of 0.23 A dispersion interaction, the relatively large van der Waals radius of the metal atom pushes it back towards the plane ˚ of the phthalocyanine, with a residual deviation of 0.04 A on average. The dispersion interaction produces substantially larger binding energies and induces significant deviations from planarity in ZnPc, deforming it towards a curved shell around the fullerene with enlarged contact area.

for 6T and DIP, the use of a minimal orbital basis in the DFTB method results in rather large deviations from the measured EA.

5. Geometry optimization of donor–acceptor pairs 5.1. Zinc phthalocyanine–C60 interface Bulk heterojunction solar cells composed of ZnPc–C60 blends are among the most carefully investigated reference systems in organic photovoltaics. As CT across the donor–acceptor interface depends heavily on the geometric arrangements between the molecules involved, model studies of this system require a careful geometry optimization of a molecule pair with dispersion-corrected DFTB [57]. In a recent study on heterojunctions between copper phthalocyanine (CuPc) and C60 , it was found that several arrangements with the spherical fullerene in contact with the central region of the planar phthalocyanine molecule are more favourable than geometries where the planar molecule stands with an edge on C60 [58]. We found five different energy minima resembling the best arrangements of CuPc and C60 [58], differing in the mutual orientation of the molecules: Lh with a hexagon of the fullerene pointing towards the central Zn site of ZnPc, La with a C atom of C60 pointing towards Zn, two arrangements Lb1 and Lb2 with a C–C edge facing Zn, and

5.2. Model geometries for donor–acceptor interfaces in solar cells Our model for the geometry of adjacent ZnPc and C60 molecules in a bulk heterojunction is inspired by the most favourable Lb1 arrangement. However, in a larger molecular assembly, stacked ZnPc molecules would planarize each 8


Topical Review

Figure 3. Geometry of different arrangements between ZnPc and C60 , optimized with DFTB using LDA: left, without dispersion interaction; right, with dispersion interaction [57]. The different orientation of the molecules towards each other are labelled as in a recent study of the interaction between copper phthalocyanine and C60 [58], see table 4. Table 5. Optimized geometries obtained with DFTB including dispersion corrections [57]: distances rdisp between the centres of mass of donor and acceptor, binding energy Edisp , and orientation of the fullerene when C60 is the acceptor. For combinations involving C60 as the acceptor, the distance ddisp is measured from the carbon atom of C60 standing most closely to the donor plane. The static dielectric constant of each donor–acceptor blend is taken as the average of the two bulk materials, with DIP: ε = 3.1 interpolated from ellipsometry [51], ZnPc: ε = 2.8 interpolated from ellipsometry [52], C60 : ε = 4.4 from capacitance measurement [53], thiophene compounds: static assumed to correspond to DIP. ˚ rdisp (A) 6T/DIP DIP/DCV3T 6T/C60 DIP/C60 ZnPc/C60 DCV4T/C60 DCV5T/C60 DCV6T/C60

3.42 3.43 6.54 6.48 6.26 6.50 6.54 6.51

˚ ddisp (A)

Edisp (eV) −1.54 −1.51 −0.64 −0.80 −1.11 −0.65 −0.64 −0.65

3.27 2.98 2.76 3.23 3.27 3.24

other, so that in the following, CT energies will be calculated on the basis of a single flat ZnPc donor and a Lb1 -oriented fullerene acceptor. This model geometry has a slightly lower binding energy of −1.11 eV, see table 5.

Configuration

Lh Lb1 Lb1 Lh Lc Lh

3.1 3.1 3.75 3.75 3.6 3.75 3.75 3.75

geometries have been constructed from optimized geometries of the monomers, and suitable dimer models were found from a scan along the distance between both molecules. The resulting minima obtained with dispersion-corrected DFTB are reported in table 5, where dimers consisting of two planar molecules are assumed to be stacked along their normal. For

Similarly, other donor–acceptor pairs have not been fully optimized with dispersion-corrected DFTB. Instead, model 9


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Figure 4. Frontier orbitals of diindenoperylene (DIP, left) and dicyanovinyl-terthiophene (DCV3T, right) together with the orbital energies (centre), calculated for each molecule separately at the B3LYP/6-31G(d) level.

˚ the lowest CT potential surface At a distance of 3.3 A, starts at 3.76 eV. A fit to a multipole expansion of the ˚ yields an Coulomb interaction over the distance range 5–20 A asymptotic CT energy of 5.71 eV at infinite distance, nearly 2 eV above the B3LYP CT energy IP(DIP) − EA(DCV3T) = 3.75 eV in figure 4. This large deviation indicates that CT excitations calculated with CIS are severely overestimated, in keeping with previous investigations of molecular ˚ dimers [16]. At the expected equilibrium distance of 3.43 A according to table 5, the CT excitation energy calculated with CIS would be 3.78 eV, far above the measured excitation energies of about 2.5 eV for DIP and DCV3T molecules in solution [41, 51], and also significantly above the lowest calculated dipole-allowed excitation of DCV3T at 3.10 eV and of DIP at 3.27 eV. Asymptotically, the CT excitation energy behaves as the Coulomb potential between opposite point charges, −e2 /(4π 0 r).

dimers involving a thiophene and a fullerene, we found the largest binding energy for a geometry where a hexagon of C60 faces a bond connecting two adjacent thiophene rings, corresponding roughly to optimized geometries found with flexible thiophene chains [59].

6. Calculation of CT states: CIS, time-dependent DFT, and constrained DFT As a model for a relatively small donor–acceptor pair allowing us to compare different computational approaches, we choose diindenoperylene and dicyanovinyl-terthiophene, see figure 4 for a visualization of the frontier orbitals, their Kohn–Sham energies, and calculated values for IP and EA. DCV3T has already been used as an acceptor in bulk heterojunction solar cells with different compounds including ZnPc [45], and as in figure 4, in combination with DIP, here it will again act as the acceptor. The open circuit voltage of photovoltaic cells based on DIP–DCV3T heterojunctions and other donor–acceptor combinations will be presented in section 9, and details of the layer sequence used are included in the supplementary information (available at stacks.iop.org/JPhysCM/25/473201/ mmedia). For the DIP–DCV3T system, we construct dimer models by superimposing the molecules in their planar geometries, with parallel orientations of the molecular planes and the stacking direction along the normal.

