Ion Implantation and Synthesis of Materials - Studium

10.2 Epitaxial Growth of Implanted Amorphous Si 135of temperature. These energy levels are estimated values but are sufficient fordemonstration purposes.The concentration of charged vacancies is governed by Fermi–Dirac statisticsand is given (using V − as an example) by−[ V ]=1 + e[ V ]T( E− − EF)/ kT(10.4)where E F is the Fermi level, E − is the energy level of the acceptor state vacancy,and [V T ] is the total vacancy concentration:[ V ] = [ V ] + [ V ] + [ V ] + [ V ](10.5)Tx− = +After some algebraic manipulation of (10.4) and (10.5) we find that−x[ V ] = [ V ]exp[( E − E )/ kT ](10.6)F−and similarly,+ x +[ V ] = [ V ]exp[( E − E )/ kT ](10.7)FUpon examining (10.6), we find that the concentration [V − ] is large compared to[V x ] only when (E F – E − ) >> kT and positive. This means that [V − ] is significantwhen the Si sample is strongly n-type (i.e., E F is located far above E − ). As the Sisample becomes less n-type and E F is far below E − , [V − ] decreases accordingly.For p-type samples where E F is far below E − , [V − ] is negligibly small comparedto [V x ].As an example of (10.6), consider Fig. 10.7 where the amorphous Si regrowthrate is enhanced by implanted P. Taking the P concentration as 2.5 × 10 20 cm −3and 550°C (823 K) as the regrowth temperature, we find that the donor density isequal to the density of states in the conduction band (Mayer and Lau), whichlocates E F at E C .Referring to Fig 10.8, at 823 K, the band gap (see the energy of the conductionband, E c ) of Si is reduced from 1.15 eV at 300 K to 0.91 (Van Vechten 1980). Theenergy level of E − tracks the band gap and is located at 0.37 eV + E v at 823 K.Therefore, the concentration [V − ] is given by (10.6).