Question

Phosphorus is diffused into a thick slice of silicon with no previous phosphorus in it. If the surface concentration of the phosphorus is 1 x 10^18 atoms/cm^3, and its concentration at 1 micrometer needs to be 1 x 10^15 atoms/cm^3, after a diffusion time of 1 hour. What is the temperature needed for the process?

Answer 1

Given: Concentration of Phosphorus at surface, C_s=1*10¹⁸ atoms/cm³=10¹⁸ atoms/cm³

Nominal or Initial Concentration of Phosphorus, C₀=0

Distance below the slice of silicon, x=1m=10^-6 m=10^-4 cm

Concentration of Phosphorus at a distance x, C_x=1*10¹⁵ atoms/cm³=10¹⁵ atoms/cm³

Time for diffusion, t=1 hr=3600 s

Temperature required for Diffusion can be calculated using the expression

where D is the Diffusion Coefficient

D₀ is the Temperature Independent pre-exponential factor

Q_d is the Activation Energy

R is Universal Gas Constant, R=8.314 J/mol K

T is the Temperature

To calculate temperature, we first have to calculate the diffusion coefficient or diffusivity of Phosphorus in Silicon.

From Fick's Second Law for Inter Diffusion,

where C_x is the Concentration at a distance x after time t

C₀ is the Nominal or Initial Concentration

C_s is the Surface Concentration

is the Gaussian Error Function

Substituting the values in above equation, we get

Let

0.9990=erf z

To find erf(0.9990) interpolate the values from the Gaussian Error Function Table, we have erf(0.9981)=2.2 and erf(0.9993)=2.4

z=2.35

Substitute the value for z and solve

D=1.2559*10^-13 cm²/s=1.26*10^-13 cm²/s

Diffusion Coefficient, D=1.26*10^-13 cm²/s

To calculate temperature, substitute the values obtained in the below expression

(or)

From the Diffusivity of Impurities in Silicon table, obtain the values of Q_d and D₀

For Phosphorus in Silicon, D₀=3.85 cm²/s and Q_d=3.66 eV=353.14 kJ/mol

Substitute the values in above expression,

T=1367.9409 K 1368 K

Temperature required for this diffusion process is 1368 K.