6.2. Time-dependent DFT The B3LYP/6-31G(d) TD-DFT excitation energies in figure 5 show a completely different distance dependence. At short distances, the calculation produces a spurious intermolecular charge transfer in the ground state, corresponding to a slightly positively charged donor with increased binding energies of the electronic states and a negatively charged acceptor molecule where they move towards the vacuum level. Hence, the difference ELUMO (DCV3T) − EHOMO (DIP) rises towards shorter distances, and other electronic excitations from the donor to the acceptor show a similar distance dependence. For electron transfer from the acceptor to the donor site, the net charges at small intermolecular distances have the opposite effect: ELUMO (DIP) − EHOMO (DCV3T) decreases towards

6.1. Configuration interaction of singles The excitation energies in a so-defined DIP–DCV3T pair calculated at the CIS/6-31G(d) level are presented in figure 5. 10


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Figure 5. Left: CIS/6-31G(d) excitation energies of a stacked DIP–DCV3T donor–acceptor pair as a function of the distance between the centres of mass, right: excitation energies calculated with time-dependent DFT at the B3LYP/6-31G(d) level. In each panel, the lowest CT excitation from donor to acceptor is shown in red, and in the right panel, the assignment of some further CT excitations between the two molecules are highlighted.

larger charge transfer in the electronic ground state when the molecules are approaching each other. Superimposed to this contribution of the Kohn–Sham orbital energies to the CT excitation energy, its asymptotic dependence on distance contains a fifth of the Coulomb potential between point charges arising from the reduction of exact non-local exchange to 20% in the B3LYP hybrid functional. Hence, the CT excitation energy from the acceptor HOMO to the donor LUMO has a steeper slope resembling the correct distance dependence of the CIS calculations, but all CT excitations from donor to acceptor still conserve the wrong sign of the slope because the influence of mutual charging outweighs the reduced Coulomb potential. In summary, CIS calculations give a correct shape of the CT excitation energy with a systematic shift of the entire curve upwards, whereas TD-DFT with B3LYP results in a qualitatively and quantitatively wrong behaviour: irrespective of distance, the lowest calculated excitation in the donor–acceptor pair is the CT transition from the HOMO of the donor to the LUMO of the acceptor, and the slope of the excitation energy as a function of distance has the wrong sign. From these failures of standard approaches, it is clear that more reliable DFT-based schemes for the calculation of CT excitations become mandatory.

also be obtained from their distribution of Mulliken charges D qD i at the sites Ri of the atoms i forming the donor molecule, A and qj at the atomic sites RA j constituting the acceptor, VMulliken =

X

A qD i qj

i,j

A 4π ε0 |RD i − Rj |

,

(14)

where the charge distribution of each ionized molecule is calculated without the molecular counterion. The respective curve in figure 6 is very close to the c-DFTB reference, demonstrating first that the charge constraint is indeed able to localize the correct excess charge on each molecule via the Lagrange multipliers, and second, that even at very small intermolecular distances, the influence of the counterion on the distribution of Mulliken charges remains marginal. At intermolecular distances below the extension of each molecule, the interaction between the charge distributions is reduced to less than half of the Coulomb attraction between point charges. The distance-dependent part of the CIS excitation energy of the lowest CT state has a similar shape as the c-DFTB calculation. Due to the somewhat different distribution of Mulliken charges and the larger polarizability of the anionic acceptor molecule, small deviations occur at very short intermolecular distances. Differences between the CT potential shape obtained with CIS and c-DFTB remain below 0.1 eV over the entire distance range, or less than 5% of the Coulomb interaction of −1.89 eV between the oppositely charged molecules at their expected equilibrium distance of ˚ 3.43 A. In the right panel of figure 6, a comparison between the c-DFTB approach and the calculation via Mulliken charges is shown for a ZnPc–C60 donor–acceptor pair in a Lb1 configuration where a hexagon–hexagon bond of C60 is oriented towards the centre of the planar phthalocyanine, see

6.3. Constrained DFT Figure 6 shows the distance-dependent part of the lowest CT potential of a pair of ionized DIP+ DCV3T− molecules calculated with the constrained DFTB scheme explained in section 2, defined as the difference between the CT excitation energy and its asymptotic limit, VCoulomb (D+ , A− , r) = c-DFTB (D+ , A− , r) − E c-DFTB (D+ , A− , ∞). The Coulomb Eexc exc interaction between the oppositely charged molecular ions can 11


Topical Review

Figure 6. Left: distance-dependent part of constrained DFTB calculation of the lowest CT energy of a DIP–DCV3T stack as a function of intermolecular distance (green), with the distance dependence calculated via the Mulliken charges of DIP+ and DCV3T− superimposed (black dots). The distance-dependent part of the lowest CT excitation calculated with CIS/6-31G(d) is shown for comparison (red). Right: distance-dependent part of c-DFTB calculation for the lowest CT state of ZnPc as donor and C60 as acceptor molecule (green), with calculation via Mulliken charges superimposed (black dots), for a Lb1 orientation of the molecules towards each other.

