Hybrid Materials: Still Challenging for Ab Initio Theory?

Hybrid inorganic/organic systems (HIOS) open new avenues for tailoring them with respect to desired features and functions by exploiting the respective advantages of their components. Therefore, these materials are actively explored in many experimental studies and devices. On the theory side, similar investigations are rather scarce as such interfaces, in addition to exhibiting large unit cells, require highest‐level theories to be described reliably. Consequently, hybrid materials pose a challenge for electronic structure theory, starting from density‐functional theory to methods beyond, particularly many‐body perturbation theory. This concerns both conceptual aspects and computational bottlenecks. In this perspective, the performance of state‐of‐the‐art theoretical approaches applied to HIOS is summarized, mainly focusing on optoelectronic properties. Recent achievements, open challenges, and urgent needs are addressed.


Introduction
Hybrid materials, i.e., those consisting of inorganic and organic components, cannot be considered to be yet another class of condensed-matter systems where state-of-the-art methodology of computational materials science can be applied in a straightforward manner.In fact, despite the variety of systems investigated experimentally, a lot less have been studied by employing highlevel theoretical approaches (e.g., see refs.[1-17]) The challenges for ab initio theory related to describing structure, electronic bands, optical excitations, polaronic effects, their dynamic behavior, and more have been critically assessed by some of us [18] several years ago.To cite from their conclusions, "In short, to fully understand hybrid materials and reach predictive power on a quantitative level, novel methodology on various levels is needed.A must for density-functional theory (DFT) and methods beyond is the ability of treating different interactions on the same footing.This includes urgent needs for a systematic validation of ab initio methodology together with highly efficient and massively parallel computer codes.Only with this level of accuracy and understanding, we can proceed to additional questions imposed by real systems, such as (un)intentional doping, band bending, disorder, grain boundaries, and more".
In this perspective, we revisit the research field in view of aforementioned challenges to monitor the progress that has been achieved meanwhile in terms of insight as well as theoretical and/or computational approaches.With selected examples, we will demonstrate a few of these aspects.They mainly concern electron-electron, electron-hole, and electron-phonon coupling (EPC).Arguably, our examples are somewhat biased toward current method and code developments that reflect the most important research questions related to the materials investigated by the Collaborative Research Center (CRC) HIOS. [19]The goal of the center is to elucidate and tailor the fundamental chemical, electronic, and photonic interactions in organic-inorganic hybrid materials, recently focusing on transition-metal dichalcogenides (TMDCs) as the inorganic component.We emphasize that our main conclusions are valid for any kind of such weakly bond materials, and to a large extent also for other interfaces.

Electronic Structure
In view of their applications in optoelectronic devices, the most important physical quantity of (semiconducting) hybrid systems is their bandgap.More fundamentally, however, first and foremost, the level alignment at the interface must be qualitatively correct.While in ref. [18] it was argued that there is a high chance that the alignment at weakly bound O/I interfaces will be wrong on the DFT level, since there is basically no exchange-correlation (xc) functional that works equally well for organic and inorganic materials, it was recently shown [16,17] that this indeed can be the case.For the examples of pyrene and pyridine on a monolayer of MoS 2 , semi-local DFT, using the Perdew Burke Ernzerhof (PBE) functional [20] predicts type II heterojunctions for both interfaces, while G 0 W 0 on top changes it to type I, as displayed in Figure 1 (left) for pyrene@MoS 2 , the material considered in the following.An issue here is that G 0 W 0 based on PBE is a rather good choice for the MoS 2 side, however, not for the molecular side.As a matter of fact, the energy diference between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) of the organic monolayer is more strongly underestimated.
An important factor in the level alignment is the mutual screening and polarization, leading to a substantial reduction of the distances between the band edges of the constituents with respect to their individual values.While it is known that an inorganic substrate impacts adsorbed molecules, [1,4,7] interestingly, it turned out that even monolayers of wide-gap materials show a non-negligible mutual effect. [21]In pyrene@MoS 2 , even the molecular monolayer causes a significant reduction of the MoS 2 gap by about 0.3 eV in the hybrid system, as obtained from G 0 W 0 @PBE. [17][24][25] Such functionals would be an ideal GW starting point, however.Ideally, some self-consistent GW variant should be applied.
The question left is whether any of these options or just a hybrid functional-like HSE [26] as a starting point would affect the qualitative behavior of the material's electronic structure in this example.In fact, there is no game change to be expected as demonstrated by employing HSE.Overall, the TMDC side is less strongly affected when going from PBE to HSE, and the increased HOMO-LUMO distance on the molecular side does not lead to a change of the alignment type.This is illustrated in Figure 1 (right).
Although the level alignment here is robust against the DFT starting point, the small energetic distance between the valenceband maximum (VBM) of the TMDC and the HOMO of the molecular component of approximately 0.3 eV still bares the risk of an uncertainty that might come in when considering zero-point vibration (ZPV) effects and temperature.This issue will be shortly addressed further later.

