Thermal equilibrium concentration of intrinsic point defects in heavily doped silicon crystals - Theoretical study of formation energy and formation entropy in area of influence of dopant atoms-

It is well known that p-type, neutral and n-type dopants affect the intrinsic point defect (vacancy V and self-interstitial I) behavior in single crystal Si. By the interaction with V and/or I, (1) growing Si crystals become more V- or I-rich, (2) oxygen precipitation is enhanced or retarded, and (3) dopant diffusion is enhanced or retarded, depending on the type and concentration of dopant atoms. Since these interactions affect a wide range of Si properties ranging from as-grown crystal quality to LSI performance, numerical simulations are used to predict and to control the behavior of both dopant atoms and intrinsic point defects. In most cases, the thermal equilibrium concentrations of dopant-point defect pairs are evaluated using the mass action law by taking only the binding energy of closest pair to each other into account. The impacts of dopant atoms on the formation of V and I more distant than 1st neighbor and on the change of formation entropy are usually neglected. In this study, we have evaluated the thermal equilibrium concentrations of intrinsic point defects in heavily doped Si crystals. Density functional theory (DFT) calculations were performed to obtain the formation energy (Ef) of the uncharged V and I at all sites in a 64-atom supercell around a substitutional p-type (B, Ga, In, and Tl), neutral (C, Ge, and Sn) and n-type (P, As, and Sb) dopant atom. The formation (vibration) entropies (Sf) of free I, V and I, V at 1st neighboring site from B, C, Sn, P and As atoms were also calculated with the linear response method. The dependences of the thermal equilibrium concentrations of trapped and total intrinsic point defects (sum of free I or V and I or V trapped with dopant atoms) on the concentrations of B, C, Sn, P and As in Si were obtained. Furthermore, the present evaluations well explain the experimental results of the so-called “Voronkov criterion” in B and C doped Si, and also the observed dopant dependent void sizes in P and As doped Si crystals. The expressions obtained in the present work are very useful for the numerical simulation of grown-in defect behavior, oxygen precipitation and dopant diffusion in heavily doped Si. DFT calculations also showed that Coulomb interaction reaches approximately 30 Å from p (n)-type dopant atoms to I (V) in Si.