Separate-path electron and hole transport across pi-stacked ferroelectrics for photovoltaic applications
aa r X i v : . [ c ond - m a t . m t r l - s c i ] M a r Separate-Path Electron and Hole Transport Across π -Stacked Ferroelectricsfor Photovoltaic Applications. Ma lgorzata Wawrzyniak-Adamczewska
Faculty of Physics, A. Mickiewicz University, ul. Umultowska 85, 61-614 Pozna´n, Poland
Ma lgorzata Wierzbowska
Institut of Physics, Polish Academy of Sciences (PAS),Al. Lotnik´ow 32/46, 02-668 Warszawa, Poland (Dated: September 6, 2018)Abstract: Electron and hole separate-path transport is theoretically found in the π -stacked or-ganic layers and columns. This effect might be a solution for the charge recombination problem.The building molecules, named 1,3,5-tricyano-2,4,6-tricarboxy-benzene, contain the mesogenic flataromatic part and the terminal dipole groups which make the system ferroelectric. The diffusionpath of the electrons cuts through the aromatic rings, while holes hop between the dipole groups.The transmission function and the charge mobilities, especially for the holes, are very sensitive tothe distance between the molecular rings, due to the overlap of the π -type orbitals. We verified thatthe separation of the diffusion paths is not destroyed by the application of the graphene leads. Thesefeatures make the system suitable for the efficient solar cells, with the carrier mobilities higher thanthese in the organometal halide perovskites. PACS numbers:
I. INTRODUCTION
The effect of charge carriers recombination is a very un-wanted phenomenon in the photovoltaic materials. Thereare various attempts to avoid this process in novel solarcell devices. One of the most popular solution, valid forelectrons as well as holes, is an addition of the chargetrapping layers. These layers selectively permit carri-ers transport across the multi–junctions. For example,the polymer solar cells equipped with the PEDOT:PSSlayers allow to extract hole carriers before they arriveat the anode. Recently, similar hole trapping functionhas been found for the graphene oxide nanoribbons. Ithas been shown, that the parameters of the metal–oxide–free methylammonium lead iodide perovskite–based solarcells are also improved by the charge transport layers. On the other hand, the simultaneous trapping of theelectrons and holes limits the photovoltaic efficiency.This is an inevitable effect in many systems. Theorganometal halide perovskites, where the liquid ions and surface states grab both types of carriers are the ex-amples. Another performance key factor is the materialstructure, particularly important in the solution pro-cessed organometal trihalide perovskite solar cells. Forthis kind of systems, planar heterojunctions are less effec-tive than the mesosuperstructured perovskites, regard-ing the occurrence of the sub gap states, but superior interms of carrier mobility. Luckily, the efficiency growsagain at very high fluences in the presence of the sub gapstates, and the trap-states density can be optimized interms of the grain size of nanocrystals. In addition, un-der temperature gradient, several structural phase tran-sitions may occur in these materials, and this fact alsoinfluences the diffusion lengths. Modern approach to increase the solar cell efficiency goes towards the so–called bulk heterojunctions, wherethe granular structure of two materials, the donor-type(e.g. polymers) and acceptor-type (e.g. fullerenes),might be regular or graded. It has been foundthat a larger grade of the composition corresponds towider charge transportation path and larger effectiveconductivity. Alternatively to the bulk heterojunctions,the ordered donor-acceptor materials – integrated hetero-junctions – have been synthesized. They are composedof two molecules integrated in the covalent organic frame-works. Recently, these kind of materials gain an attentionfor their photo-active properties.
