Kinematical x-ray diffraction in nonuniform crystalline films: Strain and damage distributions in ion-implanted garnets

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Abstract
A kinematical model for general Bragg case x-ray diffraction in nonuniform films is presented. The model incorporates depth-dependent strain and spherically symmetric Gaussian distribution of randomly displaced atoms. The model is applicable to ion-implanted, diffused, and other single crystals. Layer thickness is arbitrary, provided maximum reflecting power is less than ∼6%. Strain and random displacement (damage) distributions in He+-implanted Gd, Tm, Ga:YIG, and Ne+-implanted Gd3Ga5O12 are obtained by fitting the model to experimental rocking curves. In the former crystal the layer thickness was 0.89 μm with strain varying between 0.09 and 0.91%. In the latter crystal a wide range of strain and damage was obtained using successively higher doses. In each case layer thickness was 1900 Å, with 2.49% strain corresponding to 0.40-Å standard deviation of random displacements. The strain distributions were strictly linear with dose. The same, closely linear relationship between damage and implantation-induced strain was determined for both crystals.