Table 6. Ionization potential of donor IPD , electron affinity of acceptor EAA , calculated with the DFTB method, asymptotic value of c-DFTB c-DFTB unscreened CT energy defined as the difference between these values, Eexc (∞) = IPD − EAA , and CT energy Eexc (rdisp ) calculated at the optimized distance rdisp from table 5 according to the constrained DFTB scheme. The unscreened Coulomb energy between cationic c-DFTB donor and anionic acceptor molecule at this distance, VCoulomb (rdisp ), is calculated from the difference between Eexc (rdisp ) and c-DFTB 2 Eexc (∞), and the Coulomb potential −e /(4πε0 rdisp ) between point charges at a distance rdisp is given for comparison. All entries are in eV. D/A pair

IPD

EAA

c-DFTB Eexc (∞)

c-DFTB Eexc (rdisp )

6T/DIP DIP/DCV3T 6T/C60 DIP/C60 ZnPc/C60 DCV4T/C60 DCV5T/C60 DCV6T/C60

6.43 7.10 6.43 7.10 6.48 7.14 6.88 6.67

2.17 2.68 2.32 2.32 2.32 2.32 2.32 2.32

4.26 4.42 4.10 4.77 4.16 4.82 4.55 4.34

2.19 2.53 2.47 2.98 2.39 2.95 2.79 2.65

VCoulomb (rdisp ) −2.07 −1.89 −1.63 −1.79 −1.76 −1.87 −1.76 −1.70

−e2 /(4πε0 rdisp ) −4.21 −4.20 −2.20 −2.22 −2.30 −2.22 −2.20 −2.21

c-DFTB (r c-DFTB (∞) at the equilibrium distance Eexc disp ) − Eexc remains far below the Coulomb interaction between point charges. More specifically, pairs of planar molecules show a reduction of the Coulomb interaction by more than a factor of 2, whereas in combinations between a planar molecule and a spherical fullerene, this difference remains somewhat smaller.

table 5. Dispersion-corrected DFT for the electronic ground state gives a potential minimum at a centre-to-centre distance ˚ corresponding to 2.76 A ˚ between ZnPc and the of 6.26 A, two carbon atoms of C60 facing the centre of ZnPc. At this distance, the c-DFTB scheme gives a CT excitation energy of 2.39 eV, corresponding to a Coulomb interaction of −1.76 eV. We have investigated several further donor–acceptor pairs which have already been applied in molecular solar cells, including 6T–DIP, 6T–C60 , DIP–C60 , DCV6T–C60 , DCV5T–C60 , and DCV4T–C60 . DFTB calculations of donor and acceptor in different charge states provide reference values for IP and EA, and the respective entries IPD and EAA of table 3 are included again in table 6 together with their difference, corresponding to the asymptotic CT excitation energy from donor to acceptor at infinite distance in vacuum, c-DFTB (∞) = IP − EA . For each donor–acceptor pair, Eexc D A a calculation of the CT energy at the minimum of the ground state potential has been performed with the c-DFTB scheme, using the distances rdisp according to table 5. In all cases, the unscreened Coulomb interaction VCoulomb (rdisp ) =

7. CT energy in a polarizable medium As discussed above in connection with the voltammetry data, both the ionization potential and the electron affinity of a molecule are influenced by an embedding polarizable medium. Moreover, the Coulomb attraction between the oppositely charged ionized molecules constituting the CT state has to be screened by the dielectric constant of the surroundings according to equation (13), going beyond the usual assumption of point charges embedded into a polarizable medium [31, 60]. 12


Topical Review

Figure 7. Screened and unscreened CT excitation energy for a donor–acceptor pair: left, DIP–DCV3T; right, ZnPc–C60 . In each panel, the c-DFTB ground state potential Vg has been calculated with dispersion-corrected DFT (blue), the unscreened CT excitation Eexc with the c-DFTB screened scheme (light green), the screened CT excitation Eexc (dark green) according to equation (13), and the excited state potentials Vec-DFTB (magenta) and Vescreened (red) as the respective sums of ground state potential and CT excitation. The CT excitation energy at the minimum of the ground state potential is indicated as a double-sided arrow (dark green). Table 7. Calculated CT excitation energies based on the constrained DFT scheme and embedding into a polarizable medium: ionization potential of donor IP0D and electron affinity of acceptor EA0A , asymptotic value of CT energy defined as the difference between these values, screened (∞) = IP0D − EA0A , screened Coulomb interaction VCoulomb (rdisp )/ε at the optimized donor–acceptor distance rdisp from table 5, and ECT screened CT excitation energy Eexc (rdisp ) in the optimized donor–acceptor geometry, with dielectric constants ε of the blends according to table 2. D/A pair

IP0D (eV)

EA0A (eV)

screened Eexc (∞) (eV)

6T/DIP DIP/DCV3T 6T/C60 DIP/C60 ZnPc/C60 DCV4T/C60 DCV5T/C60 DCV6T/C60

5.46 6.12 5.38 6.04 5.48 6.13 5.92 5.75

3.15 3.67 3.28 3.28 3.26 3.28 3.28 3.28

2.31 2.45 2.10 2.76 2.22 2.86 2.64 2.48

VCoulomb (rdisp )/ (eV) −0.67 −0.61 −0.43 −0.48 −0.49 −0.50 −0.47 −0.45

screened Eexc (rdisp ) (eV)

1.64 1.84 1.66 2.28 1.73 2.36 2.17 2.02

c-DFTB (∞) = 4.42 eV asymptotic CT excitation energy of Eexc + − screened c-DFTB (∞) + of the DIP DCV3T pair to Eexc (∞) = Eexc P+ + P− = 2.45 eV, and at the expected equilibrium ˚ the screened Coulomb attraction distance of rdisp = 3.43 A, between the oppositely charged molecules is reduced from the unscreened value of −1.89 to −0.61 eV, resulting in a CT excitation energy directly at the donor–acceptor screened (r interface of Eexc disp ) = 1.84 eV. In accordance with the schematic type-II level alignment in figure 4 expected for these molecules, this CT energy is significantly below the lowest absorption bands of DIP films starting the vibronic progression at 2.25 eV [51] and of DCV3T films with the lowest vibronic band of the HOMO–LUMO transition at 2.14 eV [41, 45]. In our computational scheme, the CT excitation energy varies from 1.84 eV for molecules in direct contact to 2.45 eV at infinite separation of the ionized donor and acceptor sites, i.e. irrespective of distance, the CT energy remains always above the simple difference of 1.43 eV between the B3LYP Kohn–Sham energies visualized in figure 4. Hence, the alignment of the frontier orbitals

