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Shan Sun, Hui-zhong Ma, Xiao Zhang, Yu-chen Ma. Direct and Indirect Excitons in Two-Dimensional Covalent Organic Frameworks†[J]. Chinese Journal of Chemical Physics , 2020, 33(5): 569-577. doi: 10.1063/1674-0068/cjcp2001003
Citation: Shan Sun, Hui-zhong Ma, Xiao Zhang, Yu-chen Ma. Direct and Indirect Excitons in Two-Dimensional Covalent Organic Frameworks[J]. Chinese Journal of Chemical Physics , 2020, 33(5): 569-577. doi: 10.1063/1674-0068/cjcp2001003

Direct and Indirect Excitons in Two-Dimensional Covalent Organic Frameworks

doi: 10.1063/1674-0068/cjcp2001003
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  • Corresponding author: Yu-chen Ma, E-mail: myc@sdu.edu.cn
  • Part of the special issue on "the Chinese Chemical Society's 16th National Chemical Dynamics Symposium".
  • Received Date: 2020-01-07
  • Accepted Date: 2020-02-10
  • Publish Date: 2020-10-27
  • Highly luminescent bulk two-dimensional covalent organic frameworks (COFs) attract much attention recently. Origin of their luminescence and their large Stokes shift is an open question. After first-principles calculations on two kinds of COFs using the GW method and Bethe-Salpeter equation, we find that monolayer COF has a direct band gap, while bulk COF is an indirect band-gap material. The calculated optical gap and optical absorption spectrum for the direct excitons of bulk COF agree with the experiment. However, the calculated energy of the indirect exciton, in which the photoelectron and the hole locate at the conduction band minimum and the valence band maximum of bulk COF respectively, is too low compared to the fluorescence spectrum in experiment. This may exclude the possible assistance of phonons in the luminescence of bulk COF. Luminescence of bulk COF might result from exciton recombination at the defects sites. The indirect band-gap character of bulk COF originates from its AA-stacked conformation. If the conformation is changed to the AB-stacked one, the band gap of COF becomes direct which may enhance the luminescence.
  • Part of the special issue on "the Chinese Chemical Society's 16th National Chemical Dynamics Symposium".
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Direct and Indirect Excitons in Two-Dimensional Covalent Organic Frameworks

doi: 10.1063/1674-0068/cjcp2001003

Abstract: Highly luminescent bulk two-dimensional covalent organic frameworks (COFs) attract much attention recently. Origin of their luminescence and their large Stokes shift is an open question. After first-principles calculations on two kinds of COFs using the GW method and Bethe-Salpeter equation, we find that monolayer COF has a direct band gap, while bulk COF is an indirect band-gap material. The calculated optical gap and optical absorption spectrum for the direct excitons of bulk COF agree with the experiment. However, the calculated energy of the indirect exciton, in which the photoelectron and the hole locate at the conduction band minimum and the valence band maximum of bulk COF respectively, is too low compared to the fluorescence spectrum in experiment. This may exclude the possible assistance of phonons in the luminescence of bulk COF. Luminescence of bulk COF might result from exciton recombination at the defects sites. The indirect band-gap character of bulk COF originates from its AA-stacked conformation. If the conformation is changed to the AB-stacked one, the band gap of COF becomes direct which may enhance the luminescence.

Part of the special issue on "the Chinese Chemical Society's 16th National Chemical Dynamics Symposium".
Shan Sun, Hui-zhong Ma, Xiao Zhang, Yu-chen Ma. Direct and Indirect Excitons in Two-Dimensional Covalent Organic Frameworks†[J]. Chinese Journal of Chemical Physics , 2020, 33(5): 569-577. doi: 10.1063/1674-0068/cjcp2001003
Citation: Shan Sun, Hui-zhong Ma, Xiao Zhang, Yu-chen Ma. Direct and Indirect Excitons in Two-Dimensional Covalent Organic Frameworks[J]. Chinese Journal of Chemical Physics , 2020, 33(5): 569-577. doi: 10.1063/1674-0068/cjcp2001003
  • After Yaghi and his colleagues made the breakthrough in 2005 on the synthesis of covalent organic framework (COF) materials [1], the study of COFs grows rapidly [2-4]. In two- and three-dimensional COFs, atoms are connected by various covalent bonds to construct the framework structure, giving this new kind of material broad prospective applications. Due to its high specific surface area, low density and diverse structure, COFs may be applied in the fields of gas storage [5, 6], catalysis [7], optoelectronic devices [8-12], etc. COFs can possess wide optical absorption in the visible range. Their highly ordered arrays may facilitate charge transport and exciton migration [13]. Together with the presence of large pores in their structures, COFs might find important applications in photocatalysis [14-16].

