Inherent electronic trap states in TiO 2 nanocrystals : e ff ect of saturation and sintering †

Computational Laboratory of Hybrid/Organ Scienze e Tecnologie Molecolari, via Elce d lippo@thch.unipg.it; Fax: +39 075 585560 Dipartimento di Chimica, Università degli 06123 Perugia, Italy. E-mail: nunzi@thch.u 075 5855517 Dipartimento di Farmacia, Università G. D'A Italy Department of Chemistry, Princeton Univer Laboratory of Photonics and Interfaces Engineering, Faculty of Basic Science, Eco CH-1015 Lausanne, Switzerland † Electronic supplementary information (E DOS curves. See DOI: 10.1039/c3ee24100a Cite this: DOI: 10.1039/c3ee24100a


Introduction
2][3][4][5] DSCs are based on a dye-sensitized mesoporous oxide layer, usually composed of a network of sintered anatase TiO 2 nanoparticles (typically $20 nm in size), interpenetrated by a liquid redox electrolyte (typically I À /I 3 À or Co(II)/Co(III)-polypyridine complexes in a volatile organic solvent), Scheme 1. Upon photoexcitation of the chemisorbed dye, electrons are injected into the oxide manifold of unoccupied states and can travel across the TiO 2 nanoparticle network until they are collected at the transparent conducting glass back contact.The oxidized dye is regenerated by the redox electrolyte.The circuit is closed by electrolyte regeneration at the counter-electrode, see processes 1-4 in Scheme 1.Recently, DSC efficiencies exceeding 12% have been obtained, 6 although still higher values are needed to effectively compete with conventional photovoltaics.A main DSC efficiency bottleneck is the recombination of injected electrons with oxidized species in the electrolyte and with oxidized dye molecules, processes 5 and 6 in Scheme 1. Usually recombination with the oxidized dye is prevented by the fast dye regeneration rate by the reduced species, e.g.I À or Co(II), in the electrolyte. 3Since electron collection at the electrode competes with recombination at the TiO 2 -electrolyte interface, slow electron transport can limit the charge-collection efficiency and eventually lead to an overall diminished DSC's conversion efficiency. 7,85][16][17] For nanostructured TiO 2 lms commonly employed in DSCs, D eff values in the range 10 À8 to 10 À4 cm 2 s À1 have been measured, depending on the light intensity.][29][30] Early studies [31][32][33][34] recognized the presence of various types of traps, with energy levels ranging from 0.2 to 0.9 eV below the nominal conduction band (CB) edge.Bisquert et al. [23][24][25]35 clas-sied the electronic states in TiO 2 as: (i) conduction band states (or transport states or extended states) responsible for effective electron transport; (ii) bulk traps, i.e. localized electronic states that trap and release electrons from and to the CB only, and (iii) surface traps, i.e. localized electronic states that trap/release electrons from/to both the CB and acceptor species in solution. Hafeldt and co-workers 36 using photoelectron spectroscopy measured a broad distribution of trap states centered ca. 1 eV below the CB edge of a nanostructured TiO 2 lm.Spectro-electrochemistry measurements by Fitzmaurice and coworkers 37 evaluated the effect of the electrolyte solution on the at band potentials of mesoporous nanocrystalline TiO 2 electrodes.38 These studies suggested that in mesoporous TiO 2 lms, the CB has a low energy tail of localized states below the energy characterizing the onset of fully delocalized conduction band states (also termed mobility edge).Kavan et al. 39 and many others since then [40][41][42][43][44][45][46][47][48][49] proposed the existence of deep, surface trap states in the band gap, below the most signicant portion of the exponential tail of the DOS.In the presence of a broad distribution of localized states, electronic transport is described within the multiple trapping model, [50][51][52][53][54][55][56][57] in which transport through the extended CB states above the mobility edge is slowed down by trapping-detrapping events from the underneath localized states.Furthermore, the energetics of trap states was shown to be affected by surface complexation with electrondonating ligands.[58][59][60] For instance, if the electron density of the ligand is high enough, the electronic levels of the intraband states can be raised into the CB, implying that their trapping action will vanish.The precise shape of the tail states is a delicate issue which is still under discussion.19,20,50,[61][62][63][64][65][66] Bisquert and coworkers have found that an exponential density of states below the CB adequately ts several experimental capacitance data.67 The proposed DOS shape is: where q is the electronic charge, N L is the total number of states below the conduction band, k B is the Boltzmann constant and T is the temperature.E Fn is the Fermi level of the electrons relative to the electrolyte redox potential, and a is an adimensional parameter that, together with the conduction band energy (E C ), describes the DOS distribution.Typical a values lay in the range 0.2-0.5, 67 although values as small as $0.05 have also been reported. 68lthough the effect of trapping states on electron transport in mesoporous nanocrystalline TiO 2 has been extensively investigated, 3,23,27,36,38,42,[69][70][71][72][73][74][75][76] it is still unclear whether these states originate from defects in the bulk and surface regions, from the grain boundaries of the particles, from Coulomb trapping due to interactions of electrons with the cations of the electrolyte, or from a combination of all these factors.Also, TiO 2 nanocrystals of different shapes and sizes can present different types of defects and of trap states.In contrast to the numerous studies of anatase and rutile low index surfaces, [77][78][79][80][81][82] computational investigations of the structural and electronic properties of TiO 2 nanocrystals (NCs) are still scarce.Density Functional Theory (DFT) was used to study stoichiometric anatase clusters (up to 68 TiO 2 units) with no particular surface orientation, 83,84 while larger clusters, constituted of up to 455 TiO 2 units and with a tetragonal bipyramidal shape, were investigated by Tight-Binding (TB) calculations. 85More recently, DFT calculations have been employed to characterize the geometrical and electronic properties of anatase TiO 2 nanoparticles having up to 449 TiO 2 units. 86In another DFT study, the electronic structure of pure and Li-doped rutile TiO 2 nanoparticles of up to 61 TiO 2 units, saturated by water molecules, was investigated. 87The interfaces between two TiO 2 NCs have also been studied, but only via classical molecular dynamics simulations. 88The effect of hydrogen bonding on the photoinduced electron transfer and carrier mobility has also been investigated. 89n this paper we use quantum mechanical DFT and DFT Tight-Binding (DFTB) calculations to provide an in depth analysis of the nature of trap states in realistic anatase TiO 2 NCs of ca. 3 nm diameter.We consider the effect of the adsorption of donor ligands, specically H 2 O molecules, on the energy and density of trap states of TiO 2 NCs.We further investigate the interaction between two sintered TiO 2 NCs across different interfaces.Our study provides insight into the nature of trap states in the TiO 2 mesoporous nanocrystalline lms employed in DSCs, pointing to the presence of inherent trap states in perfectly stoichiometric and crystalline TiO 2 NCs due to undercoordinated surface Ti(IV) ions at the (100) facets.