7.1. DIP–DCV3T Any estimate of the impact of the polarizable medium requires reliable values for the dielectric constant. The values along the three principal axes of the dielectric tensor of crystalline DIP can be derived from ellipsometry measurements [51], giving an estimate of about h 1ε i−1 = 3.1 for the static limit. For the model acceptor DCV3T in the DIP–DCV3T molecule pair, we expect a dielectric constant in the same range. The radius rion of the molecular ions can be estimated by equating the volume per molecule V0 with a sphere of this radius. From the high temperature crystal phase of DIP, we deduce a molecular ˚ [47]. Assuming for DCV3T a mass density radius of 4.97 A as in the published crystal phase of DCV4T [41], the volume ˚ see table 2. per molecule gives a radius of 4.92 A, In figure 7, we present the screened potential of a zwitterionic DIP–DCV3T pair embedded into a polarizable medium with an appropriate dielectric constant, assumed to coincide with the static limit in pure DIP of ε = 3.1 [51]. According to table 7, the polarizable medium reduces the 13


Topical Review

calculated with a particular density functional for independent molecules can only serve as a rough guideline for the distance-dependent energetic range covered by the lowest CT state.

compensate. Nevertheless, the uncertainty of the dielectric constant remains a major source of uncontrolled energy offsets. screened (r The entries Eexc disp ) in table 7 correspond to the CT excitation energy at the distance of closest approach between donor and acceptor, extending over a range from 1.64 to 2.36 eV. In cases where the CT transition carries a substantial oscillator strength, a CT absorption band should be observable close to this calculated excitation energy. However, our estimate of the CT transition energy does not necessarily occur far below the lowest absorption band of either donor or acceptor, so that a weak CT transition may be hidden by stronger absorption features of either donor or acceptor.

7.2. ZnPc–C60 In the system ZnPc–C60 displayed in figure 7, the embedding scheme reduces the asymptotic CT energy in vacuum from c-DFTB (∞) = 4.16 eV to E screened (∞) = 2.22 eV. The Eexc exc Coulomb interaction of −1.76 eV at the minimum distance is screened to −0.49 eV, resulting eventually in a CT excitation energy of 1.73 eV at closest approach of donor and acceptor. As a function of distance, the CT energies cover the range from 1.73 to 2.22 eV, essentially everywhere above the rough estimate of 1.76 eV deduced from the frontier orbital energies in figure 1. The gap between donor HOMO screened (∞) = and acceptor LUMO can be identified with Eexc 2.22 eV, in excellent agreement with recent UPS and IPES measurements on similar ZnPc:C60 blends giving a peak-to-peak gap of 2.26 eV [61]. However, due to the small escape depth of the electrons emitted in UPS and the small penetration depth of electrons impinging on the sample in IPES, the molecules contributing to these spectra are localized within the first few nm from the surface [62] so that they might not be representative for donor and acceptor sites deep inside the blend. Hence, the agreement between measured gap and calculated gap might be fortuitous, and molecules deeper in the blend would be subject to larger polarization shifts, corresponding to a smaller gap energy. For perylene tetracarboxylic dianhydride (PTCDA), the change of the polarization energy between bulk and surface has been estimated to be 0.41 eV [63]. Presumably, in the donor–acceptor blends studied in the present work, the polarization energies of molecules at the surface and in the bulk will differ by a similar amount, but detailed model calculations seem to be missing.

8. Spectroscopic properties of CT states For several polymer–fullerene blends, weak absorption and electroluminescence below the absorption edges of thin films of the pure materials have been assigned successfully to CT states [8]. Concerning donor–acceptor blends consisting of small molecules, only the material combination ZnPc–C60 exhibits rather clear absorption spectra around 1.40 eV and photoluminescence (PL) spectra around 0.94 eV [11], whereas DCV4T–C60 blends show PL spectra differing significantly from neat DCV4T films [64]. Details of the photovoltaic devices analysed in the present section are summarized in the supplementary information (available at stacks.iop.org/JPhysCM/25/473201/mmedia). 8.1. ZnPc–C60 In the following, it is assumed that CT excitation and radiative recombination at a ZnPc–C60 interface are accompanied by a small intermolecular transition dipole moment µ. For a Gaussian distribution of CT transition energies arising from thermal motion around the equilibrium configuration of a donor–acceptor pair, the imaginary part of the dielectric function can be obtained as ! abs )2 (E − ECT N 2µ2 1 exp − , (15) Im (ε(E)) = √ V 30 2π σ 2σ 2

7.3. Other donor–acceptor pairs A similar embedding scheme has been applied to the donor–acceptor pairs included in table 6 using the dielectric constants from table 5, see the resulting energies reported in screened (∞) table 7. The asymptotic CT excitation energies Eexc are now in the range 2.10–2.86 eV, and the screened Coulomb interaction at the equilibrium distance is in the range −0.43 to −0.50 eV for complexes involving a fullerene, and somewhat larger in absolute for pairs of stacked planar compounds like 6T–DIP and DIP–DCV3T. As we did not find measured values of the static dielectric constant for all thiophenes, all the respective calculations rely on an assumed value of = 3.1 for the pure thiophene materials, and on an arithmetic average for blends. The uncertainty of the dielectric constant of the blend influences the reliability of the polarization shifts P+ + P− and of the screened Coulomb energy at the equilibrium distance between donor and acceptor. Both quantities are affected in opposite directions by changes of the dielectric constant, so that the respective errors partially

where N/V is the density of suitable ZnPc–C60 pairs allowing for radiative CT excitation. The absorption coefficient α can then be calculated as α(E) =

E Im (ε(E)) . h¯ cn

(16)

In photovoltaic devices, the external quantum efficiency (EQE) is proportional to the absorption coefficient. Figure 8 demonstrates that the EQE of a photovoltaic device containing a 1:1 mixture of both compounds reveals a clear signature of a CT transition in the same energy range as observed in the transmission of thin films of the same blend [11]. Using equations (15) and (16), the measured EQE can be parametrized with two Gaussian bands as ! 2 X (E − Ej )2 1 EQE(E) = aj E √ exp − . (17) 2σj2 2π σj j=1 14


Topical Review

Figure 8. Left: measured external quantum efficiency of a ZnPc–C60 bulk heterojunction solar cell with a mixing ratio of 1:1 (symbols), and fit by a sum of two Gaussians (black) according to equation (17). The larger structure at 1.81 eV (blue) coincides with the lowest absorption band of ZnPc [52], and the structure at 1.42 eV (green) is assigned to a CT excitation from ZnPc to C60 . Right: configuration coordinate diagram for CT excitation from the ground state potential of a pair of neutral donor and acceptor molecules (D0 A0 ) to the potential of the lowest CT state (D+ A− ). Boltzmann distributions around each potential minimum allow us to obtain the Gaussian DOS underlying absorption and photoluminescence, reproduced directly in the shape of Im (ε) or α/E (green), and in the PL intensity divided by E3 (red), see equation (19). The reorganization energies λ on both potential surfaces are assumed to coincide (blue arrows).