    Two-dimensional COFs (2D-COFs) with high luminescence attracted much interest in recent years as promising materials for light-emitting devices in the visible domain [17-26]. In 2015, Jiang et al. designed Ph-An-COF which had the absorption maximum at 278 nm and the emission peak at 429 nm with the fluorescence quantum yield (QY) of 5.4% [17]. One year later, Jiang and coworkers proposed TPE-Ph COF whose fluorescence QY rose substantially to 32% [18]. The absorption and emission maxima of TPE-Ph COF located at 390 nm and 500 nm respectively. These two kinds of COFs used boroxine as the linkage, being unstable upon exposure to air [17, 18]. Subsequently, Jiang et al. synthesized three kinds of COFs which were completely linked through sp$ ^2 $ C = C bonds and exhibitted excellent stability [19, 20]. The bulk solid state forms of these sp$ ^2 $c-COFs show absorption maxima between 460 nm and 500 nm and emission peaks around 600 nm, with the fluorescence QY around 10%. In 2018, Jiang et al. suggest that deprotonation of the N-H bond in the hydrazone linkage of TFPPy-DETHz-COF may enhance the fluorescence QY by 3.8 times and this feature can be employed for detecting fluoride anions in the environment [21]. In 2016, Liu et al. found that fluorescence of azine-linked COF-JLU3 (absorption and emission maxima at 430 nm and 600 nm respectively) can be quenched by Cu$ ^{2+} $ ions, with the fluorescence QY reduced from 3.86% to 0.03% [22]. Zamora et al. reported the first fluorescent solid-state imine-linked 2D-COF with QY of 3.5% [24] in 2018. In 2D-COFs, layers were stacked through van der Waals interactions. Most of the 2D-COFs synthesized were in the AA-stacked configuration, while some can be stabilized in the AB-stacked mode [24].

    Light emission in 2D inorganic semiconducting systems, such as MoS$ _2 $, MoSe$ _2 $, WSe$ _2 $ and hexagonal boronnitride ($ h $-BN), have already been extensively studied [27-40]. Their monolayer conformation has a direct band gap and can be highly luminescent. With the increasing number of layers, these materials exhibit a direct-to-indirect band gap transition. Indirect band-gap semiconductors are inefficient light emitters according to standard solid state physics textbooks. The luminescence QY of multilayer MoS$ _2 $ is four orders of magnitude smaller than that of monolayer MoS$ _2 $. However, $ h $-BN seems to be one exception. Bulk $ h $-BN has an indirect band gap, but its luminescence QY can reach 45% [35]. Origin and characteristics of the efficient light emission of bulk $ h $-BN has been widely discussed from both experiments and theoretical calculations [32-40]. Some evidences demonstrate that light emission in multilayer $ h $-BN is realized through the assistance of phonons. Similar structures of 2D-COFs to these 2D inorganic materials may make us question whether 2D-COFs also exhibit a direct-to-indirect band gap transition from the monolayer to the bulk forms and why their bulk forms are luminescent. Although much experimental work has been done to pursue highly luminescent 2D-COFs, mechanism of light emission in 2D-COFs has been seldom discussed, especially from the point of view of theory. Understanding light emission mechanism might provide us useful information for designing more efficient 2D-COF light emitters.

    In this work, we study the properties of excitons in two kinds of sp$ ^2 $ carbon-conjugated COFs (sp$ ^2 $c-COFs) developed by Jiang et al. [19, 20] theoretically through the many-body Green's function theory (MBGFT). We do find that the band gaps of bulk sp$ ^2 $c-COFs are indirect based on GW calculations within MBGFT. From the exciton energy-momentum map computed by the Bethe-Salpeter equation (BSE) within MBGFT, it seems that light emission is not phonon-assisted.