An addition of the inter-layer aligns the energy lev-els of the multijunction device, in such a way thatthey form a cascade, efficiently reduces the electron-hole recombination. The above effect is intensified ifone uses the high/low work-function material for the an-ode/cathode. This tuning might be done by an adsorp-tion of the dipole moieties. In this work, we present a new solution to the charge-carriers recombination problem – the electron and holeseparate-path transport. The considered system has beendescribed in our previous work for its ferroelectric prop-erties. The studied layers are characterized by the cas-cade energy levels alignment, gradual donor–acceptorlayers sequence, and the dipole-proximity tunable workfunction of the graphene electrodes. In Figs. 1(a)-(c),we present the system which consists of the flat benzene-based molecules, terminated with the cyano groups(NCCH ), that are responsible for the ferroelectric prop-erties, and the carboxy groups (OCOH), that form thehydrogen bonds within the planes. The molecules arenamed 1,3,5-tricyano-2,4,6-tricarboxy-benzene and theirchemical formula reads C -3(NCCH )-3(OCOH). Theband gap of the 2D layer is around 3.6 eV, while for theFIG. 1: Atomic structure of investigated systems: (a) top view at molecular layers, (b) side view at columnarstacking, (c) single molecule with the indexed atoms.columnar stacks is 2.95 eV. We focused on localizationof the diffusion paths, but also analyzed the electron andhole mobilities within the planes and across the stacks.The considered building molecules represent the mostsimplified model molecules possessing the desired prop-erties. Systems similar to the considered ones, couldbe realized in industry by modifications of the existingcolumnar liquid crystals, or via the self-assembly of the2D molecular systems, for example on graphene. Thishas been reported in the experimental and theoreticalworks. Alternatively, one could base on the productionprocesses of the covalent organic frameworks usingthe external electric field in order to orient all dipolegroups ferroelectrically, and avoid formation of the an-tiferroelectrically coupled domains. The ferroelectric hydrogen–bonded organicframeworks could be obtained by modificationsof the covalent integrated heterojunctions or bulkorganic ferrolectrics. We expect that these frame-works would be interesting for the solar cell community,opening a new door for the efficient photovoltaic devices.
II. METHOD
We calculated various material properties derived fromthe electronic states obtained within the density func-tional theory (DFT). All calculations, except for thetransmission function, were performed employing the
Quantum ESPRESSO suite of codes. The densityof states (DOS), band structures, and carrier mobili-ties for the one–, two– and three–dimensional structureswere obtained on top of the Bloch states, representedin the plane-wave basis set and the pseudopotentials forthe core electrons. For most of the calculations, if notstated explicitly, the exchange–correlation functional waschosen for the gradient corrected Perdew-Burk-Erzenhofparametrization. -4 -2 0 2 4 Energy - E F [eV] D e n s it y o f S t a t e s [ a r b . un it s ] NOC1C2C3
FIG. 2: DOS projected at groups of atoms indexed inFig. 1(c), for the top layer of the six monolayers systemin the AA-type stacking.In order to accurately interpolate the band structures,we used the wannier90 package, which enables findingthe maximally-localized Wannier functions for the com-posite bands. Next, we projected the band structureson the optimized Wannier functions, which were centeredaround chosen atoms.The mobilities µ are defined as the ratio of the conduc-tivity σ and the carrier density ρ , that is obtained fromthe quadrature of the DOS, and the system volume V, µ = σ/ ( eρ ), whereas e denotes the electron charge. Theconductivity is defined as follows: σ ij ( E ) = 1 V X n, k v i ( n, k ) v j ( n, k ) τ n k δ ( E − E n, k ) ,v i ( n, k ) = 1 ~ ∂E n, k ∂k i ,v i ( n, k ) is the band velocity, E n, k is the band dispersion,and τ n k denotes the relaxation time dependent on theFIG. 