Figure 8 reports a fit of the EQE according to equation (17), where the Gaussian band at higher energy arises from the lowest transition of ZnPc [52], and the smaller structure at lower energy is assigned to CT from ZnPc to the fullerene because it is far below the optical gaps of the pure materials. A fit of the EQE according to equation (17) abs = 1.41 eV, with σ abs = gives a CT absorption band at ECT CT abs [1 + 0.118 eV, corresponding to an average energy of ECT abs /E abs )2 ] = 1.42 eV. The full width half maximum (σCT CT (FWHM) of this CT transition is as small as 0.28 eV, far below the FWHM of 0.5 eV for the HOMO in ZnPc films observed with UPS [65] and of 1.0 eV for the LUMO of C60 with IPES [35]. The EQE peak in figure 8 remains somewhat narrower than the CT absorption band found earlier, presumably because the contribution of interference phenomena to the transmission of the thin films used has inhibited a more precise assignment of a rather small absorption feature [11]. On the other hand, the EQE band has a similar width as the photoluminescence spectra observed on a similar blend [11], indicating that both broadenings are related, and that they do not arise from the quite large width of the DOS of the donor HOMO and acceptor LUMO. An intuitive interpretation of the broadening can be obtained from the configuration coordinate diagram in figure 8. The elongation q in this diagram consists of deformations of the neutral molecules towards the charged species involved in the excited state, and small changes of the intermolecular distance. The radiative recombination rate for a sharp PL transition would be obtained as [66] 0PL =

PL n)3 µ2 (ECT

3π 0 h¯ 4 c3

,

From figure 8, it is clear that absorption and PL involve different molecular geometries, so that the Gaussian densities of states involved in the lineshapes of Im (ε) and IPL (E) are abs = E PL centred around different energies ECT CT + λ and ECT = ECT − λ, respectively, where λ is the reorganization energy. Due to the energy-dependent prefactor of the PL intensity containing E3 n3 (E) according to equation (19) and the finite Gaussian broadening, the average PL energy differs from ECT − λ, so that the observed Stokes shift is slightly smaller than 2λ. In Marcus transfer theory applied to a configuration coordinate diagram as in figure 8, the reorganization energy λ and the broadening σ are related as σ 2 = 2λkB T,

(20)

where kB is the Boltzmann constant [67]. This simple equation can be used to check the internal consistency of the configuration coordinate diagram in figure 8 when applied to the EQE spectra and PL of ZnPc–C60 blends. According to our fit of the EQE spectra in figure 8 with equation (17), we find σ = 0.118 eV, corresponding to a FWHM = 0.28 eV and a reorganization energy of λ = 0.27 eV. The observed Stokes shift between our EQE peak at 1.42 eV and previous PL spectra around 0.94 eV amounts to 0.48 eV, slightly below the expected value of 2λ = 0.54 eV. However, taking into account the prefactor E3 in equation (19) but still neglecting a possible energy dependence n(E) of the refractive index, the average observed PL transition energy is shifted to approximately PL [1 + 3σ 2 /(E PL )2 ], so that a value of E PL = 0.87 eV ECT CT CT abs = 1.41 eV compatible with an absorption or EQE band at ECT and λ = 0.27 eV would give a PL spectrum centred around 0.92 eV, in excellent agreement with the observed value of 0.94 eV [11]. The above arguments demonstrate that for the CT transition in ZnPc–C60 blends, both the observed broadening σ of the EQE in figure 8 and the Stokes shift are compatible with the configuration coordinate in figure 8. Hence, they

(18)

corresponding to a broadened PL spectrum proportional to ! PL )2 (E − ECT µ2 E3 n3 (E) 1 IPL (E) ∝ exp − . (19) √ 2σ 2 3π 0 h¯ 4 c3 2π σ 15


Topical Review

are completely independent of the width of the HOMO DOS observed with UPS and the width of the LUMO DOS observed with IPES. Moreover, the broadening of the EQE spectra does not arise from an average over several typical geometries allowing for CT absorption and PL: it is an intrinsic property of the most strongly absorbing configuration of a ZnPc–C60 pair in a bulk heterojunction. The CT contribution to the EQE arises from a Gaussian centred at 1.41 eV, somewhat below the calculated screened (r excitation energy Eexc disp ) = 1.73 eV in the most stable donor–acceptor geometry, see figure 7 and table 7. Reasons for systematic deviations of the c-DFTB approach from observed transition energies will be discussed in section 10. 8.2. DCV4T–C60 The absorption spectra of neat DCV4T films start their vibronic progression with a subband around 2.05 eV [42]. According to table 7, CT absorption at the DCV4T–C60 interface is expected at an energy around 2.36 eV, far above the lowest absorption feature of the donor. Hence, in contrast to ZnPc–C60 , photovoltaic cells based on blends consisting of DCV4T and C60 should not show any CT contribution to the EQE below the absorption onset of the constituting materials, and indeed, the EQE spectra reported in figure 9 remain structureless in the gap region. From the onset to about 3.2 eV, absorption and EQE of DCV4T–C60 blends is dominated by the thiophene donor, and at higher energy, the first strong absorption band of the fullerene contributes significantly to the measured EQE of photovoltaic devices [64]. As discussed elsewhere in more detail, the best intermixing of DCV4T and C60 blends can be achieved for codeposition at 30 ◦ C, resulting in a particularly high power conversion efficiency of η = 3.0% [64]. In these films, the PL from DCV4T is strongly quenched, revealing instead weak signals at lower energy. The PL spectra of the DCV4T–C60 blend displayed in figure 9 exhibit two subbands at 1.86 and 1.73 eV resembling PL from neat films, but with reduced intensity in the normalized spectra. In sharp contrast to this suppression of PL from donor molecules, the PL features at lower energy become more prominent, so that they can be assigned to CT transitions, including in particular the Gaussian PL subband centred at 1.58 eV. Due to an expected small contribution of the donor molecules to PL in this region, we assign a conservative error margin of 1.58 ± 0.05 eV to the position of this CT band. Presumably, the broader PL structure at 1.35 eV is composed of several somewhat narrower subbands, but a more detailed assignment is not possible because the measured data extend only down to 1.24 eV. The quite large broadening parameter σ = 0.185 eV of the PL subband at 1.58 eV would indicate a reorganization energy of λ = 0.66 eV according to equation (20), or a Stokes shift of 1.32 eV, which would result in a quite unlikely value of 2.90 eV for the CT absorption band. Therefore, we suspect that the fitted broadening of the PL subband arising from CT states cannot be understood entirely from a configuration coordinate diagram as in figure 8, but that instead an ensemble of different geometric