  • Structures of the two kinds of sp$ ^2 $c-COFs studied in this work are shown in FIG. 1. They are synthesized through the condensation of TFPPy and PDAN, and the condensation of TFPPy and BPDAN respectively in the experiments of Jiang and coworkers [20]. Following the nomenclature of Jiang et al., these two sp$ ^2 $c-COFs are also called sp$ ^2 $c-COF and sp$ ^2 $c-COF-2 respectively in this article. The bulk sp$ ^2 $c-COF and sp$ ^2 $c-COF-2 belong to the $ C2/m $ and the $ P1 $ space groups respectively. Their experimental lattice parameters are shown in Table Ⅰ and FIG. 2. The primitive cells of sp$ ^2 $c-COF and sp$ ^2 $c-COF-2 contain 204 and 122 atoms, respectively. We use the density functional theory (DFT) with the Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional to optimize the geometries of COFs by the VASP code [41-43]. Additionally, the Grimme's PBE-D2 method, which is implemented into the VASP code, is applied to model the interlayer van der Waals interactions [44]. Energy cutoff for the plane wave basis is set to 400 eV, and the $ k $-points grid is set to 1$ \times $1$ \times $3 for bulk COFs. Under these settings, the fully relaxed lattice parameters of sp$ ^2 $c-COF and sp$ ^2 $c-COF-2 agree well with the experiments, as compared in Table Ⅰ.

    Figure 1.  (a, b) Structure of sp$ ^2 $c-COF-2. (c, d) Structure of sp$ ^2 $c-COF. Rectangles in (b, d) represent the primitive cell of respective COF.

    Figure 2.  (a) Crystal structure of sp$ ^2 $c-COF. (b) and (c) are the side view and the top view of sp$ ^2 $c-COF respectively.

    Table Ⅰ.  Lattice parameters of sp$ ^2 $c-COF and sp$ ^2 $c-COF-2 from experiments and theoretical calculations.

    It has been well known that DFT-PBE has large errors in predicting the band gaps of semiconductors and insulators. We use the GW method [45, 46] within MBGFT to get more accurate band structures of COFs. In GW, the quasiparticle (QP) equation is solved to achieve the quasiparticle energies of single particles. Excitonic properties are computed via BSE [45, 46], which is a theory to describe the motion of coupled particle pairs (electron and hole here) within MBGFT. The GW method and BSE have been demonstrated to predict electronic and excitonic levels with excellent accuracy [45-47]. Our GW calculations are carried out at the level of G$ _0 $W$ _0 $ with the two key physical quantities in GW, i.e. self-energy and dielectric function, constructed by the single-particle eigenvalues and wave functions derived from DFT within the local density approximation (LDA). However, a rigid scissor shift is imposed upon the LDA eigenvalues of unoccupied orbitals in order to describe these two physical quantities more reasonably (see the supplementary materials for details). Value of the scissor shift is tuned until it equals the QP correction to the LDA band gap. The dielectric function is evaluated within the random-phase approximation, and its dynamical part is approximated by the plasmon-pole model proposed by von der Linden and Horsch [48]. Evaluation of the dielectric function and self-energy involves band summations over massive unoccupied single-particle states. To reduce the huge computational cost of MBGFT runs for COFs composed by 100-200 atoms, these band summations are truncated at the unoccupied state whose energy is 15 eV above the conduction band maximum (CBM). We have tested that the band gap could converge to about 0.10 eV under this truncation. The $ k $-point grids of 1$ \times $1$ \times $2 and 1$ \times $1$ \times $6 are employed in the calculations of dielectric function and self-energy for bulk COFs. In BSE, a 3$ \times $3$ \times $15 $ k $-point grid, six valence bands and six conduction bands are used to represent the excitonic wave functions of bulk COFs. Our GW+BSE calculations are carried out by a code with Gaussian orbitals as the basis set [49-51]. We use atom-centered Gaussian orbitals with the decay constants (in a.u.) of 0.25, 0.6, 1.6, and 4.0 are calculated for C atoms, 0.2, 0.5, 1.25, and 3.2 a.u. for N atoms, 0.15, 0.4, and 1.5 for H atoms. This set of Gaussian basis has been well tested to yield converged results. This code and similar basis sets have been applied in our previous studies on other two-dimensional organic systems [52-56].