Optical Excitations
To investigate excitonic effects, the Bethe-Salpeter equation (BSE) of many-body perturbation theory (MBPT) is the method of choice.Given its complexity, involving not only nonlocal operators but, in contrast to GW, also four-point functions, it is numerically heavy and thus a bottleneck for the materials class is discussed here.To remedy this situation, we have pursued two different approaches, both implemented in the BSE module of the exciting code. [27]For a detailed overview of the BSE formalism in exciting, the reader is referred to ref. [28].For the computations details, see ref. [29].In the first one, we simplify the computation of the dielectric screening.This part is a major bottleneck in BSE calculations of large systems because it scales to the fourth power of the number of atoms, OðN 4 Þ.Here, we compute the individual contributions to the polarizability for the two components which are then superposed to enter the screened Coulomb kernel in the direct term of the BSE Hamiltonian.This additive approach allows us to compute the contribution of MoS 2 in its unit cell and to subsequently map it onto the supercell, thereby reducing the computational cost by Oðm 6 Þ for an m Â m supercell. [30]The so-obtained optical spectrum (labeled easBSE, where eas stands for expanded and added screening) is shown in the middle panel of Figure 2. Compared to the full calculation, the differences are overall very small, and we conclude that this procedure is a good approximation for complex weakly bonded interfaces like the one considered here.32] Considerable speedup is also gained by our second approach.We apply interpolative separable density fitting (ISDF) [33,34] to set up the interaction kernel of the BSE Hamiltonian and use a Lanczos algorithm for its diagonalization.This procedure results in a scaling of the two parts with OðN 2 e N k Þ and OðN k log N k Þ, respectively, where N e is the number of electrons and N k is the number of k-points. [35]For comparison, the standard BSE approach scales with OðN 6 e N 3 k Þ, making the calculations for large systems unfeasible.The ISDF technique was shown to reduce the computational cost in BSE calculations for molecules [36] and demonstrated as a proof of principle for solids. [35]In Figure 2 (bottom panel), we compare the here-obtained result with the standard BSE solution for the case of pyrene@MoS 2 .

Spin-Orbit Coupling
While spin-orbit coupling (SOC) typically does not play a key role in organic molecular systems owing to only light atoms being involved, it has an important effect in TMDCs, splitting the VBM of MoS 2 by 0.13 eV. [37]Although this is a small perturbation to the overall electronic structure, it is crucial when it comes to optical excitations, by dramatically reshaping the lowest-energy excitations. [37]Moreover, photoluminescence experiments show that the two prominent peaks at the absorption onset arise from transitions across the direct gap to the maxima of the split valence bands. [38]Hence, including SOC in ab initio approaches is a necessity for a reliable description of the optical properties.For hybrid materials formed with the TMDC as the inorganic counterpart, this is equally important as-given the type I alignment pointed out for the example provided earlier-the lowest excitations are dominated by the MoS 2 side.Even such ground-state calculations can be very demanding as they may require a large basis-set size when SOC is treated in a secondvariational approach.To make these calculations feasible for complex materials, some of us have recently developed an The computational parameters are the same as in ref. [17].
optimal basis set suitable for treating SOC in such demanding cases. [39]In short, it makes use of the fact that SOC effects arise close to the atomic nuclei, for which local orbitals provide an efficient solution.

