United model for formation kinetics of oxygen thermal donors in silicon

Abstract

A new model is proposed for the formation and decay kinetics of the thermal donors in silicon. This model is an extension of the Kaiser-Frisch-Reiss (KFR) model [Phys. Rev. 112, 1546 (1958)] and includes the effect of the electronic environment hitherto neglected. The concentration of the thermal donors $n_{\mathrm{TD}}\mathrm{}\left(t\right)\mathrm{}$ is expressed by $n_{\mathrm{TD}}\mathrm{}\left(t\right)=\left(\frac{a}{b}\right){\left[{\mathrm{O}}_i\right]}^3{n}^{-2\mathrm{}}\{1-\exp (-b{D}_i\left[{\mathrm{O}}_i\right]t\left)\right\}$, where [${\mathrm{O}}_i$] denotes the initial oxygen-interstitial concentration; $n$, the electron concentration; $D_i$, the diffusion coefficient of oxygen interstitials; and $a$ and $b$, constants. Based on this equation, the maximum concentration ${\left[n_{\mathrm{TD}}\right(t=\infty \left)\right]}_{\mathrm{eq}}$ and initial formation rate ${\left[\frac{d{n}_{\mathrm{TD}}\mathrm{}\left(t\right)\mathrm{}}{\mathrm{dt}}\right]}_{t=0\mathrm{}}$ are expressed, respectively, by ${\left[n_{\mathrm{TD}}\right(t=\infty \left)\right]}_{\mathrm{eq}}=\left(\frac{a}{b}\right){\left[{\mathrm{O}}_i\right]}^3{n}^{-2\mathrm{}}$ and ${\left[\frac{d{n}_{\mathrm{TD}}\mathrm{}\left(t\right)\mathrm{}}{\mathrm{dt}}\right]}_{t=0\mathrm{}}=a{D}_i{\left[{\mathrm{O}}_i\right]}^4{n}^{-2\mathrm{}}$. The equations are used to derive the annealing-temperature and dopant-concentration dependences as well as the oxygen-concentration dependence of thermal-donor formation. Enhancement of the thermal donor formation in heavily doped, $p$-type materials and its suppression in heavily doped, $n$-type materials are theoretically deduced and the latter is experimentally confirmed by deep-level transient (capacitance) spectroscopy (DLTS). It is found that the aggregates with four oxygen atoms are doubly charged donors. It is further suggested that the early aggregates with two or three oxygen atoms would be doubly charged donors and would play the role of the predominant thermal donors in the initial formation stage at a temperature lower than 450°C.