Structures of the two kinds of sp$ ^2 $c-COFs studied in this work are shown in FIG. 1. They are synthesized through the condensation of TFPPy and PDAN, and the condensation of TFPPy and BPDAN respectively in the experiments of Jiang and coworkers . Following the nomenclature of Jiang et al., these two sp$ ^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 . 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 . 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].