Effect of weak positional disorder on the miniband structure of spherical quantum dot chains

A theoretical framework is developed for the electron miniband structure in one-dimensional chains of spherical quantum dots subjected to weak positional disorder. Within the tight-binding approximation and effectivemedium approach, stochastic fluctuations of the inter-dot spacing are mapped onto the renormalization of key Hamiltonian parameters: the hopping integral B, overlap integral Q, and on-site energy shift M. Analytical expressions for these parameters are derived by performing an ensemble average over a narrow Gaussian distribution of positional deviations (σ ≪ a). The resulting generalized dispersion relation reveals that weak positional disorder broadens the minibands. For typical fabrication fluctuations σ = 0.1a, the miniband width increases by 8-12% , depending on the mean inter-dot distance a. This sensitivity to disorder decreases rapidly with increasing lattice period due to the exponential decay of electron wave functions. In this weak-disorder regime, the Anderson localization lengh significantly exceeds the lattice constant, meaning states remain delocalized.