Figure 9. Measured external quantum efficiency of a DCV4T–C60 bulk heterojunction solar cell with a mixing ratio of 2:1 (blue), and normalized PL spectra IPL (E)/E3 for neat films of DCV4T (green) and of a DCV4T–C60 blend codeposited at 30 ◦ C (red) [64]. The latter spectra can be fitted by a sum of 4 Gaussians (black dots), where the two structures at higher energy (orange dashed) reproduce the subbands of neat DCV4T with lower intensity, and the enhanced PL spectra at lower energy are assigned to CT states at the DCV4T–C60 interface (magenta dashed). EQE spectra are given in absolute units, and PL spectra are normalized to the respective peak values.

arrangements of donor–acceptor interfaces contributing to radiative recombination enhances the PL width significantly. The difference between the calculated CT absorption band at 2.36 eV and the PL band at 1.58 eV would indicate a Stokes shift of 0.78 eV. This is somewhat larger than, but still in a similar range as, the observed Stokes shift of 0.48 eV between the peaks of CT absorption and PL in ZnPc–C60 blends, see section 8.1.

9. Comparison of calculated CT energies and observed open circuit voltages Assemblies of molecules as used in photovoltaic devices exhibit a DOS of their frontier orbitals with a significant Gaussian broadening. As discussed already in section 4, UPS and IPES can in principle determine the respective DOS, resulting however in much broader features assigned to the LUMO than to the HOMO. At the present stage, it is unclear whether the larger width of the LUMO DOS is really an intrinsic feature of molecular materials because the experimental method used for IPES measurements might contribute to the broadening. Therefore, in the following we assume that donor HOMO and acceptor LUMO have the same broadenings, taking for simplicity a value of FWHM = 0.4 eV, somewhat below values measured recently on blends of ZnPc and C60 [61]. The open circuit voltage of a photovoltaic device is given by the difference of the electron and hole Fermi energies, 16


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Figure 10. Comparison of electron and hole distributions in a ZnPc–C60 photovoltaic device with the energy ECT of an electron–hole pair forming a CT state at the donor–acceptor interface, on a logarithmic scale, comparing the situation at room temperature (T = 300 K, left) and at low temperature (T = 30 K, right). Gaussian DOS of ZnPc HOMO (black, centred around −5.41 eV) and C60 LUMO (black, centred around −3.51 eV), Fermi energies (black dash–dotted) together with their difference eV oc (black arrow), energy ECT of the CT state (blue arrow), and parts of the DOS occupied by thermalized electrons (red) or holes (green). For simplicity, the broadening of each DOS relies on the same value of FWHM = 0.4 eV, and the energy difference ECT is visualized symmetrically around the midgap. The open circuit voltage at room temperature corresponds to the measured value of 0.52 V, and the value of 1.09 V at T = 30 K is obtained from linear interpolation towards eV oc (T → 0) = ECT = 1.15 eV.

carriers deep within each donor or acceptor grain is expected to be much longer than the lifetime of a CT state localized around the interface. Our scenario at low temperature relies on the experimental observation that the open circuit voltage grows linearly when the temperature decreases, reaching eventually the CT energy for T → 0 [68, 69]. A linear relation between open circuit voltage and temperature follows in a natural way from the diode characteristics of a donor–acceptor heterojunction [70] and from the equilibrium between drift and diffusion currents at open circuit [71]. The connection between the open circuit voltage and the CT energy is more subtle because it relies on thermodynamic considerations involving detailed balance between photogenerated charge pairs and thermal radiation from the solar cell [9, 68]. The temperature-dependent open circuit voltage of photovoltaic devices containing ZnPc–C60 indicates a low-temperature limit of 1.08 ± 0.02 V [69], in good agreement with the value ECT = 1.15 eV in figure 10 deduced from spectroscopic observables. The visualization of the situation at T = 30 K in the right panel of figure 10 relies on an open circuit voltage of 1.09 V found from linear interpolation between room temperature and an assumed value eV oc (T → 0) = ECT = 1.15 eV. The activation energy towards CT recombination decreases accordingly, but due to the steeper Boltzmann tail of the Fermi–Dirac distribution, the fraction of charge pairs with an energy difference above ECT remains again quite small. This recombination bottleneck results in an increased lifetime of the photocarriers which are flooding the device up to a much larger charge carrier density, corresponding to an increased difference of the Fermi energies of electrons and holes defining a larger open circuit voltage. Due to the asymmetry of the occupied part of the DOS, at high temperature the average energy of the electrons exceeds the respective Fermi energy, whereas at low temperature, it remains below. The high temperature Boltzmann limit of the

