We investigate photoluminescence (PL) transitions of MAPbX$_{3}$ (X = I, Br and Cl) organic-inorganic hybrid perovskite single crystals under magnetic fields of up to 60 T. In these materials, sharp free-exciton transition peaks emerge at a low temperature (4.2 K). Under strong magnetic fields, the free-exciton PL transitions of three different halogens show dramatic differences. The free-exciton transitions of the MAPbCl$_{3}$ crystal undergo negative energy shifts, while those of the MAPbBr$_{3}$ crystal show normal diamagnetic shifts. To obtain the variation from Cl to Br, we attempt to measure PL transitions of MAPbCl$_{x}$Br$_{3-x}$. For MAPbI$_{3}$, the transition-energy shifts for both $sigma^{+}$ and $sigma^{-}$ transitions at 4.2 K exhibit a power-law dependence on the magnetic field. Such inconsistent magnetic-field effects on different halogens make it difficult to understand the transition-energy behavior through a unified model. We propose a possible mechanism for the field effects that is based on a combination of the Rashba effect induced by strong spin-orbit coupling and the polaron effect caused by the polar nature of the inorganic elements.
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