Page 48 - Microsensors, MEMS and Smart Devices - Gardner Varadhan and Awadelkarim

30 ELECTRONIC MATERIALS AND PROCESSING
2.5.1 Diffusion

In a diffusion process, the dopant atoms are placed on the surface of the semiconductor by
deposition from the gas phase of the dopant or by using doped oxide sources. Diffusion
of dopants is typically done by placing the semiconductor wafers in a furnace and passing
an inert gas that contains the desired dopant through it. Doping temperatures range from
800 to 1200°C for silicon. The diffusion process is ideally described in terms of Pick's
diffusion equation
2
dC d C
dt dx 2 (2.21)
where C is the dopant concentration, D is the diffusion coefficient, / is time, and x is
measured from the wafer surface in a direction perpendicular to the surface (Figure 2.16).
The initial conditions of the concentration C(x, 0) = 0 at time t = 0 and the boundary
conditions are that surface concentration C(0, t) = C s at surface and that a semi-infinite
medium has C(oo, t) = 0. The solution of Equation (2.21) that satisfies the initial and
boundary conditions is given by

C(x, t) = C serfc (2.22)
where erfc is the complementary error function and the diffusion coefficient D is a function
of temperature T expressed as
D = D 0 exp(—E a /kT) (2.23)

where E a is the activation energy of the thermally driven diffusion process, k is
Boltzmann's constant, and DO is a diffusion constant. The diffusion coefficient is

t 2 > t 1 > 0

Figure 2.16 Theoretical diffusion profile of dopant atoms within a silicon wafer