Supplementary Figure 1. Picture of crystallization setup. This picture demonstrates, that all 8 vessels containing 1 M MAPbBr 3 in DMF produced less

Transcription

1 Supplementary Figure 1. Picture of crystallization setup. This picture demonstrates, that all 8 vessels containing 1 M MPbBr 3 in DMF produced less than 5 crystals, in particular, 3 of them produced an individual MPbBr 3 crystal. Time-accelerated video of one of these individual crystals growth is presented in Supplementary Movie 2.

2 Supplementary Figure 2. SEM images of the surface and cleaved crystals. SEM of (a) surface of MPbBr 3, (b) cleaved MPbBr 3, (c) surface of MPbI 3 and (d) cleaved MPbI 3. Excess precursors from solution can be observed on the surface of the crystals. No grain boundaries are found on the surface or core of cleaved crystals indicating the single crystalline nature of both crystals. The dusts in (b,d) indicate the good focus of electron beam.

3 Supplementary Table 1. Single crystal XRD data. ompound MPbBr 3 MPbI 3 D calc. / g cm /mm Max Size/mm Mid Size/mm Min Size/mm T/K 250(2) 250(2) rystal System cubic tetragonal Space Group Pm3m I4/m a/å (6) (5) b/å (6) (5) c/å (6) (7) / / / V/Å (6) (12) Z 1 4 Z' min / max / Measured Refl Independent Refl Reflections Used R int Parameters 4 9 Restraints 0 0 Largest Peak Deepest Hole GooF wr 2 (all data) wr R 1 (all data) R Single crystals of both materials have been obtained and their unit cells have been verified by single crystal X-ray diffraction. The unit cell dimensions as well as the space groups (cubic, space group Pm3m, a = (6) Å for MPbBr 3 and tetragonal, space group I4/m, a = b = (5) Å, c = (7) Å for MPbBr 3 ) are in excellent agreement with literature reports 1-3.

4 Supplementary Note 1 The free energy of a system of unit volume containing a number n S of solvent molecules in the solution and numbers n and n of, respectively, isolated -molecules, and the complexes is P P ln ln S ln S S ln G n S n Tn n v Tn n v Tn n v T n n n n v n v n v S S S, (1) where is the cohesive energy of -molecules in the particle, is the surface energy per molecule, is the binding energy of the complexes, S P is the number of -molecules on the surfaces of - particle, T is the solution temperature and vs, v and v are the characteristic volumes of the solvent molecule, the complex, and the -molecule, respectively. It is convenient to introduce total concentrations of - and solvent molecules m n np n and ms ns j n, respectively, where j is the number of solvent molecules in the -solvent complex. Expressing n S and n in terms of the total concentrations allows rewriting equation (1) in the form ln ln S ln S S S P ln S S P S G n S n Tn n v T m n n m n n v P P P P T m jn m jn v T m m n jn v m v m v n n v v jv (2) We are interested in both the equilibrium size of -particles and the concentration n of complexes with solvent. Therefore we choose n P and n as independent variables. Minimization of the free energy with respect to these variables leads to the equations

5 G n P G n 2 T ln nv 0 R T ln n v T ln n v jt ln n v 0 S S (3) The first line gives an equality of the chemical potentials of -molecules in the particle and in the solution T ln n v P 2 /R. The second line states that chemical S potential of the complex is equal to the sum of chemical potentials of -molecule and j solvent molecules involved in complex formation T ln n v. Solving the equations SS S S 2 T ln nv R T ln n v T ln n v jt ln n v S S (4) with respect to n and n gives n n exp v T R 1 j 1 2 n v exp v T R S S (5) These equations contain the particle radius R, which can be expressed in terms of n P. Therefore, these equations can be solved numerically. However, we can understand qualitatively the temperature effect on n P by considering the particle size to be far from its critical value (where the surface energy is really important) when we may set 0. In that limit the number fraction of precipitated molecules is v j npv mv nsvs exp exp. (6) v T T

6 Supplementary References 1. Shi D, et al. Low trap-state density and long carrier diffusion in organolead trihalide perovskite single crystals. Science 347, (2015). 2. Dong Q, et al. Electron-hole diffusion lengths > 175 μm in solution-grown H 3 NH 3 PbI 3 single crystals. Science 347, (2015). 3. Baikie T, et al. Synthesis and crystal chemistry of the hybrid perovskite (H 3 NH 3 ) PbI 3 for solid-state sensitised solar cell applications. J. Mater. hem. 1, (2013).