Computational details and calibration
To model realistic TiO 2 NCs containing up to ca. 1500 atoms, we adopt a multi-step computational strategy.We rst carry out geometry optimizations by employing self-consistentcharge density-functional tight-binding (SCC-DFTB) methods within the DFTB program. 90,91We then perform single point electronic structure calculations on the optimized structures by means of semi-local (i.e.GGA) and non-local (i.e.hybrid) DFT within the ADF and PWSCF program packages, [92][93][94] employing various combinations of basis sets and exchangecorrelation functionals, as detailed below.Previous works have shown that DFTB 95 can predict band structures, geometrical parameters and cohesive energies of anatase polymorphs in good agreement with reference DFT and available experimental data. 96,97or ADF calculations, the local density approximation of Vosko, Wilk and Nusair 98 (LDA-VWN) augmented with the gradient corrections of Becke 99 and Perdew 100 (exchange and correlation, respectively, BP86) was employed, together with single-z (SZ), double-z (DZ) and triple-z plus polarization functions (TZP) basis sets.Solvent effects have been evaluated through the COSMO solvation model as implemented in the ADF program.A comparison between the GGA-BP86 and the hybrid B3LYP functional was also performed with the ADF program.For PWSCF calculations, we employed the GGA-PBE exchange-correlation functional, 101 together with ultraso pseudopotentials, as implemented in the Quantum Espresso package. 94Plane-wave basis set cutoffs for the smooth part of the wave functions and the augmented density were 25 and 200 Ry, respectively.A supercell was employed ensuring a minimum separation of 5 Å between periodic images.All the DOS curves have been obtained by a Gaussian convolution of width s, as specied in the various cases (FWHM ¼ 2.35 s).
To check the accuracy of DFTB geometry optimizations, we compared DFT (BP86/DZ) and DFTB optimized structures for a relatively small (TiO 2 ) 161 -H 6 cluster model, see Fig. S1, ESI, † nding average differences in the Ti-O distances within $1%.We further checked the quality of DFTB results for the electronic structure of the same (TiO 2 ) 161 -H 6 cluster model by comparing the DOS obtained by DFTB and BP86/DZ on the DFTB optimized geometry and the DOS obtained by BP86/DZ on the BP86/DZ optimized geometry.The results, Fig. S2, ESI, † indicate very similar DOS curves for the three methods (the DFTB and BP86/DZ single point energy calculation on the same DFTB geometry give curves that are essentially indistinguishable).We also consider the effect of the basis set dimension and of polarization functions on the unoccupied states of TiO 2 NCs by comparing, for the same (TiO 2 ) 161 -H 6 cluster model at the DFTB-optimized geometry, the DOS obtained by DFTB, PBE/PW, BP86/DZ and BP86/TZP levels of theory.The results, Fig. S3, ESI, † show very similar DOS curves for the four methods.In particular, the PB86/DZ and BP86/TZP DOS curves are very similar to the PBE/PW DOS curve.What is most relevant to our study is that the shape of the DOS tail is essentially the same in the four cases, despite the fact that the four methods deliver different band gaps for (TiO 2 ) 161 -H 6 (2.76, 1.64, 1.61 and 1.95 eV for DFTB, BP86/DZ, BP86/TZP and PBE/ PW, respectively).
The effect of a polarizable continuum model of solvation on the unoccupied DOS was checked by employing again the (TiO 2 ) 161 -H 6 cluster model at the BP86/DZ level of theory.The results (Fig. S4, ESI †) show a band gap increase of 0.13 eV and an energy up-shi of the HOMO and LUMO (by 0.18 and 0.31 eV, respectively) upon solvation; still, the scaled DOS curves are essentially identical.As an additional check of our approach, we compared the GGA-BP86 and hybrid-B3LYP results for the (TiO 2 ) 161 -H 6 cluster model.Due to the high computational cost of B3LYP calculations, a SZ basis set was used in this case.The resulting DOS curves are shown in Fig. S5, ESI.† We can see that B3LYP predicts a slightly enhanced DOS in the low energy region.Finally, additional tests show that the agreement between DFTB and PBE/PW results is also maintained for the largest models employed, see Fig. S6 and S7, ESI.† Based on the above calibration studies, we have adopted the PBE/PW method to describe the electronic structure of individual NCs, while the less computationally intensive DFTB method has been condently used to study the electronic structure of water-saturated models and of sintered NCs.