3: A column of four molecules between the graphene leads: (a) top view at the AA-type stacked system, (b)R30 ◦ -stacked system, (c) AB-stacked case, (d) side view, (e) DOS for the top molecule of the AA-stacked system,projected at the groups of atoms indexed in Fig. 1(c).band index n and the reciprocal space k . We used theconstant τ approximation within this work, with its valueset to 10 fs. The first derivatives of the energy bands inthe Brillouin zone were obtained using the BoltzWannpost-processing code from the wannier90 package.The total relaxation time is a sum of many contribu-tions: elastic (due to acoustic phonons), nonelastic (dueto the optical phonons), and the ionic impurities effects.1 τ = 1 τ ac + 1 τ opt + 1 τ imp , The elastic scattering theory and the Fermi golden rule reads 1 τ ac k = X k ′ π ~ δ ( ǫ k − ǫ k ′ ) | M k , k ′ | (1 − cos ( θ )) | M ac k , k ′ | = k B T E C ii . The elastic constant, C ii , can be obtained from the to-tal energies, E and E , of the strained and equilibriumsystem with respect to the dilation (∆ l/l ), following theformula ( E − E ) /V = C ii (∆ l/l ) /
2. The value E canbe obtained from the valence and conduction band en-ergy change ∆ ǫ V BT ( CBM ) - for the holes and electrons,respectively - with respect to the dilation (∆ l/l ). Thetadenotes the angle between the vectors k and k’ , and for the 1D structure can be equal only 0 or 180 deg. Con-tribution to the relaxation time is nonzero only for thebackward scattering.The nonelastic scattering due to the electron-phononcoupling with the optical phonons is much lower thanthat from the acoustic phonons. This fact is highlightedin the work about the naphthalene crystal. Impor-tance of the acoustic phonons has been reported also forthe organic perovskites. For the ordered quasi-1D sys-tems, in absence of the elastic and ionic contributions,purely electron-phonon scattering might lead to the mo-bilities, which are three orders of magniture larger thanthese measured for the 3D-disordered crystals of the samemolecules. In contrast to the nonelastic effects, the scatteringin the presence of the charge impurities often predom-inates in the organic crystals.
In this case, thescreened Coulomb potential leads to the scattering ma-trix elements | M imp k , k ′ | = n Z ion e V ( ε r ε ) ( L D + | k ′ − k | ) − L D = p ε r ε k B T N / e . where n is the concentration of the ionic impurities, Z ion is the charge of the impurity, L D is the Debye screen-ing length with the free charge concentration N , therelative permitivity of the material ε r and the dielectricconstant of the vacuum ε . The dielectric constant of the1D molecular systems was obtained using the QuantumEspresso postprocessing tool epsilon.x. The transmission functions were calculated usingthe DFT method, as implemented in the siesta -3.0package.
The software is based on the pseudopoten-tial approach and uses a finite range localized basis sets.The transport computations use the non–equilibriumGreen’s functions, represented in terms of the solutions ofthe Kohn–Sham Hamiltonian. Both the local-densityapproximation (LDA) for the electron exchange and cor-relation in the Ceperley-Alder parametrization, as wellas the PBE parametrization, were used to compare thetransmission functions for the system geometries opti-mized in each method.The system geometry, used for the transport calcula-tions, has been arranged as follows: the molecule wassandwiched between the two graphene planes (upper andlower). The upper and lower graphene sheets were ori-ented in the AA-type stacking with each other and withthe molecular aromatic-ring. In order to provide thecharge flow, for biased system, from the upper graphenesheet across the molecule to the lower graphene sheet(or vice versa), both graphene sheets were terminated.The edge atoms of graphene were additionally passivatedwith the hydrogen atoms. The extensions of the up-per and lower graphene planes formed the left and rightelectrodes. Thus, the scattering region consists of themolecule, as well as the upper and lower discontinu-ous graphene sheets. The supercell contains 850 atoms.The interplanar graphene distance in the geometry opti-mized with the LDA scheme is 8.6 ˚ A , and in the GGA-structure, it is 9.8 ˚ A . The maximal examined value of the k –sampling of the Brillouin zone, along the passivatedgraphene–edge direction, was k = 10. Further details areincluded in the supporting information. III. RESULTS AND DISCUSSION
