Dynamics of self-trapped excitons in layered Pb$_{\mathrm{1-X}}$Cd$_{\mathrm{X}}$I$_{2}$ semiconductors

The dynamics of self-trapped excitons, localized on stretched Pb-I chemical bonds which are formed on PbI$_{2}$ nanocluster surface was investigated. It should be noted that these nanoclusters are naturally formed in Pb$_{\mathrm{1-X}}$Cd$_{\mathrm{X}}$I$_{2}$ (X~$=$~0.5 and X~$=$~0.7) layered semiconductor solid solutions. They have different sizes (from several nm to several tens of nm). The measurements of photoluminescence (PL) spectra and kinetics of PL intensity decay for those materials were performed at T$=$300 K. The kinetic dependencies were obtained for the maximum of PL band (600 nm) and for its short-wave shoulder (550 nm). It was shown that PL decay kinetics is approximated by Kohlrausch-Williams-Watts (KWW) function, i.e. by stretched exponential function I(t)~$=$~I$_{0}$exp(-(t/$\tau _{\mathrm{ef}})^{\beta})$. Obtained values of $\tau _{\mathrm{ef}}$ and $\beta $ for X~$=$~0.5 are equal about 800~ns and 0.76 at 600~nm. At 550 nm these values are 700~ns and 0.74, respectively. Similarly for X~$=$~0.7 these values correspond about 800~ns and 0.80 at 600~nm. At 550 nm they are 800~ns and 0.82. Application of the Inverse Laplace Transformation (ILT) to our experimental data gave us opportunity to estimate the probability density function of self-trapped exciton state lifetimes for Pb$_{\mathrm{1-X}}$Cd$_{\mathrm{X}}$I$_{\mathrm{2}}$ (X $=$ 0.3, 0.5 and 0.7). The position of the maximum of F($\tau )$ gives us the average decay time \textless t\textgreater which is about 1250~ns which significantly differs from $\tau_{\mathrm{ef}}$ which is about 800~ns (for $\lambda $~$=$~550~nm). This complex dynamics of excitons is associated with strong heterogeneity of the investigated system.