Energy-Level Alignment at the Interface of Graphene Fluoride and Boron Nitride Monolayers: An Investigation by Many-Body Perturbation Theory

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1 Supporting Information Energy-Level Alignment at the Interface of Graphene Fluoride and Boron Nitride Monolayers: An Investigation by Many-Body Perturbation Theory Qiang Fu, Dmitrii Nabok, and Claudia Draxl Institut für Physik and IRIS Adlershof, Humboldt-Universität zu Berlin, Zum Großen Windkanal 6, Berlin, Germany. S-0

2 1. Convergence tests for the G 0 W 0 calculations 1.1 Vacuum thickness Applying a truncation to the Coulomb potential Vc leads to well converged G0W0 band gaps by using relatively small vacuum sizes. In Fig. S1, we show the G0W0 band gaps of with different supercell sizes along the vertical direction when the Coulomb cutoff is applied (black). For comparison, the corresponding results obtained without the truncation are shown (gray). Note that the lattice of is not compressed here. As expected, the default setting (unmodified Vc) leads to a steady increase of the G0W0 gap, while applying a truncation to Vc produces well-converged band gaps truncated V c default Thickness (bohr) Figure S1: G 0W 0 band gap ( ) of the monolayer, calculated for slabs with different vacuum thickness, with (black) and without (gray) applying a truncation to the Coulomb potential V c. The calculations are performed using a coarse k-mesh of 4x6x1 and 100 empty states. Corresponding tests are also performed for the monolayer. As shown in Fig. S2, good convergence of the band gap is achieved by truncating the Coulomb potential truncated V c Thickness (bohr) Figure S2: G 0W 0 band gap ( ) of the monolayer calculated for slabs with different vacuum thickness, by applying a truncation to V c. The calculations are performed using a coarse k-mesh of 3x5x1 and 100 empty states. S-1

3 1.2 Number of k-points By using the truncated Coulomb potential, a dense k-grid is required to converge the G0W0 band gap [1]. In Fig. S3, we present the calculated G0W0 band gaps of both monolayers with different k-grids. Convergence results of 7.24 ev for and 6.95 ev for are achieved when using a 9x15 k-grid. Note that for, the CB level at K is obtained by using an interpolation scheme for the k-grids of 3x5, 4x7, 6x10, and 8x14, since the K point is not included in these k-meshes (using a rectangular unit cell) x5 4x7 5x9 6x10 7x12 8x14 9x15 k-mesh Figure S3: G 0W 0 band gaps of ( ) and (K ) calculated with different k-meshes. In Fig. S4, we plot the G0W0 band gaps as a function of 1/ Nk, with Nk being the number of k-points, expressed as [2]: E g (N k ) = a N k + b The value b in this relation corresponds to an infinite number of k-points. Very good linear relations are obtained for both materials. We estimate the converged gaps to be 7.15 ev for and 6.85 ev for, and the absolute error, coming from employing a finite Nk, is within 0.1 ev. The relative error, i.e., the band gap reduction upon interface formation, is expected to be even less, owing to error cancellations /N k Figure S4: Linear relations between the G 0W 0 band gaps and 1/ N k, for the (green) and the (blue) monolayers. For, the fundamental gap from the 8x14 mesh (1/ N k = ) is not S-2

4 shown, because the K point is not included in this k-mesh. 1.3 Number of empty states Nempty The number of empty states, Nempty, is a key parameter in G0W0 calculations. Here, we carefully check the convergence of Nempty on the band gaps and the quasiparticle energies. Results for the monolayer are shown in Fig. S5. The band gap of is not so sensitive to Nempty, and its value changes only by 0.07 ev when Nempty is increased from 100 to For the quasiparticle energies, however, the situation is different. The VBM and the CBm energies decrease by about 0.9 ev with the increase of Nempty in this range. This indicates that in G0W0 calculations, quasiparticle energies may converge much more slowly than band gaps. Thus, attention should be paid to this issue when energy-level alignment is the primary focus of such calculations VB ( ) CB ( ) N empty Figure S5: VB (black) and the CB (gray) quasiparticle energies of the monolayer at the point, for different values of N empty. A k-grid of 3x5x1 is used in these calculations. The vacuum level is set to 0 ev N empty Figure S6: VB (black) and CB (gray) quasiparticle energies of the monolayer at both the K point (up triangle) and the point (down triangle), for different values of N empty. A k-grid of 3x6x1 is used in these calculations. The vacuum level is set to 0 ev. S-3 VB (K) VB ( ) CB (K) CB ( ) The corresponding results of are shown in Fig. S6. Not only all the eigenvalues