Electron-Phonon Coupling
Ab initio approaches for EPC and its impact on the electronic structure, spectroscopy, and transport have experienced a huge boost in the last decade.For an overview of the developments, we refer to a recent review article by Giustino. [40]Even more recently, [41] the many-body theory of phonon-induced bandstructure renormalization has been unified with the description of polarons which, in turn, sets the stage for a parameterfree description of charge-carrier transport.While these developments will be state of the art in future calculations, they are currently not feasible for complex materials like O/I interfaces, [42] the target materials of this work.
The importance of EPC-related effects on the electronic structure of TMDCs is, nevertheless, evident from a variety of investigations.For the MoS 2 monolayer, the bandgap renormalization due to temperature effects has been estimated to be around 60-90 meV both from calculations [43,44] and experiments. [45,46]ecent angle-resolved photoemission spectroscopy (ARPES) experiments together with calculations of the electron spectral function have enabled a combined description of distinctive spectral signatures, such as plasmonic polarons. [47]Moreover, ultrafast electron diffraction and ultrafast electron diffuse scattering provide a momentum-resolved picture of the phonon dynamics. [48]From a theoretical perspective, the combination of EPC calculations with the solution of the time-dependent Boltzmann equation has resulted in a novel approach toward nonequilibrium lattice dynamics. [49]Here, we show, in Figure 3, the EPC-derived spectral function in the Fan-Migdal approximation projected onto the electronic band structure for the MoS 2 monolayer at zero temperature.The broadening of the electronic bands reveals strong ZPV effects, together with non-negligible shifts in certain parts of the Brillouin zone, indicating a lowering of the bandgap.At the K point of the Brillouin zone, it amounts to 60 meV, in good agreement with refs.[43,44].Expecting similar or even more pronounced effects on the molecular side, as observed experimentally for different molecules, [12,50,51] also the level alignment could be affected, as argued in ref. [18].This might indeed happen in pyrene@MoS 2 where the HOMO-VBM distance in the G 0 W 0 band structure is only about 0.1 eV [17] (see also Figure 1), and an upward shift of the HOMO might bring it above the VBM of the TMDC.

Benchmarking
Benchmarks for excited-state properties based on many-body techniques are basically absent in general, and the more so for hybrid materials.So far, large-scale comparison of different  codes [52] has been limited to simple materials (elemental solids), functionals (PBE), and properties (equation of state).It is quite natural to assume that increasing the complexity on either side will bring issues to light.Among the few data collections for excitations, we mention GW100, [53][54][55] a project to benchmark the GW approach for molecules.Another example is a recent benchmark of X-ray absorption spectra for titanium compounds. [56]Systematic evaluations on different levels of theory are overall missing.
As obvious from the examples provided earlier, in the CRC HIOS, the focus lies on TMDCs as the inorganic component.Interestingly, even for the TMDC itself, there are intriguing computational issues to be solved.For example, for the MoS 2 monolayer, an indirect bandgap has been reported from several G 0 W 0 calculations.The reason for this has been controversially discussed in literature, [57][58][59][60] claiming the underlying geometry (optimization), the use of a 2D Coulomb truncation, or the amount of the vacuum size to be responsible for discrepancies.Therefore, we are currently systematically assessing the electronic structure of MoS 2 to get a profound understanding on how computational details, including the choice of the xc functional, the so-obtained geometry, etc., impact the results.Obviously, it is an important prerequisite to get the building blocks right before entering complex materials like O/I interfaces.