  • The three angles in the lattice parameters of sp$ ^2 $c-COF-2 are close to 90$ ^\circ $ (Table Ⅰ). To simplify our MBGFT calculations and also to facilitate discussions in the first reciprocal space, the three angles are fixed at 90$ ^\circ $. After optimized by VASP, a new set of lattice constants is gotten which deviates a little from the fully optimized one and the experiment (Table Ⅰ).

    FIG. 3 presents band structures of the monolayer and bulk sp$ ^2 $c-COF-2. From monolayer to bulk, sp$ ^2 $c-COF-2 undergoes a direct-to-indirect band gap transition. For the monolayer, the band gap is 1.37 eV in LDA and 3.98 eV in GW respectively, with the valence band maximum (VBM) and the conduction band minimum (CBM) both lying at the $ \Gamma $ point. For the bulk, VBM locates at the $ \Gamma $ point, while CBM locates at the Z point. The direct band gap at the $ \Gamma $ point and the indirect band gap are 1.11 eV and 0.65 eV in LDA and 2.95 eV and 2.40 eV in GW. GW correction affects little the dispersion of band for both monolayer and bulk sp$ ^2 $c-COF-2. The GW indirect band gap is close to the electrochemical band gap (2.07 eV) measured in experiments [20]. VBM is mostly localized on the pyrene knot, while CBM is shared by the pyrene knot and the biphenyl linker (FIG. 4). VBM-1 and CBM+1 are localized on the biphenyl linker.

    Figure 3.  Band structures of sp$ ^2 $c-COF-2. (a, b) Bulk sp$ ^2 $c-COF-2. (c) First Brillouine zone of bulk sp$ ^2 $c-COF-2, (d, e) monolayer sp$ ^2 $c-COF-2. (a, d) are calculated by DFT-LDA, while (b, e) are calculated by GW.

    Figure 4.  Charge density distributions (purple and yellow isosurfaces) of four bands in sp$ ^2 $c-COF-2.

    We first discuss excitons with the momenta $ \vec q $$ \rightarrow $0. FIG. 5 shows optical absorption spectra of the monolayer and bulk sp$ ^2 $c-COF-2. In the experimental measurements [20], the incident direction of light with respect to the surface of bulk sp$ ^2 $c-COF-2 is not well defined and might be randomly distributed. To make a better comparison with experiments, absorption spectra for light irradiating along the three lattice vector directions, i.e. $ \vec q $//$ \vec a $, $ \vec q $//$ \vec b $ and $ \vec q $//$ \vec c $, are computed for bulk sp$ ^2 $c-COF-2. Two sets of spectra are provided, one being Gaussian-broadened by 0.10 eV and the other by 0.30 eV. The 0.30 eV broadening is chosen in order to reproduce the spectral width in the experiments [20]. Finer structures can be discerned in the spectra broadened by 0.10 eV.

    Figure 5.  Optical absorption spectra of sp$ ^2 $c-COF-2 with the exciton momentum $ q $$ \rightarrow $0. (a, b, c) Spectra of bulk sp$ ^2 $c-COF-2 with the direction of exciton momentum parallel to lattice vectors $ \vec a $, $ \vec b $, and $ \vec c $ respectively. (d) Spectrum of monolayer sp$ ^2 $c-COF-2. Black and red curves represent the spectra which are broadened artificially by 0.1 eV and 0.3 eV respectively. T$ _1 $ represents position of the lowest triplet exciton. S$ _1 $ and S$ _2 $ represent positions of the lowest and the second lowest singlet excitons. The blue dashed curves show the experimental absorption spectra from Ref.[20].

