Single-Layered Hittorf s Phosphorus: A. Wide-Bandgap High Mobility 2D Material

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1 Single-Layered Hittorf s Phosphorus: A Wide-Bandgap High Mobility 2D Material Georg Schusteritsch,,, Martin Uhrin,, and Chris J. Pickard, Department of Physics and Astronomy, University College London, Gower Street, London WCE 6BT, United Kingdom gs55@cam.ac.uk Supporting Information Further calculations and computational details A full list of the relative energies of all structures considered here can be found in Table S. Using PBE the lowest energy bulk structure is Hittorf s phosphorus, followed by black and A7 phosphorus. This is consistent with previous theoretical work by Bachhuber et al, who consider several metastable allotropes of phosphorus and we reproduce the energetic differences for the three allotropes we consider here very closely. Hittorfene is only marginally less energetically stable than Hittorf s phosphorus (by.3 ev/atom), whilst black and A7 To whom correspondence should be addressed Department of Physics and Astronomy, University College London, Gower Street, London WCE 6BT, United Kingdom Current address: Department of Materials Science and Metallurgy, University of Cambridge, 27 Charles Babbage Road, Cambridge CB3 FS, United Kingdom Current address: Theory and Simulation of Materials (THEOS), and National Centre for Computational Design and Discovery of Novel Materials (MARVEL), École Polytechnique Fédérale de Lausanne, CH-5 Lausanne, Switzerland Current address: Department of Materials Science and Metallurgy, University of Cambridge, 27 Charles Babbage Road, Cambridge CB3 FS, United Kingdom and Advanced Institute for Materials Research, Tohoku University 2-- Katahira, Aoba, Sendai, , Japan S

2 phosphorene are less stable by.4 and.44 ev/atom. We also consider the two dispersioncorrected xc-functionals, PBE with TS and G6 corrections. As can be anticipated, this results in stabilizing the bulk structures with respect to the layered structures. Hittorf s, black and A7 phosphorus are then very close in energy for both TS and G6 results. Our results are qualitatively similar to the results for the dispersion-corrected calculations by Bachhuber et al, who used PBE with the Grimme correction (PBE+G6) and similarly found Hittorf s, black and A7 phosphorus to be extremely close in energy. It is important to note that giving a very accurate ordering of the bulk phosphorus allotropes is not the main issue addressed here, but rather the ordering of layered phosphorus of various structures is being considered. We also include results using the MBD* approach, which shows A7 to be the most stable bulk structure, followed by Hittorf s and black phosphorus. Table S: Energetic order for different single-layered, multi-layered, and bulk phosphorus allotropes using PBE, PBE+TS, PBE+MBD* and PBE+G6 xc-functionals. Multi-layered structures are labeled as n=2 to 4 in the case of black phosphorene. Values in brackets are approximate values for the energetic ordering of bulk Hittorf s, black and A7 phosphorus, extracted from Fig. (c) of Bachhuber et al who used PBE and the GGA+G6 xc-functionals. E [ev/atom] PBE +TS MBD* PBE+G6 PBE Hittorf s...4 (.). (.) A7.7.. (.7).72 (.8) Black..2. (.).3 (.3) Hittorfene bilayer Black phosphorene n= Black phosphorene n= Hittorfene Black phosphorene n= Black phosphorene n= A7 phosphorene We assess the importance of including vdw corrections to describe the intra-layer interactions of hittorfene by considering the cleavage energy of hittorfene based on the different xc functionals. That is we cleave the single layer of hittorfene along its center plane such S2

3 that two individual levels of flat fiber-networks remain. We do not allow for relaxations after cleavage and consider the fully relaxed PBE structure as the starting point for the entire set of calculations. We then find a decohesion energy ( E = E initial E cleaved ) of 2.24 ev for PBE. This energy changes to 3.34 ev and 3.5 ev when performing the calculations for the same structures with PBE+TS and PBE+G6, respectively. This shows that only approximately 26-33% of the total decohesion energy stems from vdw interactions. In comparison, the decohesion energy of a bilayer of hittorfene is dominated by vdw interactions ( 9%), with similar numbers for cleaving a bilayer of black phosphorene. The following on-the-fly-generated pseudopotential strings were used in our calculations: :3:32[] and N:3N:32N[] Hittorf s phosphorus The crystal structure of violet or Hittorf s phosphorus was first determined by Thurn and Krebs (ICSD code 29273). 2 We show the crystal structure in Fig. S using the same color coding as for hittorfene in the main text. The lattice constants and angles are shown for PBE, PBE+TS and PBE+G6 in Table S2. Table S2: Crystal structure of bulk violet or Hittorf s phosphorus. PBE PBE +TS PBE+G6 a [Å] b [Å] c [Å] α [ ] β [ ] γ [ ] In contrast to hittorfene, including vdw corrections is much more important in calculations of bulk Hittorf s phosphorus. Fig. S2 a) and b) show the band structure of bulk Hittorf s phosphorus using PBE+TS and PBE, respectively. We find significant differences: The band gaps are.45 ev and.82 ev for PBE+TS and PBE, respectively, a difference of S3