Observation of the charge carrier paths, for the elec-trons and holes separately, sets a challenge for the exper-imental techniques. Hence, only a hypothesis has beenderived from the experimental work on the reduced bi-molecular charge recombination in the metal-halide per-ovskite system. In contrast, the theoretical methods allow to picturethe carrier density in the real space by projection onthe chosen atomic orbitals. Several molecular layers inthe AA-type stacking pattern (one exactly on top of theother) absorb light better than the single one, whichwould be transparent and photovoltaicly inefficient. Ad-ditionally, the electronic transport across the films ofthe certain thickness might be affected by the stack–termination effects. For the polar layers, terminated withthe nitrogen atoms on the top and mostly hydrogens atthe bottom, one could expect different density of statesfor the outermost and inner layers. Surprisingly, the anal- ysis of the projected DOS (PDOS) for the six layers sys-tem leads to the conclusion, that the hole and electronproperties do not differ depending on the layer positionin the stack.Fig. 2 shows the PDOS for the top layer only. The cor-responding pictures for all the following layers do not dif-fer in shape one from another. Although, as described inour previous work they are shifted in the energy, accord-ing to the Stark effect. The PDOS for all six layers ispresented in Fig. S1 in the supporting information. ThePDOS analysis indicates, that the electrons and holes lo-calize at different parts of the building molecules. Theelectrons tend to localize at the central ring, while holesat the dipole terminal groups. This space separation isstrongly pronounced for the electrons and light holes, andto some extend for the heavier holes. Similar space sepa-ration has been reported for the covalent organic frame-works where two different molecules play the role of ac-ceptor and donor. We model the effect of the electrodes on the terminallayers of the optically active material with the graphenesheets. This material is a good candidate for both thecathode and anode, due to its high polarizability andinduced change of the work function. The hydrogenbonds within the 2D networks are weak, and in fact, thereare also possibilities to pattern at graphene without anyintermolecular bonding.
Hence, we consider the sepa-rated molecular stacks with three different arrangementsbetween the molecules and graphene. The top and sideviews are presented in Figs. 3(a)-(d). By breaking theplanar connections, we gain the additional path for theholes, which now can move between the neighboring car-boxy groups. As seen in Fig. 3(e), the application of thegraphene electrodes do not affect the electron–hole pathseparation. The effect of molecular orientation with re-spect to the graphene axes, as for the cases presented inFigs. 3(a)-(c), turned out to be negligible. The PDOSfor all other considered orientations is presented in Fig.S2 in the supporting information.Breaking the planar bonds, as well as addition ofthe graphene leads, modifies the valence band charac-ter. Comparison of Fig. 2 and Fig. 3(e), shows that thetop of the valence band in the 2D case (Fig. 2) is builtof the states localized at nitrogens and oxygens - whichmeans that the NCCH2- and COOH-group contribute tothe states at the same energy below the Fermi level. Incontrast, in the 1D case (Fig. 3(e)), the PDOS is builtof the oxygen states close to the Fermi level and ener-getically deeper (around 2-3 eV below the valence bandtop) nitrogen states. This is a fingerprint of the energeticseparation of the light and heavier hole. The same prop-erty is exhibited by the infinite 1D molecular stack, forwhich the band structures projected at the near atomic–centred Wannier functions are shown in Figs. 4(a) and(b) for two lattice constants. The projection onto oxy-gens (the red color in the middle panel) shows that thecontribution of the COOH group is energetically locatedjust below the Fermi level. While the projection to ni-
G A (a)
N, C3 E - E F [ e V ] G A
O, C2