    The lowest singlet excited state (S$ _1 $) of bulk sp$ ^2 $c-COF-2 locates at 2.10 eV, which agrees very well with its optical band gap measured in the experiments (2.14 eV) [20]. S$ _1 $ originates from the transition between the uppermost valence band and the lowest conduction band, but with the photoelectron and hole localized around the $ \Gamma $ point (FIG. 6). The S$ _1 $ exciton has a pronounced oscillator strength for $ \vec q $//$ \vec b $ and $ \vec q $//$ \vec c $, but is optically forbidden for $ \vec q $//$ \vec a $. The spectrum for $ \vec q $//$ \vec a $ has a strong absorption peak at 2.60 eV which involves transitions related to lower valence bands and higher conduction bands. With the 0.10 eV broadening, there is a prominent absorption peak at 2.22 eV for $ \vec q $//$ \vec b $ and $ \vec q $//$ \vec c $. If increasing the broadening to 0.30 eV, the several peaks below 3.00 eV for $ \vec q $//$ \vec c $ merge into a broad absorption band peaked at 2.63 eV. Combining the spectra for $ \vec q $//$ \vec a $, $ \vec q $//$ \vec b $ and $ \vec q $//$ \vec c $ with the 0.30 eV broadening, the overall absorption spectrum might exhibit a broad band peaked around 2.60 eV. This is in line with the experiments in which the absorption peak of bulk sp$ ^2 $c-COF-2 is at 2.61 eV. Optical absorption of bulk sp$ ^2 $c-COF-2 below 2.65 eV is constituted by excitations in which the binding energy between the photoelectron and hole is huge ($ \sim $1.10 eV). FIG. 7(a, b) show spatial distribution of the photoelectron for one of them, the S$ _1 $ state. It can be seen that the electron is mainly excited into the $ \pi^* $ orbitals of the layer, where the hole resides in, and those of the two most adjacent layers. Above 2.65 eV, excitations with weak exciton binding energies dominate the optical absorption of bulk sp$ ^2 $c-COF-2. FIG. 7(c, d) display spatial distribution of the photoelectron for the excitation at 2.67 eV whose exciton binding energy is 0.46 eV. In this state the electron is excited out of the layer accommodating the hole and transferred to the next-nearest-neighbor layers mainly.

    Figure 6.  Main compositions of the lowest singlet exciton of bulk sp$ ^2 $c-COF-2 in the first Brillouine zone depicted in (a), a 3$ \times $3$ \times $15 $ k $-point grid is used to represent excitonic wave functions in our BSE calculations. (b), (c), and (d) illustrate the contributions from $ k $ = ($ k_x $, $ k_y $, $ k_z $) points distributed on the planes colored in blue ($ k_z $ = 0), red ($ k_z $ = 1/15$ |Z| $), and green ($ k_z $ = 2/15$ |Z| $) shown in (a), respectively.

    Figure 7.  Real-space distributions of photoelectrons (red isosurfaces) for excitons in bulk sp$ ^2 $c-COF-2. (a, b) The lowest exciton at 2.10 eV, and (c, d) the exciton at 2.67 eV. (a, c) Top views and (b, d) side views. The hole is fixed at atoms enclosed inside the rectangle depicted in (a, c).

    The BSE absorption spectrum of monolayer sp$ ^2 $c-COF-2 has two peaks at 2.97 and 3.63 eV (see FIG. 5(d)). They originate from the lowest two excited sates respectively (S$ _1 $ and S$ _2 $ in FIG. 5(d)). Perfect monolayer sp$ ^2 $c-COF-2 is unavailable yet in the experiment. Optical absorption of the sp$ ^2 $c-COF-2 thin film with crumpled shape exhibits two peaks at 2.68 and 2.97 eV in the experiment [20]. Exciton binding energies of the S$ _1 $ and S$ _2 $ states are both around 1.20 eV, differring little from those of the lower-energy excitons in bulk sp$ ^2 $c-COF-2. In both monolayer and bulk sp$ ^2 $c-COF-2, the lowest triplet exciton (T$ _1 $) is lower than S$ _1 $ by 0.64 eV (FIG. 5), and the exciton binding energy of T$ _1 $ is 1.95 eV. These similarities between monolayer and bulk sp$ ^2 $c-COF-2 implie comparable electronic screening environment in them. In fact, the dielectric constant of bulk sp$ ^2 $c-COF-2 is calculated to be 1.22 by the GW method, which is close to that of the vacuum. This might be due to the large pores inside the COF.