4 approximately 26% (the difference for hittorfene is only %). More importantly the type of band gap changes from direct to indirect when dispersion corrections are added. The significant change in the value of the band gap can be understood in terms of the strain behavior of Hittorf s phosphorus. Fig. S3 c) shows the band gap as a function of strain perpendicular to the layers: A significant change in the value of the band gap can be induced. This plot also shows the value of the PBE band gap as an x at 7.4% strain, which is the difference of the perpendicular lattice constant in comparison to the results with PBE+TS. Strain in the xy plane of Hittorf s phosphorus has a significantly smaller effect on the band gap (Fig. S3 (b)), whilst xy strain has a much stronger effect on the band gap of hittorfene (Fig. S3 (a)). The latter may be of importance for lattice matching of hittorfene to substrates. Figure S: Crystal structure of bulk violet or Hittorf s phosphorus. Violet spheres represent phosphorus atoms, whilst the yellow tubes are added as guides to the eye only to emphasize the fibrous nature of Hittorf s phosphorus. The unit cell is shown as a black wireframe. S4

5 (a) (b) Energy (ev) Energy (ev) H C Y H M E Γ X D H 2 Z A Y D M - - Γ Y H C E M A XH M D Z Y D Γ Y HC E M A X H M D Z Y D Figure S2: (a) PBE+TS and (b) PBE band structure and density of states for bulk Hittorf s phosphorus. A schematic of the reciprocal lattice with the high symmetry points indicated is shown. The band gap changes from indirect to direct band gap (marked by grey arrows). (a).85 (b).5 (c) 2 band gap [ev] biaxial strain x-direction strain y-direction strain band gap [ev] x-direction strain biaxial strain y-direction strain band gap [ev] PBE results strain [%] strain [%] strain [%] Figure S3: Band gap as a function of strain for (a) hittorfene with strain in the xy plane, (b) Hittorf s phosphorus with strain in the xy plane and (c) Hittorf s phosphorus perpendicular to the xy plane. All results using PBE+TS except for the result of the band gap of fully relaxed Hittorf s phosphorus calculated with PBE shown as an x and labelled in (c). Solid squares in (c) are for indirect band gaps C Z, whilst empty squares change to other indirect band gaps. S5

6 (a) 3 (b). Band gap [ev] Hittorf (HSE) Hittorf (PBE+TS) black (HSE) energy [ev/atom] Hittorf black black (TS) black (G6) Hittorf (TS) Hittorf (G6) Bulk n 2 3 n 4 (c) Band gap [ev] Hittorf (HSE) Hittorf (PBE+TS) black (HSE) (d). energy [ev/atom] black (TS) black (G6) Hittorf (TS) Hittorf (G6) /h [/Å] /h [/Å] Figure S4: a) Band gap of Hittorf s and black phosphorus as a function of layers, n. Both HSE and PBE+TS results are shown for Hittorf s phosphorus. The HSE data of black phosphorus shown are taken from Qiao et al. 3 b) Relative energy of Hittorf s and black phosphorus with n layers with respect to their bulk forms (as in Fig.2 b) of the main text). c) Band gap of Hittorf s and black phosphorus as a function of inverse thickness (as in Fig.2 a) of the main text). Both HSE and PBE+TS results are shown for Hittorf s phosphorus. The HSE data of black phosphorus shown are based on the band gap results of Qiao et al. 3 in combination with their respective thickness taken from our own PBE+TS results. d) Relative energy of Hittorf s and black phosphorus with respect to their bulk forms as a function of inverse thickness. S6

7 References () Bachhuber, F.; von Appen, J.; Dronskowski, R.; Schmidt, P.; Nilges, T.; Pfitzner, A.; Weihrich, R. The extended stability range of phosphorus allotropes. Angewandte Chemie (International ed. in English) 24, 53, (2) Thurn, H.; Krebs, H. Über Struktur und Eigenschaften der Halbmetalle. XXII. Die Kristallstruktur des Hittorfschen Phosphors. Acta Crystallographica Section B Structural Crystallography and Crystal Chemistry 969, 25, (3) Qiao, J.; Kong, X.; Hu, Z.-X.; Yang, F.; Ji, W. High-mobility transport anisotropy and linear dichroism in few-layer black phosphorus. Nature communications 24, 5, S7