We report the development and application of a new method for carrying out computational investigations of the effects of mass and force-constant (FC) disorder on phonon spectra. The method is based on the recently developed typical medium dynamical cluster approach (TMDCA), which is a Greens function approach. Excellent quantitative agreement with previous exact diagonalization results establishes the veracity of the method. Application of the method to a model system of binary mass and FC-disordered system leads to several findings. A narrow resonance, significantly below the van Hove singularity, that has been termed as the boson peak, is seen to emerge for low soft particle concentrations. We show, using the typical phonon spectrum, that the states constituting the boson peak cross over from being completely localized to being extended as a function of increasing soft particle concentration. In general, an interplay of mass and FC disorder is found to be cooperative in nature, enhancing phonon localization over all frequencies. However, for certain range of frequencies, and depending on material parameters, FC disorder can delocalize the states that were localized by mass disorder, and vice-versa. Modeling vacancies as weakly bonded sites with vanishing mass, we find that vacancies, even at very low concentrations, are extremely effective in localizing phonons. Thus, inducing vacancies is proposed as a promising route for efficient thermoelectrics. Finally, we use model parameters corresponding to the alloy system, Ni1-xPtx, and show that mass disorder alone is insufficient to explain the pseudogap in the phonon spectrum; the concomitant presence of FC disorder is necessary.
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