Open Challenges and Perspectives
Why can't we just apply the best possible methodology for hybrid materials?The biggest problem is the unfortunate scaling of many-body methodology that limits us to a system size of about 100 atoms for most codes implementing MBPT.Entering the exascale era, however, there are new opportunities opening.As pointed out in a recent Roadmap article, [61] covering the concepts of 14 electronic structure codes in view of exascale computing, the community has to tackle a number of challenges to make use of new architectures and hardware accelerators to achieve optimal scaling and performance.In this context, we point to the NOMAD Centre of Excellence, [62] undertaking a joint effort toward developing exascale libraries for code families with representative implementations for different basis-set types.
Overall, in addition to the fact that these computational developments are just ongoing, the outlook to massive parallel code with exascale performance does not solve issues of missing or unsatisfactory methodology.
So, where are current bottlenecks?As mentioned earlier, local hybrid functionals are most promising to deliver a DFT ground state with a proper level alignment.This would not only represent a good starting point for methods beyond but could also allow for staying on the G 0 W 0 level compared to a self-consistent treatment.Challenges here [23] are the gauge problem and the need of efficient analytical/numerical integration techniques.Nevertheless, advances are being made in the applicability of such local hybrid functionals to weak and strong correlated systems, [22] and in the evaluation of dipole moments and polarizabilities. [24]oncerning charge-neutral spectroscopy, BSE is a valid state-of-the-art approach for static optical [63,64] and core-level excitations, [28,65,66] being feasible for moderately large system sizes.
New grounds are currently explored for describing dynamical processes as counterparts to significant developments of various time-resolved (tr) experimental probes, realized in pump-probe experiments, such as, trARPES, etc. Extensions of ab initio methodology in all these aspects are required, e.g., to capture time scales for exciton formation and recombination, [14,[67][68][69] effects of EPC, [70][71][72][73] or exciton diffusion processes. [74]In addition to the challenges to treat various interactions and processes on the same footing, making method development extremely complex, the unfavorable scaling with system size makes such approaches computationally too expensive to be applied to large systems.This is, in particular, so if they require supercells big enough to accommodate excitons that may be delocalized in nature.Therefore, also clever approximations as well as new algorithms and/or interpolation techniques are highly desirable.
Urgently needed would also be benchmarks that consider all kinds of aspects, i.e., approximations, starting points, numerical implementations, and computational parameters.Such data would allow one not only to assess what the true result for a given level of theory is, but also what is needed for a trustworthy solution of a given problem.
In summary, during the last years, the community has gained substantial understanding of electronic and optical properties of hybrid materials, though not all problems have been solved.Substantial progress has been made in applying hybrid functionals and the GW approach to determine the electronic properties of HIOS, e.g., pointing out DFT issues with the level alignment in interfaces.We note that though these high-level calculations are still not fully quantitative as due to the numerical complexity, the computational parameters cannot be taken to the same stringent settings as typical for simpler materials.In this perspective, we have given an outlook to two approaches that make BSE calculation for HIOS feasible, thus allowing for a proper description of optical spectra.Also the inclusion of temperature effects through many-body EPC theory is coming into reach for complex systems like inorganic/organic interfaces.For capturing dynamical processes, most relevant for applications in optoelectronic devices, new methodology is currently under development.However, despite all these efforts, there is still no method available that is ready to be used for the interface systems discussed here.We conclude that hybrid materials remain a challenge for ab initio theory and require more advanced methodology and substantial progress in high-performance computing to understand all interactions and processes taking place in such systems.

Figure 1 .
Figure 1.Level alignment in pyrene@MoS 2 as obtained by G 0 W 0 using PBE (left) and HSE (right) as the starting point.The gray bars indicate the density-functional theory (DFT) starting points, the colored bars the resulting quasiparticle levels.In both panels, they are projected onto pyrene at the left side and onto MoS 2 at the right side, and the valence-band maximum (VBM) of MoS 2 obtained from DFT is taken as the reference.The computational parameters are the same as in ref.[17].

Figure 3 .
Figure 3. Electronic band structure (red lines) of the MoS 2 monolayer and the impact of zero-point vibration effects in terms of the electronic spectral function A k ðω, TÞ at T = 0 K, as highlighted by the color code.

Figure 2 .
Figure 2. Top: schematic representation of how to compute the additive screening, for the example of pyrene@MoS 2 .The contributions of the pristine systems to the polarizability are obtained independently and then superposed to further compute the screened Coulomb interaction in the Bethe-Salpeter equation (BSE) Hamiltonian.Middle: comparison of the resulting in-plane component of the dielectric function (red line) to the full calculation (black line).Bottom: in-plane component of the dielectric function, calculated with the interpolative separable density fitting (ISDF)-BSE approach (blue) and compared with the standard BSE solution (black).In both panels, the results are obtained on top of the G 0 W 0 quasiparticle band structure.