Structural properties
We generated our starting NC structures by cleaving bulk anatase TiO 2 according to the typical bipyramidal Wulff shape.The resulting NCs expose (101) surfaces on the lateral facets, (001) surfaces on the truncation facets and (100) surfaces at the junction of the two pyramids, see Fig. 1.We considered NC models of various sizes, up to 3 nm.Aer verifying the impossibility to generate perfectly crystalline and stoichiometric, (TiO 2 ) n , truncated bipyramidal NCs, we chose to focus on two types of models, obtained by: (i) saturating all the under-coordinated dangling oxygen atoms on the (001) surfaces by hydrogen atoms; (ii) removing selected atoms from the (101)  surfaces to keep the cluster neutral and stoichiometric.In particular, two specic models are discussed in detail in the following: (i) a (TiO 2 ) 411 -H 16 NC (structure 1 in Fig. 1 The facets of our NC models are characterized by undercoordinated Ti and O atoms with respect to the bulk, where Ti and O atoms have pseudo-octahedral (Ti 6c ) and pseudo-trigonal (O 3c ) coordination, respectively.In particular, for all the considered NC models, ve-fold Ti 4+ (Ti 5c ) and two-fold coordinated O 2À (O 2c ) sites are present on the ( 101) and (001) facets, as found on extended surfaces.In addition, four tetracoordinated Ti 4+ sites (Ti 4c ), Fig. 1, occur on the vertices at the intersection of the four (101) cleavage planes.The under-coordinated Ti 4c sites represent an intrinsic characteristic of TiO 2 NCs; e.g., Ti 4c sites have been found also in simulated spherical nanoparticles. 105The presence of under-coordinated Ti 4c sites in the TiO 2 mesoporous lms and their correlation with trap states for electronic transport has been the subject of various papers by Teng et al. [74][75][76] Upon geometry optimization of 1 and 2, the largest structural distortions with respect to the bulk crystalline structures occur at the Ti 4c under-coordinated sites, which rearrange from the under-coordinated octahedral conguration characteristic of the bulk-truncated structure to a distorted tetragonal conguration.This feature is found in all investigated NC models at all levels of theory employed, thus suggesting that the presence of pseudo-tetrahedral Ti 4c sites is a typical feature of these NC structures.

