Phonon Green’s Function Method for Multiscale Modeling of Color Centers in Nanodiamonds

Nanodiamonds are materials of strong topical interest because of their unusual properties and diverse potential applications in biomedical, communications, and sensing devices. Their dimensions range from a few to several hundred nanometers. There is special interest in color centers in nanodiamonds, which give them unique photonic and spin characteristics. These characteristics are useful for their applications in optical and quantum devices.<br/>Color centers are lattice defects such as vacancies. A lattice defect causes a break in the translation symmetry of the lattice. Consequently, it causes a distortion or strain in the lattice, which also distorts the electronic wave functions and perturbs the associated energy levels. This effect is likely to be a serious material issue, which can affect the reliability and the performance of the finished device based upon the use of nanodiamonds. It is, therefore, of paramount importance to develop modeling and measurement techniques of the lattice distortion/strain field due to a color center in nanodiamond, which is the objective of this work.<br/>A unified theory for lattice defects in nanodiamonds is a challenging problem because the nanodiamonds can be too small or too big for a conventional theoretical treatment. A 5 nm nanodiamond is too small for a standard bulk material continuum model to be accurate. One must use a discrete lattice theory such as molecular dynamics (MD). On the other hand, modeling a 100 nm nanodiamond, containing up to a billion atoms, using ordinary MD will be a formidable task, even with modern computer.<br/>Further, in addition to variations in the size, there is also a need for seamless linking of the length scales in the same nanodiamond. Lattice distortion is a discrete function expressed as the displacement of atoms from their positions of equilibrium and needs to be calculated by accounting for the atomistic structure of the lattice. On the other hand, strain is a continuum model parameter defined in terms of the derivatives of the continuous displacement field. The inherent assumption in a continuum model is that the discrete lattice distortion is smeared out into a continuum, which is valid only at distances much larger than the interatomic separations. For the final results to be physically realistic, the near field and far field formulation must be seamlessly linked.<br/>We describe a Green’s function (GF) method for modeling a color center in a nanodiamond of arbitrary size. The model is computationally efficient and can simulate even several hundred million atoms on an ordinary computer. It links the different length scales smoothly and seamlessly so can be used to estimate the nanodiamond size effect. We calculate the lattice GF by using a simple Born-von Karman model, in which each atom interacts with up to its second neighbor atoms. We also assume the harmonic approximation and the adiabatic approximation.<br/>In this model, the main parameters are the interatomic force constants. These are given by the derivatives of the total potential energy, evaluated at the positions of equilibrium for each atom. In practice, the force constants are treated as adjustable parameters, which are determined phenomenologically by fitting them to elastic constants and phonon dispersion. Hence, an explicit knowledge of the interatomic potential is not needed. Our selected set of the force constants correctly reproduces the measured elastic constants of bulk diamond and gives a reasonable agreement with the observed phonon spectrum over the first Brillouin zone.<br/>Numerical results will be presented for the strain field due to a color center, specifically a single vacancy, in a nanodiamond. The size effect on the strain field and the limitations and applicability of our model to interpretation of measurements will be briefly discussed.