G A -1.5-1.2-0.9-0.6-0.30 C1 (b) N, C3 E - E F [ e V ] O, C2 C1 FIG. 4: Band structure, projected at the groups of atoms indexed in Fig. 1(c), of the columnar structure for twointermolecular distances: (a) 5.2 ˚ A optimized for the GGA method, and (b) 4.6 ˚ A optimized for the LDA method.The color scale represents a sum of the coefficients of the chosen Wannier functions in the wavefunction expansion. -1-0.8-0.6-0.4-0.20 G A E - E F [ e V ] A G M K H A L M G M K G FIG. 5: Band structure of: (a) single molecular layer,(b) bulk structure of molecular layers, (c) molecularwire.trogens (the orange color in the leftmost panel) displaysthe energetic location of the contribution of the NCCH2group, which is well below that of the COOH moieties.The bandwidth is a signature of the carrier mobility,and it is sensitive to the strength of the p z -orbitals over-lap. This can be seen from a comparison of the two above described band structures. The bandwidth of the lighthole is about 0.5 eV, for the larger separation between the π -rings (Fig. 4(a)), and about 1 eV, for the more closelystacked case (Fig. 4(b)). The heavier hole extends toabout 1.4 eV or 1.85 eV for the GGA- or LDA-optimizedring separations, correspondingly. These bandwidths aredirectly connected to the hole mobilities, which are thederivatives of the energy levels with respect to the linesin the k-space (the change of the point in the Brillouinzone). In other words: fast carriers move along thickbands. The effect of the inter-ring separation for theelectron mobilities is less pronounced that that for theholes.The energy barriers for the rotation of the dipolegroups are expected to be small especially at the ele-vated temperatures, when the devise is exposed to Sunshine. But in the technological reality, the columnar liq-uid crystals are densely packed, as well as the columnsstudied in this work are also placed closely - on the hydro-gen bond distance in the 2D case. The 1D model is goodenough for the analysis of the mobilities, but the workingdevice will be more similar to the 2D case, where flippingthe orientation of the dipole group is not as easy as inthe 1D case.The transport properties of the columnar and planarsystems differ too. Therefore, in Fig. 5, we compare theelectronic structures of the 1D and 2D systems, includingalso the model 3D bulk material, which connects the pa-rameters of those two. For the efficient solar cell devices,it is desirable to achieve high conductivity across the lay- -0.8 -0.6 -0.4 -0.2 0 Energy - E F [eV] D µ [ - c m V - s - ] hole mobilityhole density (a) D ρ [ / c m ] Energy - E F [eV] D µ [ - c m V - s - ] electron mobilityelectron density (b) D ρ [ / c m ] -0.8 -0.4 0 Energy - E F [eV] D µ [ c m V - s - ] D = 5.2 Å (10x)D = 4.6 ÅD = 5.2 Å D = 4.6 Å (2x) (c) D ρ [ / c m ] Energy - E F [eV] D µ [ c m V - s - ] D = 5.2 Å (4x)D = 4.6 ÅD = 5.2 ÅD = 4.6 Å (d) D ρ [ / c m ] FIG. 6: 2D carrier mobilities at 300 K and τ =10 fs and carrier densities for (a) holes and (b) electrons, as well as 1Dcarrier mobilities and densities for (c) holes and (d) electrons. For better visibility, we scaled some curves as denotedin legends. -0.8 -0.4 0 Energy - E F [eV] D µ [ c m V - s - ] total, τ = 0.0216 pselastic (0.1x), τ ac = 0.36 ps impurity, τ imp = 0.023 ps (a) Energy - E F [eV] D µ [ c m V - s - ] total, τ = 0.0226 pselastic (0.05x), τ ac = 0.8 psimpurity, τ = 0.023 ps (b) FIG. 7: 1D carrier mobilities, at the intermolecular distance 4.6 ˚ A , for (a) holes and (b) electrons, calculated at 300K with the relaxation times obtained with the elastic scattering theory ( τ ac ), the scattering on the ionic impurities( τ imp ), and the both effects ( τ ). For better visibility, we scaled some curves as denoted in legends.ers, whereas insulating properties within the planes – forthe purpose of the energy dissipation reduction. Indeed,the band dispersions of the 2D system are very flat withrespect to the 1D case. The band gap of the 2D molecu-lar layer is very similar to the HOMO-LUMO gap of anisolated molecule, and decreases with the thickness of thefilm, to be the smallest for the infinite molecular wire.The number of carriers (holes and electrons) per theenergetic range decreases for the low dimensional sys-tems. The mobilities are defined as the ratio of the con- ductivity, proportional to the band dispersion, and thecarrier density. This is one of the reasons for the high car-rier mobilities obtained in graphene, for example. Figs.6(a)-(b) present the mobilities obtained for the 2D andFigs. 