    Bulk sp$ ^2 $c-COF-2 is an indirect-band-gap material and should be non-luminescent in principle. However, bulk sp$ ^2 $c-COF-2 is luminescent in experiments [20]. Inspired by recent discussions in the community on the high luminescence of the indirect-band-gap bulk $ h $-BN, we further investigate excitons with finite momenta in bulk sp$ ^2 $c-COF-2. Photoelectron (hole) would relax to CBM (VBM) of the system. CBM and VBM of bulk sp$ ^2 $c-COF-2 sit at the Z and $ \Gamma $ points respectively (FIG. 3). Recombination of the photoelectron at CBM and hole at VBM requires the assistance of a phonon with the momentum $ \vec q $ = $ Z $-$ \Gamma $. The coordinate of the $ Z $ point is (0, 0, 0.840) Å$ ^{-1} $. We compute the optical spectra for $ q $ = 0.168, 0.336, 0.504, 0.672 and 0.840 Å$ ^{-1} $ by BSE, as plotted in FIG. 8, to examine the dispersion of excitons with respect to their momenta. Energy of the S$ _1 $ state descends gradually with the increase of $ q $, from 2.10 eV for $ q $ = 0 to 1.59 eV for $ q $ = 0.840 Å$ ^{-1} $. For all $ q $, the hole of S$ _1 $ is localized around VBM ($ \Gamma $ point), as shown in FIG. 9(a) for the distribution of hole in the reciprocal space for $ q $ = 0.336 Å$ ^{-1} $ as the example. Photoelectron of S$ _1 $ spreads over the lowest conduction band for all $ q $. The redshift of S$ _2 $ from $ q $ = 0 to $ q $ = 0.840 Å$ ^{-1} $ is less than half that of S$ _1 $. From $ q $ = 0 to $ q $ = 0.504 Å$ ^{-1} $, the photoelectron of S$ _2 $ is confined in the region around CBM ($ Z $ point), as shown in FIG. 9(b) for the distribution of photoelectron in the reciprocal space for $ q $ = 0.336 Å$ ^{-1} $ as the example, at the same time the hole occupies the highest valence band. However, for $ q $ = 0.672 Å$ ^{-1} $ and $ q $ = 0.840 Å$ ^{-1} $, composition of S$ _2 $ is complicated, with the contributions from single-particle orbitals beyond the highest valence band and the lowest conduction band dominating the S$ _2 $ transition. The exciton binding energy of S$ _1 $ remains at 1.10 eV for all $ q $. This leads to the continuous redshift of S$ _1 $ with the increase of $ q $. This is different from bulk $ h $-BN in which the exciton binding energy of S$ _1 $ weakens with the increase of $ q $ [32]. This might be due to the difference in the direction of exciton momentum. In $ h $-BN the exciton momentum is parallel to the BN layer, while in sp$ ^2 $c-COF-2 it is perpendicular to the COF layer. More realistic theoretical prediction of the spectrum shape for phonon-assisted luminescence needs to consider the electron-phonon coupling based on BSE, just like that for bulk $ h $-BN [32, 35]. However, this is beyond the scope of this article and might also be unachievable presently due to the extremely huge computational demand for COFs which contain more than one hundred of atoms in the primitive cell. However, some important information can still be acquired from the BSE spectrum without electron-phonon coupling. The S$ _1 $ state is optically allowed for $ q $ = 0.840 Å$ ^{-1} $, although its oscillator strength is more than one magnitude weaker than that for $ q $ = 0. If phonon could assist the recombination of indirect exciton in bulk sp$ ^2 $c-COF-2, there should be light emission around 1.60 eV. In the experiment [20], bulk sp$ ^2 $c-COF-2 emits at 2.05 eV. The large gap between theoretical calculations and the experiment might demonstrate that luminescence of bulk sp$ ^2 $c-COF-2 is not phonon-assisted. Possible origin of the luminescence of bulk sp$ ^2 $c-COF-2 then might be the recombination of direct excitons at defect sites.

    Figure 8.  Theoretical optical absorption spectra of bulk sp$ ^2 $c-COF-2 for excitons with finite momenta (in Å$ ^{-1} $). Directions of the exciton momenta are parallel to the $ \Gamma $-$ Z $ direction in the first Brillouin zone. $ q $ = 0.840 corresponds to the $ Z $ point. Right panels are enlarged views of the corresponding left ones. Vertical lines in the left panels mark energies of the lowest five excited states.

    Figure 9.  Distributions of the photoelectron and hole in the first Brillouine zone for the exciton with the momentum $ q $ = 0.336 Å$ ^{-1} $ in bulk sp$ ^2 $c-COF-2. A 3$ \times $3$ \times $15 $ k $-point grid is used to represent excitonic wave functions in our BSE calculations. (a) Distribution of hole for the lowest exciton. (b) Distribution of photoelectron for the second lowest exciton. Each red horizontal column illustrates the sum of contributions from the $ k $ points distributed on the corresponding blue plane in the first Brillouine zone.