Electronic structure of individual TiO 2 NCs
The electronic structures of NCs 1 and 2 computed at the PBE-PW level are compared in terms of DOS in Fig. 2, focusing on the manifold of unoccupied states.
Similar band gaps of 1.68 (2.71) and 1.79 (2.83) eV were computed for 1 and 2 by PBE-PW (DFTB), respectively.The PBE-PW calculated band gaps are considerably underestimated compared to the 3.2 eV experimental band gap of anatase.This discrepancy could be reduced by the use of computationally convenient DFT+U methods, 81,106 that however are strongly dependent on the choice of the U value, or by employing a hybrid functional with a variable fraction of Hartree-Fock exchange, [107][108][109] which is however too computationally expensive to be used here.Since the focus of the present work is mainly on comparing the density of unoccupied states of different NC models, we expect the present level of theory to be sufficiently accurate for comparative purposes.The slightly smaller, 0.11 eV (0.12 eV), band gap of 1 compared to 2 is due to an energy up-shi of the valence band (VB) maximum, which is mainly caused by the electron-donor properties of the OH À groups.In the following we focus on NC 1, resorting again to 2 for the discussion of the effect of sintering.
To compare the calculated DOS for 1 with available experimental capacitance measurements on dye-sensitized TiO 2 , we tted our data to eqn (1) in its logarithmic form.As shown in Fig. 3, very good linear ts (R 2 ¼ 0.99) were obtained in an energy range ca.0.1-0.6 eV above the LUMO.Below 0.1 eV, a few localized states are found, while above this energy range the DOS of our models changes its shape and slope due to the nite system's dimension.Also notice that although a Gaussian broadening has been used to calculate the DOS, this should not directly inuence the t, which does not include the tail below the LUMO originated by the broadening.The resulting a and E C values (cfr.eqn (1)) of NC 1 are 0.13 and 0.35 eV (at T ¼ 300 K).Although this value of a is at the lower edge of measured data, the result is remarkable, considering the relative simplicity of our model.The unoccupied states of lowest energy for NC 1, of titanium t 2g character, are localized within the central part of the NC, mainly at the intersection of the ( 100) and ( 101) surfaces, see the le panel of Fig. 4. We note that a different LUMO was found in ref. 86 for sharp-and at-shaped NCs, using TiO 2 NC models with a different saturation scheme of dangling Ti and O atoms.At higher energy, the unoccupied states are progressively more delocalized, with the lowest energy state completely delocalized over the NC structure (right panel of Fig. 4) being found ca.0.3-0.4eV above the LUMO, in agreement with both electrochemical and spectro-electrochemical results, [23][24][25]35 and with the data t analysis presented above.
To check the sensitivity of the low-energy states at the bottom of the DOS to external ligands, we optimized the geometry of model 1 aer adding 154 surface-adsorbed H 2 O molecules.We found that the surface saturation in 1/H 2 O gives rise to an energy up-shi of both the VB and CB edge of ca.0.5 eV at the DFTB level, see Fig. S8, ESI.† Such a shi is perfectly coherent with the decrease of the work function experimentally observed upon hydration of TiO 2 surfaces, 110 and was also found for rutile nanoparticles in ref. 87.Surface-adsorbed H 2 O molecules raise the energy and reduce the number of localized states at the bottom of the unoccupied DOS, Fig. 5.In fact, the DOS curve for 1/H 2 O is not simply shied to higher energy compared to that calculated for 1, but has an evidently different curvature.We also note that some localized trap states are still found in 1/H 2 O, since the Ti 4c sites on the (100) surfaces are not entirely saturated by water molecules and maintain a weak 5-fold coordination, with average computed Ti-OH 2 distances of 2.90 Å.
To provide a quantitative picture of the unoccupied state energy localization within the considered TiO 2 NCs, we report in Fig. 6 a contour plot of the space/energy diagram for system 1.This diagram is obtained by scanning the NC along the main longitudinal axis and summing up the contributions of each atom to a given unoccupied state as a function of the state energy.In line with the qualitative analysis presented above, this diagram shows a substantial contribution to the low-energy portion of the DOS from states which are mainly localized on the central NC part.By increasing the energy, the states become progressively more delocalized over the entire NC structure, with the top/bottom (001) facets maximally contributing to the high energy portion of the DOS.Similar plots are obtained for NC 2, see Fig. S9, ESI.† The results can be visualized in the pictorial representation of the space/energy distribution in NC 1 shown on the right of Fig. 6.The localized surface states constituting the bottom of the DOS for the observed TiO 2 NC clearly constitute trapping sites for electron transport, and may further represent recombination sites between injected electrons and oxidized species in the electrolyte.