eV oc = EFermi,e − EFermi,h . Figure 10 visualizes the situation for a photovoltaic device consisting of a 1:1 blend of ZnPc and C60 , relying on the measured open circuit voltage at room temperature and illumination by 1 sun, Voc = 0.52 eV, CT abs = 1.41 eV, absorption with a Gaussian DOS centred at ECT PL = 0.89 eV PL from CT states with a Gaussian centred at ECT giving a PL band around 0.94 eV [11], and an average CT energy of ECT = 1.15 eV corresponding to the arithmetic abs and E PL . The HOMO–LUMO gap of 1.90 eV average of ECT CT is assigned to the CT excitation energy at infinite distance, abs = 1.41 eV and the screened Coulomb compatible with ECT interaction of −0.49 eV between a neighbouring cationic ZnPc donor and an anionic fullerene acceptor. This gap energy is 0.36 eV below the value of 2.26 eV deduced from UPS and IPES [61], the difference giving a rough estimate for the magnitude of the polarization shift between molecules deep in the blend and sites rather close to the surface where they can contribute to UPS and IPES. In an open electric circuit, no external current can extract charges from the photovoltaic device, so that the charge pairs created by exciton dissociation at the donor–acceptor interface have to recombine. At room temperature, the energy difference between the parts of each DOS occupied by electrons or holes is quite low, but as the tails of the DOS are dominated by states deep within the donor or acceptor grains, direct electron–hole recombination involving these tails is not expected because the respective molecular sites are rather far apart. Instead, electrons and holes are forced to recombine via CT states at donor–acceptor interfaces, requiring a much larger energy ECT = 1.15 eV for the initial state before recombination. Hence, radiative or non-radiative recombination via CT states is a process with an activation energy of the order of ECT − eV oc . Only a tiny fraction of the electron and hole distributions can generate pair energies above the threshold ECT , so that the lifetime of the free charge 17


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Table 8. Comparison of calculated CT excitation energy, observed absorption and PL from CT states, and measured open circuit voltage. abs PL The energies hECT i and hECT i assigned to spectroscopic features are measured average energies, not the underlying parameters of the Gaussian DOS discussed in section 8. All entries are in eV. Flat heterojunction (FHJ) D/A pair 6T/DIP DIP/DCV3T 6T/C60 DIP/C60 ZnPc/C60 DCV4T/C60 DCV5T/C60 DCV6T/C60

screened Eexc (rdisp )

1.64 1.84 1.66 2.28 1.73 2.36 2.17 2.02

abs hECT i

PL hECT i

eV oc a

1.42

0.94 1.64

1.38 0.98b 0.45a 0.93a 0.55e 1.13f 1.10f 0.93f

Bulk heterojunction (BHJ)

screened Eexc (rdisp ) − eV oc

eV oc

screened Eexc (rdisp ) − eV oc

0.26 0.86 1.21 1.35 1.18 1.23 1.07 1.11

1.08b 0.58c 0.85d 0.52b 0.98g 0.95h 0.90i

0.76 1.08 1.43 1.21 1.38 1.22 1.12

a

Reference [72, 35]. This work, see supplemental information (available at stacks.iop.org/JPhysCM/25/473201/mmedia). c Reference [74]. d Reference [75]. e Reference [73]. f Flat heterojunction [42]. g Reference [64]. h Reference [76]. i Reference [77]. b

average electron energy involving a shift of −σ 2 /(2kB T) with respect to the maximum of the Gaussian DOS describing the LUMO does not even apply at room temperature. For all 8 donor–acceptor combinations investigated in the present work, photovoltaic devices based on flat heterojunctions (FHJ) have been presented, see table 8 for the open circuit voltages at room temperature under an illumination intensity corresponding approximately to 1 sun. In cases where several measures of the open circuit voltage under similar conditions have been published, we have selected the highest value of the open circuit voltage because this is an indication that the respective layer sequence and layer thicknesses are chosen in the most suitable way, including electrodes, doped transport layers, electron and hole blocking layers, and the photoactive donor and acceptor layers. Most of these donor–acceptor pairs have also been used in bulk heterojunction (BHJ) solar cells, achieving open circuit voltages in the same range, with deviations from flat heterojunctions between −0.15 and +0.13 eV. A comparison of the calculated CT energies with absorption or PL spectra involving CT states was only possible for ZnPc–C60 and for DCV4T–C60 because these seem to be the only blends where such spectra have become available. The energy eV oc corresponding to the open circuit voltage of the respective photovoltaic devices at room temperature is 0.42–0.60 eV below the PL arising from the CT state, and in ZnPc–C60 , it is about 0.9 eV below the absorption or EQE band assigned to CT. For the systems where both flat and bulk heterojunctions have been investigated, on average our computational scheme gives CT excitation energies between neighbouring donor and acceptor sites 1.16 eV above the observed value of eV oc at room temperature. In polymer–fullerene blends, this difference is around 0.9 eV [10], indicating that the basic CT physics are similar.

As discussed above in connection with figure 10, thermodynamic models of recombination via CT states and the assumption of detailed balance rely on the average CT energy ECT , halfway between the absorption energy abs screened (r Eexc disp ) = ECT and the corresponding PL energy PL ECT [68]. Hence, a more comprehensive comparison of calculated CT states and observable quantities would require calculation of the Stokes shift between CT absorption and PL. For various molecular systems, the respective reorganization energies between neutral and ionized molecules have been calculated with high precision, requiring however a careful choice of the density functional and the variational basis set for the electronic orbitals. This task is beyond the scope of the present work.

