Supplementary Information for: On the Exfoliation and Anisotropic Thermal Expansion of Black Phosphorus.

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1 Eletroni Supplementry Mteril (ESI) for ChemComm. This journl is The Royl Soiety of Chemistry 218 Supplementry Informtion for: On the Exfolition nd Anisotropi Therml Expnsion of Blk Phosphorus. Giuseppe Snsone, 1 Antti Krttunen, 2, 3, 4, Jn Gerit Brndenurg, Denis Usvyt, 5 Mrtin Shütz, 5 nd Lorenzo Mshio 1, 1 Diprtimento di Chimi nd NIS (Nnostrutured Interfes nd Surfes) Centre, Università di Torino, vi Giuri 5, I-11 Torino, Itly 2 Deprtment of Chemistry nd Mterils Siene, Alto University, FI-76 Alto, Finlnd 3 Thoms Young Centre, University College London, Gower Street, London WC1E 6BT, United Kingdom 4 Institute for Physil Chemistry, University of Göttingen, Tmmnnstr. 6, 3777 Göttingen, Germny 5 Institut für Chemie, Humoldt-Universität zu Berlin, Berlin, Germny (Dted: July 2, 218) I. FURTHER COMPUTATIONAL DETAILS All quntum hemil lultions were performed within the frmework of Density Funtionl Theory (DFT), using the the CRYSTAL17 ode[1] nd the VASP5.4 softwre suite.[2, 3] CRYSTAL17 lultions were rried out using Γ-entered k-meshes ording to the Monkhorst-Pk method to smple of the first Brillouin zone (FBZ).[9] The pplied TZVPP level sis set is reported in Ref. [1]. A lrger sis set, nmed TZVPP2 s reported in Ref. [4] hs een lso used for evluting ny sis set defiieny. From the results in Tle S1, nd from Figures S4, S6 nd S1, it is ler tht the use of different sis set hs smll effet on the solute vlues, nd lmost negligile on generl trends. Signifint differenes in the therml expnsion re oserved only t tempertures higher thn 4 K. The five thresholds T i, whih ontrol the truntion riteri of the Coulom nd exhnge infinite lttie series, hve een set to 8 (T1-T4) nd 16 (T5).[11]. In Figure S2 we show tht the PBE-D3(zd) yields lmost identil results on the therml expnsions s the PBE-D3(zd,tm) vrint. VASP lultion were rried out y using hrd pseudopotentils (PAW PBE P 6Sep2 ), n energy utoff of 1 ev. 8x8x8 Gmm-entered k-points were used for the ulk, nd 8x8x1 for the lyer lultions. In the ltter se, -xis of Å ws hosen, whih ws tested to e erged with respet to 4 Å. An urte FFT grid ws dopted. The hoie of the superell expnsion mtrix[13] nd the eletroni k-meshes re reported in T. S2. II. ADDITIONAL RESULTS As omplement to the results reported in the min text, in the following some dditionl results re reported in Figures S3, S5, S7, S8. These inlude the therml expnsion oeffiients α V nd α L, the temperture dependene of the ulk modulus, nd the het pity C p. These plots show lso the dependene on the superell size nd shpe nd, s previously mentioned, the stility of the results with respet to hnge in the sis set hoie. All results, if not differently stted, hve een otined t the PBE-D3(zd, tm) level. [1] R. Dovesi, A. Er, R. Orlndo, C. M. Ziovih-Wilson, B. Civlleri, L. Mshio, M. Ferrone, M. Rért, S. Csss, J. Bim, et l., Wiley Interdisiplinry Reviews: Computtionl Moleulr Siene p. e136 (218). [2] G. Kresse nd J. Hfner, Phys. Rev. B 47, 558 (1993). [3] G. Kresse nd J. Furthmüller, J. Comp. Mt. Si. 6, 15 (1996). [4] G. Snsone, L. Mshio, D. Usvyt, M. Shütz, nd A. Krttunen, The journl of physil hemistry letters 7, 131 (215). [5] A. Brown nd S. Rundqvist, At Crystllogr. B 19, 684 (1965). Eletroni ddress: ntti.j.krttunen@iki.fi Eletroni ddress: gerit.rndenurg@thh.uni-onn.de Eletroni ddress: lorenzo.mshio@unito.it