Sintered nanocrystals
To investigate whether the boundaries between sintered NCs can introduce electronic trap states at the bottom of the unoccupied states manifold, we constructed models of sintered NCs by attaching two TiO 2 NCs at their available surfaces.The optimized structures of two NCs of type 2 with 101/101, 101/001, 001/001 and 100/001 interfaces are shown in Fig. 7.The same   picture holds for NCs of type 1, for which we modeled only the 101/101 interface, see Fig. S10, ESI.† We found that sintering of two NCs at their 101/101 surfaces is the most effective, leading to a DFTB-calculated stabilization energy of 28.6 eV relative to two non-interacting NCs.This value reduces to 9.1, 7.1 and 6.6 eV for 101/001, 001/001 and 100/001 interfaces, respectively.
The larger stabilization energy found for the 101/101 interface is due to the larger accessible surface available and to the almost optimal structural matching that is observed at the interface between the two interacting NCs, see Fig. 7.When normalizing the calculated interaction energies by the surfaces area, we nd values of 0.14, 0.05, 0.08 and 0.09 eV ÅÀ2 for 101/ 101, 101/001, 001/001 and 001/001 interactions, respectively, indicating that the 101/101 interaction is the most favorable one in relation to the available surface area.The strength of the interaction is due to the number of Ti-O bonds that are formed upon sintering, within the limits imposed by the morphology of the NCs.We also note that the formation of Ti-O bonds between two NCs causes the saturation of only a few under-coordinated Ti sites, so that the effect of sintering on the distribution of the surface trap states of individual NCs is expected to be quite limited.
The computed DOS for the sintered congurations conrms that the shape of the DOS tail is not largely affected by the NC boundaries, while subtle differences in the distribution of the trap states are found for those congurations where some under-coordinated Ti sites are saturated, such as the 101/101 and 100/001 models in Fig. 7.These results indicate that, upon full relaxation, the structure of two interacting NCs tends to become similar to a bulk-like structure.Obviously, this might not always be the case under the high temperature/short time conditions used experimentally for NC lm sintering (500 C for ½ hour), also considering that at that temperature the ligands surrounding the NCs are destroyed, thus adding an additional level of disorder to the NC interactions.

Conclusions
We have used quantum mechanical calculations based on DFT and DFTB methods to investigate the nature of electronic trap states in realistic models of sintered and individual anatase TiO 2 NCs of ca. 3 nm diameter.We found the unoccupied states of lowest energy, of titanium t 2g character, to be specically localized within the central part of the investigated NCs.These states originate from under-coordinated 4-fold coordinated surface Ti atoms mainly lying at the (100) edges found at the intersections between (101) surfaces.At higher energy, the unoccupied states get progressively more delocalized, with the lowest energy state completely delocalized over the NC structure signaling the system's conduction band (CB).The localized states give rise to an exponential DOS tail which is found to be 0.3-0.4eV below the fully delocalized CB states, in excellent agreement with electrochemical, spectro-electrochemical and capacitance data.
The effect of the adsorption of donor ligands (specically H 2 O molecules) on the energy and density of trap states of TiO 2 NCs was also considered.We found that surface saturation  substantially alters the NC electronic structure, with a computed energy up-shi of both the VB and CB edge of ca.0.5 eV, in agreement with the decrease of the work function experimentally observed upon hydration of TiO 2 surfaces.The adsorbed water molecules also reduce the number of localized states at the bottom of the manifold of unoccupied states, thus modifying the DOS shape.
Finally, the interaction between two sintered TiO 2 NCs was investigated by considering attachment at all the possible surface combinations.Interestingly, no major effects on the joint DOS of two interacting NCs were found compared to that of the individual, constituent NCs.Although our calculations did not consider the role of defects, e.g.oxygen vacancies, and for sintered NCs did not include the disorder which is expected to characterize the TiO 2 lm formation under typical DSC fabrication conditions, our results clearly point at the presence of inherent trap states even in perfectly stoichiometric and crystalline TiO 2 NCs due to the unavoidable presence of 4-fold coordinated surface Ti(IV) ions.Our results constitute the basis for building specically tailored TiO 2 nanostructures which may lead to enhanced DSC efficiency, by virtue of enhanced transport properties, and provide insight into the effect of surface-passivating layers in reducing recombination reactions in DSCs.