10. Discussion Our calculated CT energies rely on DFT values for the ionization potential of the donor and the electron affinity of the acceptor, involving sums and differences of altogether four DFT calculations applied to molecules in different charge states. For the combinations chosen in the present work, the computed differences for electron transfer from a donor to an acceptor in vacuum, IPD −EAA , lie in the range 4.10–4.82 eV. Some deviations between measured and calculated values for ionization potentials IP0 and electron affinities EA0 of molecules embedded into a polarizable medium were discussed in section 4. They arise from systematic deficiencies of the density functionals used, the convergence of the variational basis sets for the electronic orbitals, the choice of the dielectric function of the embedding medium, and a possible influence of molecular orientation on measured UPS and IPES spectra [56]. Based on the values in table 1 relying on the dielectric constants in table 2, B3LYP/6-31G(d) gives rms deviations of 0.43 eV for IP0 and 0.42 eV for EA0 , whereas 18


Topical Review

model for polymer–fullerene solar cells would be particularly interesting. Due to the very restricted set of observed spectroscopic features arising from CT states in donor–acceptor blends of small molecules, a final assessment of the precision of our approach remains difficult. Therefore, more comprehensive spectroscopic studies of CT states would be required to bring our understanding of the open circuit voltage to a similar level as in polymer–fullerene solar cells where quite detailed models of recombination via CT states have been proposed and corroborated by observed spectroscopic features assigned to CT states [8, 9]. Together with UPS and IPES spectra measured on blends of small molecules, a more systematic comparison of measured data with the ingredients of a microscopic computational scheme like ours could reveal whether the residual deviations arise mainly from the DFT method used or from the embedding into a polarizable medium.

according to table 3, in the DFTB scheme the rms deviations amount to 0.25 eV and 0.79 eV, respectively. When excluding 6T and DIP from the comparison, both DFT methods would give a much lower scatter for EA0 , reducing to 0.19 eV for B3LYP and 0.15 eV for DFTB. In this reduced set of six reference materials, a somewhat optimistic estimate for the uncorrelated deviations of IP0D and EA0A on the difference IP0D − EA0A would then result in a rms deviation of 0.46 eV in B3LYP and 0.29 eV in DFTB, respectively, whereas in the full set of 8 materials, the scatter of the calculated energy differences would be significantly larger. The embedding into a polarizable medium relies on a suitable choice of an isotropic dielectric constant ε of the donor–acceptor blend and polarization energies calculated for spherical molecular ions, resulting in red shifts P+ + P− between −1.86 and −2.01 eV, see tables 6 and 7. The screening of the Coulomb interaction is again obtained with the same isotropic model for the dielectric constant, giving values in the range from −0.43 to −0.67 eV at the distance rdisp of the most stable donor–acceptor pair in the electronic ground state. Hence, the calculated CT excitation energy according to equation (13) arises from a difference between three quite large energies, where the first introduces some uncertainty arising from the density functional and variational basis set involved, and the other two rely heavily on the assumption of an isotropic polarizable medium and a suitable choice of the static dielectric constant. It is very likely that more realistic assumptions for the embedding scheme would modify the electrostatic corrections, e.g. by accounting for a granular medium consisting of spherical fullerenes and planar ZnPc molecules with anisotropic polarizabilities. Hence, we do not expect that our embedding scheme can estimate P+ + P− + VCoulomb (rdisp )/ε in a donor–acceptor blend with a precision better than 0.3 eV. Together with the uncertainty of 0.29 eV for the constrained DFT values of IPD − EAA when excluding 6T and DIP from the test set, this results in an error margin of 0.42 eV, more than doubling when including all 8 reference compounds.

Acknowledgments We thank Novaled for providing dopants, the University of Stuttgart for diindenoperylene, and the University of Ulm for dicyanovinyl-quaterthiophene. This work has received funding by the European Community’s Seventh Framework Programme under Grant Agreement No. FP7267995 (NUDEV), by the DFG through priority program SPP 1355, and Center for Advancing Electronics Dresden (CfAED).

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11. Conclusion, outlook In this work, we have presented a constrained DFTB scheme together with an embedding scheme accounting for an isotropic dielectric constant of a donor–acceptor blend, allowing us to compute the CT excitation energy of a donor–acceptor pair in direct contact and the energy required for separating the opposite charges to infinite distance. On average, the calculated values for the CT excitation energy are 1.16 eV above the energy corresponding to the open circuit voltage measured at room temperature under an illumination of 1 sun. This energy difference is in the same range as in photovoltaic devices based on polymer–fullerene blends where it amounts to about 0.9 eV. From this semi-quantitative agreement, we conclude that the basic photophysics in photovoltaic devices containing either small molecules or polymer–fullerene blends rely on similar considerations concerning CT energies. Hence, an application of similar constrained DFT schemes to larger oligomers as a 19


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21

Acceptor Interfaces in Organic Solar Cells.

Charge Transfer Excitons at van der Waals Interfaces.

metal interfaces.

Ultrafast charge transfer in operating bulk heterojunction solar cells.

Efficient charge generation by relaxed charge-transfer states at organic interfaces.

Metal-organic frameworks at interfaces in dye-sensitized solar cells.

Interfaces in perovskite solar cells.

spiro-OMeTAD interfaces: uncovering the detailed mechanism behind high efficiency solar cells.

Probing Charge Transfer and Hot Carrier Dynamics in Organic Solar Cells with Terahertz Spectroscopy.

Charge Transfer Dynamics at Dye-Sensitized ZnO and TiO2 Interfaces Studied by Ultrafast XUV Photoelectron Spectroscopy.

Hybridization-controlled charge transfer and induced magnetism at correlated oxide interfaces.

Organic Interfaces: The Role of Dielectric Interlayers.

Charge regulation at semiconductor-electrolyte interfaces.

Molecular excited states: accurate calculation of relative energies and electronic coupling between charge transfer and non-charge transfer states.

electrode interfaces studied by core-hole clock spectroscopy.

Charge Transfer at Hybrid Interfaces: Plasmonics of Aromatic Thiol-Capped Gold Nanoparticles.

Anomalous coarsening driven by reversible charge transfer at metal-organic interfaces.

TCNE.

Resonance Raman and excitation energy dependent charge transfer mechanism in halide-substituted hybrid perovskite solar cells.

High-surface-area architectures for improved charge transfer kinetics at the dark electrode in dye-sensitized solar cells.

Charge generation mechanism in organic solar cells.

organic interfaces in organic solar cells.

Fullerene-Based Photoactive Layers for Heterojunction Solar Cells: Structure, Absorption Spectra and Charge Transfer Process.

Charge asymmetry at aqueous hydrophobic interfaces and hydration shells.