  • Two angles in the lattice parameters of sp$ ^2 $c-COF, $ \alpha $ and $ \gamma $, are exactly 90$ ^\circ $ (Table Ⅰ). Another angle $ \beta $ is around 106$ ^\circ $. Instead of this stack mode of sp$ ^2 $c-COF, we study the AA and AB stack modes of sp$ ^2 $c-COF. Table Ⅰ lists the optimized lattice constants for AA-stacked sp$ ^2 $c-COF. We use the same lattice constants for AB-stacked sp$ ^2 $c-COF. FIG. 10(d) gives the geometry of AB-stacked sp$ ^2 $c-COF. From AA to AB, we can know how the electronic structures and therefore optical excitations evolve with the stack mode.

    Figure 10.  Band structures of bulk sp$ ^2 $c-COF. (a, b) AA-stacked sp$ ^2 $c-COF, (c) AB-stacked sp$ ^2 $c-COF. (a, c) are calculated by DFT-LDA, while (b) is calculated by GW. (d) Top view of AB-stacked sp$ ^2 $c-COF where the brown and the blue atoms belong to different layers.

    In DFT-LDA, AA- and AB-stacked sp$ ^2 $c-COF are an indirect and a direct band-gap material respectively (FIG. 10). Just like sp$ ^2 $c-COF-2, VBM and CBM of AA-stacked sp$ ^2 $c-COF locate at $ \Gamma $ and $ Z $ points respectively. The direct band gaps at the $ \Gamma $ point for AA- and AB-stacked sp$ ^2 $c-COF are very close, differing by just 0.10 eV. The main difference in the band structures between these two stack modes lies in the band dispersion along the normal direction of the COF layer. Dispersions along $ \Gamma $$ \rightarrow $Z and R$ \rightarrow $S directions are less than 0.10 eV for AB-stacked sp$ ^2 $c-COF, while reach 1.00 eV for the AA-stacked one. The huge dispersion in AA-stacked sp$ ^2 $c-COF is induced by strong interlayer interactions. GW correction affects a little the band dispersions of AA-stacked sp$ ^2 $c-COF. The GW band gap of AA-stacked sp$ ^2 $c-COF is 2.16 eV, close to the electrochemical band gap of 1.90 eV measured in the experiment [20]. GW calculation is not carried out here for AB-stacked sp$ ^2 $c-COF due to its huge primitive cell which contains more than 400 atoms. We think that GW correction should not modify the band dispersions in AB-stacked sp$ ^2 $c-COF either. The perfect sp$ ^2 $c-COF should be non-luminescent in the AA-stacked mode according to discussions above on sp$ ^2 $c-COF-2, but may become luminescent in the AB-stacked one. Most of 2D-COFs synthesized till now are in the AA-stacked mode [15, 21, 57-59]. If they can be converted into the AB mode, such as by exerting external pressure along the $ \vec c $ direction, luminescence might be enhanced.

    Three peaks, located at 2.50 eV, 2.80 eV and 3.40 eV, can be discerned from the optical absorption spectrum of bulk sp$ ^2 $c-COF measured in experiments [19, 20]. Our BSE calculations on AA-stacked sp$ ^2 $c-COF with light irradiating along the $ \vec c $ direction yield three peaks at 2.38, 2.63, and 3.13 eV respectively (see FIG. 11), agreeing well with the experiments. The lowest excited state calculated by BSE is at 2.22 eV which is also close to optical gap of 2.05 eV measured in the experiment.

    Figure 11.  Optical absorption spectrum of AA-stacked sp$ ^2 $c-COF with the exciton momentum $ q $$ \rightarrow $0 and the direction of exciton momentum parallel to the lattice vector $ \vec c $. S$ _1 $ represents position of the lowest singlet exciton. The blue dashed curve shows the experimental absorption spectrum from Ref.[20].