6(c)-(d) for the 1D molecular system, respectively.The results for the 1D system are compared for the twointermolecular distances, see Figs. 6(c)-(d). As expected,the carrier mobilities for the 2D case (with moleculesconnected via the hydrogen bonds) are much lower thanthese for the 1D system (with the molecules in the π - t r an s m i ss i on E-E F (eV)(b) t r an s m i ss i on E-E F (eV)(c) FIG. 8: Geometry of the simulated system with thegraphene leads (a), the transmission obtained with theGGA method (b), and with the LDA method (c). Red(green) lines correspond to the stronger (weaker)coupling between molecule and graphene sheets, i.e. 8.6and 9.2 ˚ A , repsectively.stacking); even at larger (the GGA) lattice constant.This effect is enhanced for the holes, due to the verticalgeometry of the dipole groups, which makes the inter-molecular distances in the terminal parts smaller than inthe central part. On the other hand, the inter–ring dis-tances are equal to the lattice constant of the chain, thusmuch larger than for instance the interplanar distances ingraphite ( ≈ A ). Decreasing the lattice constant, theelectron mobilities grow about 4 times, while the corre-sponding increase for the holes is about 15 times.Additionally, in the supporting information, we showthe conductivities scaled with the constant relaxation time and plotted for several temperatures, for the 2Dand 1D cases; see Figs. S4 and S5 in the supporting in-formation, respectively. For the 2D system, a decreaseof the conductivity with the temperature is much morestrongly pronounced with respect to the 1D case. More-over, we observe a visible anisotropy of the hole conduc-tivity along the zigzag and armchair directions in ourhexagonal molecular lattice. It is worth to point out,that it is not the case for the electron mobilities.In order to go beyond the mobilities parametrized withthe relaxation time, for the 1D infinite molecular column,we estimate the scattering effects according to the elasticand the ionic impurity contributions. We abandon thecalculation of the optical-phonons contribution, becausein the organic systems this kind of scattering is muchlower than that caused by the acoustic phonons. Taking into account only the elastic scattering, the cal-culated relaxation time for the molecular pillar with theintermolecular distance of 4.6 ˚ A is 0.36 ps for holes and0.8 ps for the electrons. These values of τ correspondto the moblities achieving 5000 cm V − s − for the holesand 1360 cm V − s − for the electrons.In order to calculate the scattering from the ionic im-purities, we used the formulae given in the methods sec-tion and assumed one ionic impurity with the charge 1 or-1 (which leads to the same result) per ten molecules inthe wire (n=0.1/(4.6˚ A ), Z ion =1). The density of free car-riers is assumed 1 per the unit cell for the holes and elec-trons ( N =1/(4.6˚ A )). The relative dielectric constant ε r of our wire was found 1.2 - accoring to the static com-ponent of the real part of the dielectric function. Thecalculated relaxation time for the ionic impurity scatter-ing only is 23 fs for the holes and the electrons. Thus, themaximal values of the mobilities achieve 300 cm V − s − and 39 cm V − s − for the holes and electrons, respec-tively.The ionic scattering takes over the acoustic contribu-tion. Hence, the total relaxation time is 21.6 fs for holesand 22.6 fs for the electrons. The mobilities rescalledaccording to the obtained relaxation time are printedfor holes and electrons in Figs. 7(a) and 7(b), respec-tively. Concluding, the elastic effects in our molecularwires are compared to these in the organometal halideperovskites. The addition of the ionic defects - whichare often in the molecular crystals – drastically in-creases the scattering and decreases the mobilities. Ouranalysis shows that, despite the presence of the ionic im-purities, the mobilities in our 1D system are high in com-parison to other organic systems. The previous worksshow that the mobilities in some organic materials excess100 cm V − s − at low temperatures. We underline,that the systems studied in this work are highly ordered,which is another reason for the low scattering and it isknown that the mobilities in organic crystals are veryanisotropic. We compare the mobilities for our systems with theseof the organometal halide perovskites.
