DIFFUSION FROM A isfactory agreement is found between the predictions of the model and measured parameters such as the junction depth. In the particular samples studied here the concentration profile in the silicon is expected to be nearly Gaussian, especially at the longer diffusion times. A Gaussian profile is not always to be expected, however, and the theoretical model shows that the important quantity is the diffusion length k/Dlt, in the oxide. When the diffusion length is small compared with the thickness of the oxide (i.e., short diffusion times) an ERFC profile will result.

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The disagreement between experimental measurement of the reflectivity minimum of low-mobility semiconductors and simple theory is accounted for by inclusion of the imaginary part of the dielectric constant in deriving the carrier-concentration/plasma-wavelength relationship. Data on the plasma wavelength of p-type gallium arsenide are presented. INTRODUCTIONT HE carrier concentration of heavily doped semiconductors can be inferred from the minimum in the infrared reflection spectrum (the plasma frequency) of polished samples, which is usually defined by the following approximation:where q is the electronic charge, m* is the effective mass, N is the carrier density, EO is the free-space permittivity, and EL is the dielectric constant in the absence of carriers. Although this equation is accurate for high-mobility materials such as n-type InSb, GaAs, and Si, the carrier concentrations deduced from Hall measurements and from Eq. (1) differ by an order of magnitude in low-mobility semiconductors such as p-type GaAs, n-type GaP, and p-type Si. This paper derives the relationship between plasma frequency and carrier concentration for these materials. THEORETICALThe general expression for the dielectric constant E of a material having a carrier concentration N is given by This equation is valid in the range of frequencies for which EL is a constant and real, i.e., below the absorption edge and above the reststrahlen.The real and imaginary parts of Eq.(2) can be expressed as follows:where r, the relaxation time, is equal to m*p./q. For high-mobility samples, therefore, r is large, wr»1, 2nk is small, and the index of absorption is negligible; as a result, Eq. (3) reduces to Eq. (1) when n 2 = 1 (at Wmin)'In the case of low-mobility samples, the term 2nk becomes appreciable and must be included in the dielectric constant. Although the relaxation time has been * Portions of the work described were supported by the Air Force Aero Propulsion Laboratory, Research and Technology Division, Air Force Systems Command, U. S. Air Force.shown to be an energy-dependent quantity, Brooks 2 has shown that this dependence can be taken into account by multiplying Eq. (4) by a constant 'Y which is determined by the dominant scattering mechanism. In the impure samples used in the work described, ionized impurity scattering dominates and "I is approximately 3.4. Equations (3) and (4) can be solved simultaneously with the inclusion of "I, as follows:where a= (41rq2)/(m*EO)=3.2X10 9 (m/m*). The reflectivity R is then given by R=[(n-1)2+k 2 ]/[(n+l)2+k 2 ]. (7) Equation (7) can be differentiated to produce the following results: dR 4[(n 2 -,k2-1) (dn/dw)+2nk(dk/dw)] (8) dw and where W=Wmin, w 2 r2»1, and f=l at the reflection minimum. For a complete solution for Wmin, dk/dn must be evaluated. Differentiation of Eq. (6) with respect to wavelength provides the equation for dk/dn and gives rise to a complex set of terms, which can tben be simplified by use of the following assumptions: a2N2'Y2/wr2<3aN'Y2/2w2r2