    After optical excitation, higher-energy excitons would decay quickly to the lowest excited state via nonadiabatic transitions. Light emission occurs through the recombination of electron and hole near the potential energy minimum in the lowest excited state. From the experiments [20], the absorption energy of the lowest excited state, i.e. the optical gap and the emission peak of bulk sp$ ^2 $c-COF-2 are 2.14 eV and 2.05 eV, respectively, while those of bulk sp$ ^2 $c-COF are 2.05 eV and 2.00 eV, respectively. This demonstrates that the Stokes shifts of these two COFs are less than 0.10 eV. Using the constrained DFT, in which populations of the highest occupied molecular orbital and the lowest unoccupied molecular orbital are both set to one, to simulate the excited state approximately, we relax the geometry of sp$ ^2 $c-COF in its lowest excited state. Band gaps of both the monolayer and the AA-stacked bulk sp$ ^2 $c-COF at the optimized excited-state geometries narrow by around 0.10 eV compared to those at the ground-state geometries, and the Stokes shift of monolayer sp$ ^2 $c-COF is calculated to be 0.08 eV (see FIGs. S1-S3 in supplementary materials), which are consistent with the experiments. Therefore, structure relaxation in the excited state is insignificant for COFs and should not be responsible for the huge energy gap between their absorption and emission peaks reported in the experiments.

  • Our first-principles calculations based on the GW method and BSE reproduce well the electronic gap, optical gap and optical absorption spectrum of two kinds of sp$ ^2 $ carbon-conjugated COFs. The COFs have an indirect band gap, which may account for why the optical gap is smaller than the electronic gap as reported in experiments. The energy of indirect exciton calculated from BSE is so low compared to the experimental fluorescence spectrum that we may exclude the possibility that light emission of COFs is caused by phonon-assisted recombination of indirect excitons. Light emission of COFs might result from recombination of direct excitons at defect sites. Changing the stack mode of COFs from AA to AB can convert the band gap from indirect to direct and therefore may enhance the fluorescence quantum yield.

    Supplementary materials: Details of the GW method, band structures and optical absorption spectra of sp$ ^2 $c-COF at the excited-state geometry are shown.

  • This work was supported by the National Natural Science Foundation of China (No.21833004, No.21573131 and No.21433006) and the Natural Science Foundation of Shandong Province (No.JQ201603).

  • In the GW method, electronic structures are calculated via the quasiparticle (QP) equation

    The self-energy operator Σ(r, r′, E) describes the exchange and correlation interactions among electrons, which equals the convolution of one-particle Green's function (G) and the screened Coulomb interaction (W), i.e.

    The screened Coulomb potential is computed by

    where ε and υ are the dielectric function and the bare Coulomb potential respectively. The dielectric function ε = 1 − υP, with P being the polarizability. Within the random-phase approximation, polarizability takes the form

    The one-particle Green's function in Eq. (2) is represented as

    The summations in Eq. (4) and Eq. (5) cover all the occupied and unoccupied molecular orbitals in principle. In our G0W0 calculations, we use the single-particle energies Ei and wave functions ψi calculated by DFT-LDA to construct Eq. (4) and Eq. (5). However, since LDA underestimates the band gap greatly, using LDA single-particle energies directly may cause some errors in P and G. To reduce the error from this respect, we add a rigid scissor shift ∆ to the energies of unoccupied orbitals, i.e. Ei becomes Ei + ∆ for unoccupied orbitals. The value of ∆ is tuned until it equals the difference between the QP band gap computed by Eq. (1) and the LDA band gap approximately.

    Figure S1.  GW band structure of the monolayer sp2c-COF in the ground-state geometry optimized by DFT (a) and that in the excited-state geometry optimized by the constrained DFT (b). In the constrained DFT, populations of the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbitals are both set to one. The GW band gap in (b) is smaller than that in (a) by 0.12 eV.

    Figure S2.  Optical absorption spectrum of the monolayer sp2c-COF in the ground-state geometry optimized by DFT (black curve) and that in the excited-state geometry optimized by the constrained DFT (red curve). The exciton momentum q→0. The direction of exciton momentum is parallel to the lattice vector $\vec c$. Absorption peaks in the excited-state geometry redshift by 0.08 eV compared to those in the ground-state geometry.

    Figure S3.  LDA band structure of the AA-stacked bulk sp2c-COF in the ground-state geometry optimized by DFT (a) and that in the excited-state geometry optimized by the constrained DFT (b). The band gap in (b) is smaller than that in (a) by 0.09 eV.

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