Constant-Source Diffusion

 

 

Fig 2 Constant-source diffusion results in a complementary error function impurity distribution. The surface concentration N0 remains constant and the diffusion moves deeper into the silicon wafer as the Dt product increases. Dt can change as a result of increasing diffusion time, increasing diffusion temperature, or a combination of both. [1]

 

A constant-source diffusion or constant surface concentration or solid solubility limited predeposition / diffusion results in a complementary error function impurity distribution. The surface concentration N0 remains constant and the diffusion moves deeper into the silicon wafer as Dt product increases. Dt can change as a result of increasing diffusion time, increasing diffusion temperature, or a combination of both.

 

Initial Condition:

 

At time t=0, impurity concentration at depth x and time t, that is, N(x,t) is given by,

 

N (x, 0) = 0

 

Boundary Conditions:

 

At surface we have N(0,t) = N0 and N(¥,t) = 0 for x=¥

 

The solution of Fick's second law that satisfies the initial and boundary conditions is given by

 

                  

 

Where N0 is the surface concentration (atoms/cm2)

          D is constant diffusivity in cm2/s

          x is the distance in cm

          t is the diffusion time in seconds

 

The table below gives some values of z= x/2(Dt)1/2 and erfc(z)

                          z

              erfc(z)

                          0

                 1.0

                          0.5

                 0.5

                          1.0

                 1.7

                          1.5

                 0.35

                          2.0

                 0.005

                          2.5

                 0.0004

                          3.0

                 0.00002

                           ¥

                 0