next up previous contents
Next: 3.1.2 Sources and Sinks Up: 3.1 Diffusion in Silicon Previous: 3.1 Diffusion in Silicon

3.1.1 Intrinsic Point Defects

One of the fundamental achievements in the understanding of point-defect diffusion mechanisms in silicon was the realization that both types of point-defects, interstitials and vacancies, are involved in the self-diffusion process [See68] [Fra84]. The formation of intrinsic point defects is governed by the principle of minimizing the total Gibbs free energy of the silicon crystal. Any destruction of the perfect silicon lattice increases the Gibbs free energy. The existence of point-defects on the other hand increases the randomness of the crystal and hence can decrease the Gibbs free energy via entropy. The thermal equilibrium concentrations of the point-defects tex2html_wrap_inline5005 and tex2html_wrap_inline5007 are given by a balance of entropy and enthalpy (3.1-1) - (3.1-3), where H and S are the formation enthalpy and entropy, respectively, T is the lattice temperature and k the Boltzmann constant. Quantity tex2html_wrap_inline5017 gives the number the equivalent defect locations.

    eqnarray686

Up to now there is still a lack of determing these equilibrium concentrations by experiments. The most common technique for measuring the equilibrium concentrations is to use the silicon self-diffusion phenomenon, where the diffusion of radioactive silicon isotopes is traced until equilibrium is reached. By assuming that both interstitials and vacancies diffuse, the diffusivity of silicon is given by (3.1-4), where tex2html_wrap_inline5019 and tex2html_wrap_inline5021 are the diffusivities of interstitials and vacancies, respectively, tex2html_wrap_inline5023 is the number of lattice sites, and tex2html_wrap_inline5025 and tex2html_wrap_inline5027 are correlation factors of diffusion for each mechanism involved [Hu85].

   eqnarray699

The temperature dependence of the self-diffusion coefficient tex2html_wrap_inline5029 shows good agreement which an Arrhenius expression (3.1-5). However, the tracer experiments cannot distinguish between the contribution of interstitials and vacancies to the self-diffusion. It was found that dopants which diffuse mainly by an interstitial interchange process can be used to monitor the self-diffusion of interstitials. Some researches performed experiments using gold [Sto84] [Zim89] and others used platinum in-diffusion for silicon self-diffusion studies [Zim92] [Zim93]. These experiments lead to a fairly rough estimation for tex2html_wrap_inline5031 (3.1-6). Data for the vacancy concentrations (3.1-7) came from the positron annihilation study of Dannefaer [Dan86] or tracer experiments [Mor83].

   eqnarray725

It should be pointed out that no measurement technique has directly extracted the equilibrium concentrations of interstitials and vacancies in silicon. It is still facing research today to determine tex2html_wrap_inline5005 and tex2html_wrap_inline5007 over the range of temperature involved in nowadays thermal processing.

Point-defects exist in silicon either charged or uncharged. Even multiple charge states are possible, e.g. by a doubly charged vacancy tex2html_wrap_inline5037 or during precipitation and clustering processes (see Section 3.1.5). The energy associated with the presence or absence of an electron is directly related to the background energy of the crystal. Therefore, each charged species has different equilibrium values. The equilibrium concentration of the charged species depends upon the Fermi level and can be expressed, e.g. for doubly positively charged vacancies, by (3.1-8), where tex2html_wrap_inline5039 is the Fermi energy and tex2html_wrap_inline5041 the respective energy level in the band gap for the charged species.

  equation745

It is possible for a non-degenerate semiconductor, to sum up the r charged intrinsic point defect states to one equilibrium level (3.1-9),

  equation755

where tex2html_wrap_inline5045 gives the intrinsic carrier density, n and p the electron and hole concentration, respectively.

Each charge state may also be related to a different diffusivity tex2html_wrap_inline5051 . Like the equilibrium concentrations, we can also lump the diffusivities of the different charged point defects together to one effective diffusivity tex2html_wrap_inline5053 (3.1-10).

  equation768

(3.1-10) implies that electronic interactions are occurring on a much faster time scale than chemical diffusion, hence, charge neutrality is accomplished.


next up previous contents
Next: 3.1.2 Sources and Sinks Up: 3.1 Diffusion in Silicon Previous: 3.1 Diffusion in Silicon

IUE WWW server
Fri Jul 5 17:07:46 MET DST 1996