Scheme 1 3 À
Scheme 1 Schematic representation of the constituent materials and energy levels of a DSC along with forward (green lines) and backward (dotted red lines) electronic processes.The energy levels roughly correspond to those of a DSC based on the N3 dye (red spots), TiO 2 nanoparticles (grey spheres) and I À /I 3 À ) with Paper Energy & Environmental Science Downloaded by University of Perugia on 22 February 2013 Published on 14 January 2013 on http://pubs.rsc.org| doi:10.1039/C3EE24100AView Article Online perfectly crystalline (101) surfaces and OH-saturated (001) surfaces; (ii) a (TiO 2 ) 367 cluster (structure 2 in Fig. 1), with some missing atoms ('holes') on the (101) surfaces.The saturation pattern in 1 is justied based on the experimental evidence that anatase surfaces are controlled by acid-base equilibria involving Ti-OH surface hydroxyl groups, 102-104 while structure 2 is convenient to model the sintering of two NCs and their grain boundaries, since this structure has equally accessible (101) and (001) surfaces.Furthermore, structure 1 is representative of sharp, faceted NCs, while structure 2 is closer to spherical nanoparticles.

Fig. 1 (
Fig. 1 (a and b) Schematics of the truncated bipyramidal NCs, showing (101), (100) and (001) surfaces from different views.(c) Optimized geometry of the TiO 2 NC model 1.(d) Optimized geometry of the TiO 2 NC model 2. (e) Under-coordinated Ti 4c at the vertex of the square basal plane joining two bipyramids together with the neighboring atoms, corresponding to the highlighted region in (c) and (d).Ti atoms are in grey, O in red and H in white.

Fig. 2
Fig.2DOS profile (300 lowest unoccupied states) for models 1 (red line) and 2 (green line) calculated at the DFTB optimized geometry with the PBE/PW level of theory (s ¼ 0.18 eV).The two DOS have been aligned at their maximum.The zero of the energy is set at the LUMO of 1.The inset shows a magnification of the bottom region (s ¼ 0.08 eV).The arrows in the inset show the region of maximally localized states.

Fig. 3
Fig. 3 Linear fit of log data obtained from the DOS of NC 1 calculated at the PBE/PW level of theory (s ¼ 0.08 eV).The zero of the energy is set at the LUMO.

Fig. 4
Fig. 4 Representative unoccupied states characterizing the LUMO (left) and the higher energy CB states at ca. 0.3-0.4eV above the LUMO (right).

Fig. 5
Fig. 5 DOS profile (300 lowest unoccupied states) for models 1 (blue line) and 1/H 2 O (red line) calculated at the DFTB level of theory (s ¼ 0.18 eV).The zero of the energy is set at the LUMO of 1.The inset shows a magnification of the bottom region (s ¼ 0.08 eV).The arrows in the inset show the region of maximally localized states.

Fig. 6
Fig. 6 Contour plot of the space-energy (eV) diagram for the DOS of unoccupied states of NC 1, scanned along the length of the NC.Three DOS areas are identified, corresponding to the central, intermediate and top/bottom NC regions.The right panel shows a pictorial representation of such a space/energy diagram, with colors shifting from red to green to blue indicating localization in the related colored region of states of increasing energy.

Fig. 7
Fig. 7 Optimized geometries for two interacting NCs (model 2) with the 101/ 101, 101/001, 001/001 and 100/001 interfaces along with their corresponding DOS (curves of different colors) compared to those of the isolated model 2 (red curves) calculated at the DFTB level of theory (s ¼ 0.18 eV).