The relax-tion time calculated for our 1D system leads to the mo-bilities - 300 and 39 cm V − s − for the holes and elec-trons, respectively - and these values are larger than forthe perovskites. For instance, for CH NH PbI3 in thecubic structure, reported in Fig. 4 in Ref. [54], the max-imal values for the holes achieve 12 cm V − s − and 8cm V − s − for the electrons - with the assumed relax-ation time of 1 ps for the both types of charges. Thesevalues agree with the measured mobilities. However,the other ab initio calculations with the inclusion of thespin-orbit coupling, reported in Table 1 in Ref. [38], leadto largely overestimated values of 1400-2200 cm V − s − for the holes and 570-800 cm V − s − for the electrons.The above mobilities were obtained with the same meth-ods which were applied in this study. The relaxationtimes obtained by Zhao et al. are an order of magni-tude larger than our values reported in Figs. 7(a) and7(b). This overestimation results probably from large in-crease of the bandwidths when the spin-orbit coupling isincluded and the quasiparticle corrections not. A single molecule sandwiched between the terminated,H-passivated graphene planes is shown in Fig. 8(a). Thisis the minimal model which represents the π -stackingorder between the optical spacer (molecules) and theelectrodes. The zero-bias transmission function calcu-lations are performed within the GGA and LDA schemes– see Figs. 8(b) and (c), respectively. The two couplingstrengths, related to the distances between the moleculeand the planes, are considered. The stronger (weaker)coupling corresponds to the case, where the distances areset to 4.6 and 4.0 ˚ A (5.2 ˚ A and 4.6 ˚ A ) for the upper andlower graphene-molecule separations, respectively. Thevalue of the width of the band gap visible in the transmis-sion function stays in a good agreement with the gap sizein the PDOS obtained for the molecular stack, see Fig.3(e). The coupling strength determines the widths of thetransmission peaks, i.e. increasing the coupling broad-ens the peaks. These facts correspond to the discussedmobility features, as well as the band structure derivedconclusions. For the weaker coupling cases the positionsof the peaks are slightly shifted towards the Fermi en-ergy. This effect is related to the renormalization of theenergy levels of the system. The threshold bias for the’dark current’ is about 3 eV for the holes, while for theelectrons it is approximately 2 times higher. IV. CONCLUSIONS
We found and discuss the unique transport propertiesin the π -stacked aromatic-rings with the terminal groupspossessing the dipole moment – the separate paths forthe electrons and holes. This feature could be a solutionfor the charge recombination problem in the solar celldevices. The electronic diffusion paths cut through thearomatic rings, whereas the holes hop between the neigh- boring dipole groups. We verified theoretically, that con-necting the system to the graphene electrodes does notdestroy the above separation. The ferroelectric and path-separation properties are not necessarily directly con-nected to each other. The driving force for the electron-hole separation is not the dipole moment itself, but theexcitonic character of the system - with the donor statesat the p -electron rich atoms and the acceptor states atthe aromatic ring.The proposed materials can be realized in 2D or 1D,as the layers or molecular wires, correspondingly. Forthe simplicity, we use the smallest model: one benzenering, and the cyano groups for the ferroelectric proper-ties, and the carboxy groups for the formation of the hy-drogen bonds within the planes. This choice of the planarbonds is dictated by the requirement of a high planar re-sistance. On the other hand, the carrier mobilities, forthe electrons and holes across the layers, respectively, areone and two orders of magnitude larger than the planarones.In comparison with the organometal halide per-ovskites, reported in Ref. [54], our mobilities along the π -stacked 1D columns are an order of magnitude largerfor holes and about 5 times larger for electrons. Thesevalues are obtained in the presence of one charged im-purity per ten molecules, which gives large contributionto the scattering mechanisms. Supporting Information Description
Further details of the calculations, (S1) DOS projectedat groups of atoms indexed in Fig. 1(c), for each of sixmonolayers in the AA-type stacking without the leads,(S2) the same for four monolayers between the grapheneleads, (S3) comparison of that for the first molecularlayer in the three stackings: AA-type, R30-deg-type andAB-type, (S4) conductivity function plots in the 2Dcase, (S5) the same for the 1D case.
Acknowledgements
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