5 move down in energy with increasing N empty, but also, the amount of decrease is not the same for different energy levels. For example, the VB position at K moves down by 0.86 ev over the full range of the tests, while that of the NFE-like state (CB at ) decreases only by 0.12 ev. With Nempty being 1000, good convergence is achieved. 2. Comparison of the G 0 W 0 band gaps of with previous results The G0W0 band gaps of from our calculations are 0.37 ev (K ) and 0.23 ev (K K) larger than the values reported in Ref. 3. It may be attributed to the DFT starting point (PBE vs LDA), computational parameters or methodology (all-electron vs pseudopotential code). At the DFT level, the band gap (K ) from PBE is 0.13 ev larger than that from LDA, which may influence the corresponding quasiparticle energies. Besides, the different k-grid used in our calculations and in Ref. 3 leads to a difference in the band gap by additional 0.1 ev, according to Fig. S4. 3. Configuration of at the interface In addition to the chair structure, the boat and zigzag configurations of have also been considered. The lattice constants always adopt the same value of Å. For the isolated monolayer, the chair geometry is the most stable one, while the boat and the zigzag structures are 0.75 ev and 0.30 ev higher in energy, respectively (for a unit cell with eight atoms). The energetic order of the three isomers are the same as in Ref. 4, although the values are slightly different (being 0.60 ev and 0.29 ev in Ref. 4) due to a compressed lattice. At the interface, the vertical distance between and is also optimized. The same value (6.0 bohr) is obtained for all configurations via a potential-energy scan (at the vdw-df level). Both the vdw-df functional and the DFT-D2 method are employed to describe the van der Waals interactions. In both cases, the interface composed by the chair configuration of is the most stable geometry. This is to be expected due to the weak vdw bonding that should not change the energetic order of the different isomers at the interface. The corresponding results are listed in Table S1. Isolated (PBE) Interface (vdw-df) Interface (DFT-D2) chair 0.00 ev 0.00 ev 0.00 ev boat 0.75 ev 0.82 ev 0.82 ev zigzag 0.30 ev 0.22 ev 0.36 ev Table S1: Relative energies of an isolated monolayer and its interface with for three different isomers. The value of the most stable configuration is set to 0. S-4

6 4. Band structure and density of states In Fig. S7, we present the Kohn-Sham band structure for the interface (black lines) and the two isolated monolayers (blue and green symbols) together with the density of states (DOS). All the calculations are performed at the DFT level with the PBE functional, using the same rectangular unit cell. One can see that the VBM of the interface is that of the monolayer, while the CBm is provided by. The comparison of the bands of the interface with those of and demonstrates the very weak interaction between the two components, indicative of the van der Waals nature of the heterojunction. Figure S7: Band structure (left) and density of states (right) for the interface and the two components. The black lines, the green dots, and the blue dots represent the interface, the, and the BN monolayer, respectively. The Fermi energy (E F) of the interface is set to 0 ev. 5. Estimation of the effective thickness for and Figure S8: Plane-averaged density of valence electrons for the individual and monolayers. The effective thickness (in bohr) is determined by the respective full-width-at-half-maximum value. References [1] Hüser, F.; Olsen, T.; and Thygesen, K. S. Phys. Rev. B 2013, 88, S-5

7 [2] Puschnig, P.; Amiri, P.; and Draxl, C. Phys. Rev. B 2012, 86, [3] Hüser, F.; Olsen, T.; and Thygesen, K. S. Phys. Rev. B 2013, 87, [4] Leenaerts, O.; Peelaers, H.; Hernández-Nieves, A. D.; Partoens, B.; and Peeters, F. M. Phys. Rev. B 2010, 82, S-6