2 2 Tle S1: Struturl dt nd exfolition energy of lk-p for different first-priniples methods. Some theoretil nd experimentl dt ville in literture re reported too. sis set TZVPP2[4] TZVPP[4] MINIX def2-msvp def2-msvp def2-mtzvp method V E rlx exf funtionl D3 version (Å) (Å 3 /tom) (mev/tom) zd zd+tm j PBE-D3 j+tm zd zd+tm j j+tm PBE zd HF-3 zd+tm j j+tm PBEH-3 zd+tm j+tm HSE-3 zd+tm j+tm B97-3 zd+tm j+tm Plne Wve utoff=1 KeV PBE SCAN D3 (zd+tm) TS MBD D3 (zd+tm) D3 (j+tm) Literture Exp. Theo. powder diffr. Ref. [5] powder diffr TOF neutron diffr. Ref. [6] QMC Ref. [7] -8 rccd Ref. [8] -92 p-lmp2 Ref. [4] Strting from the experimentl struturl dt t room temperture. Strting from the p-lmp2 optimised dt. vlues t K temperture, estimted y orreting the results in the line ove for temperture effets s estimted y the QHA method in this pper. Tle S2: The hoies of the Super Cells (SCs) nd k smpling for eh lultion. Superells Prmeters 2x2x2 3x3x3 SC expnsion mtrix k-mesh 4x4x4 3x3x3 2x2x2 The SC expnsion mtrix of the lk phosphorus hs een set in order to perform the lultions on the rystllogrphi unit ell nd not on the primitive unit ell. [6] L. Crtz, S. R. Srinivs, R. J. Riedner, J. D. Jorgensen, nd T. G. Worlton, J. Chem. Phys. 71, 1718 (1979). [7] L. Shulenurger, A. Bzewski, Z. Zhu, J. Gun, nd D. Tománek, Nno letters 15, 817 (215). [8] M. Shütz, L. Mshio, A. J. Krttunen, nd D. Usvyt, The Journl of Physil Chemistry Letters 8, 129 (217). [9] H. J. Monkhorst nd J. D. Pk, Phys. Rev. B 13, 5188 (1976). [1] G. Snsone, L. Mshio, nd A. J. Krttunen, Chem. Eur. J. 23, (217). [11] R. Dovesi, V. R. Sunders, C. Roetti, R. Orlndo, C. M. Ziovih-Wilson, F. Psle, B. Civlleri, K. Doll, N. M. Hrrison, I. J. Bush, et l., Crystl17 user s mnul (university of torino, torino, 217), [12] C. Stephenson, R. Potter, T. Mple, nd J. Morrow, J. Chem. Thermodynmis 1, 59 (1969). [13] A superell is otined y defining the new unit ell vetors s liner omintions of the primitive ell unit vetors.[11]

3 3.6.4 TZVPP TZVPP2 (x-x )/x (%) NO QHA Figure S1: Reltive lttie prmeter therml expnsion s otined for the 2x2x2 superell with TZVPP nd TZVPP2 sis sets, respetively x2x2 (x-x )/x (%) NO QHA Figure S2: Reltive lttie prmeter therml expnsion s otined for the 2x2x2 nd superell with PBE-D3(zd) funtionl.

4 4 2 2x2x2 3x3x3 α V (1-6 /K) Figure S3: Volume therml expnsion oeffiient s otined with the TZVPP sis set for the 2x2x2,, nd 3x3x3 superells, respetively. 2 TZVPP TZVPP2 α V (1-6 /K) Figure S4: Volume therml expnsion oeffiient s otined for the 2x2x2 superell with the TZVPP nd TZVPP2 sis sets, respetively.

5 α L (1-6 /K) x2x2 3x3x3 Figure S5: Liner therml expnsion oeffiients of eh lttie prmeter s otined with the TZVPP sis set for the 2x2x2,, nd 3x3x3 superells, respetively α L (1-6 /K) TZVPP TZVPP2 Figure S6: Liner therml expnsion oeffiients of eh lttie prmeter s otined for the 2x2x2 superell with the TZVPP nd TZVPP2 sis sets, respetively.

6 6 28 3x 2x B (GP) NO QHA Figure S7: Bulk modulus s otined with the TZVPP sis set for the 2x2x2,, nd 3x3x3 superells, respetively. C p (J K -1 mol -1 ) HA- QHA- QHA-3x QHA-2x Exp Figure S8: Het pity s otined with the TZVPP sis set nd QHA pproh for the 2x2x2 (tringles), (squres), nd 3x3x3 (irles) superells, respetively. For omprison, the HA results for the superell (pluses) nd those reported experimentlly y Stephenson et l. (rosses) in Ref. [12]