Explanation Given: The boundary conditions are: Depth = 0.25 μm . The conduction temperature is 1000 ° C . Concentration of P at depth of 0.25 μm is 10 13 atoms / cm 3 . Diffusion coefficient is 2.5 × 10 − 4 cm 2 / sec . Total time is 8 hr . Calculation: The given function is, c ( x , t ) = C s [ 1 − e r f ( x 4 D t ) ] Integrate the above function with limits as follows: c ( x , t ) = C s [ 1 − 2 π ∫ 0 x 2 D t e − y 2 dy ] c ( x , t ) = C s × e r f c ( x 2 D t ) Thus, the equation considered for calculation is, c ( x , t ) = C s × e r f c ( x 2 D t ) Consider the depths values as follows: x = 0.25 μm = 2.5 × 10 − 5 cm x = 0.50 μm = 5 × 10 − 5 cm x = 0.75 μm = 7.5 × 10 − 5 cm x = 1 μm = 1 × 10 − 4 cm x = 1.25 μm = 0.000125 cm x = 1.5 μm = 0.00015 cm x = 1. 75 μm = 0.000175 cm x = 2 μm = 0.0002 cm x = 2.25 μm = 0.000225 cm x = 2.5 μm = 0.00025 cm Concentration with respect to depth and time is calculated as follows: Considering depth as x = 0.25 μm = 2.5 × 10 − 5 cm Value of diffusion coefficient (D) is 2.5 × 10 − 4 cm 2 / sec . Total time for carrying out the process is 8 hr = 8 hr × 60 sec hr = 480 sec . Substituting the given value in the above relation, c ( x , t ) = C s × e r f c ( x 2 D t ) c ( x , t ) = C s × e r f c ( x 2 D t ) c ( x , t ) = 10 21 × e r f c ( 2.5 × 10 − 5 2 2.5 × 10 − 14 × 480 ) c ( x , t ) = 10 21 × e r f c ( 3.60 ) Using table for calculation of error function, e r f c ( 3.60 ) = 0.99 c ( x , t ) = 10 21 × 0.99 c ( x , t ) = 9.9 × 10 20 atoms /cm 3 Thus, concentration at a depth of 0.25 μm is c ( x , t ) = 9.9 × 10 20 atoms /cm 3 . Concentration with respect to depth and time is calculated as follows:. Considering depth as x = 0.50 μm = 5 × 10 − 5 cm Value of diffusion coefficient (D) is 2.5 × 10 − 4 cm 2 / sec . Total time for carrying out the process is 8 hr = 8 hr × 60 sec hr = 480 sec . Substituting the given value in the above relation, c ( x , t ) = C s × e r f c ( x 2 D t ) c ( x , t ) = C s × e r f c ( x 2 D t ) c ( x , t ) = 10 21 × e r f c ( 5 × 10 − 5 2 2.5 × 10 − 14 × 480 ) c ( x , t ) = 10 21 × e r f c ( 7.216 ) When the value of error function exceeds the range of limit 4. The formula used for calculation of error function is, e r f ( x ) = 1 − e − x 2 π x x = 7.216 e r f ( 7.216 ) = 1 − e − 7.216 2 π × 7.216 e r f ( 7.216 ) = 1 − 2.43 × 10 − 23 12.790 e r f ( 7.216 ) = 1 e r f ( 7.216 ) = 0.8427 c ( x , t ) = 10 21 × 0.8427 c ( x , t ) = 8.427 × 10 21 atoms / cm 3 Thus, concentration at a depth of 0.50 μm is c ( x , t ) = 8.427 × 10 21 atoms /cm 3 . Concentration with respect to depth and time is calculated as follows:. Considering depth as x = 0.75 μm = 7.5 × 10 − 5 cm Value of diffusion coefficient (D) is 2.5 × 10 − 4 cm 2 / sec . Total time for carrying out the process is 8 hr = 8 hr × 60 sec hr = 480 sec . Substituting the given value in the above relation, c ( x , t ) = C s × e r f c ( x 2 D t ) c ( x , t ) = C s × e r f c ( x 2 D t ) c ( x , t ) = 10 21 × e r f c ( 7.5 × 10 − 5 2 2.5 × 10 − 14 × 480 ) c ( x , t ) = 10 21 × e r f c ( 10.825 ) When the value of error function exceeds the range of limit 4. The formula used for calculation of error function is, e r f ( x ) = 1 − e − x 2 π x x = 10.825 e r f ( 10.825 ) = 1 − e − 10.825 2 π × 10.825 e r f ( 10.825 ) = 1 − 1.2855 × 10 − 51 19.186 e r f ( 10.825 ) = 1 e r f ( 10.825 ) = 0.8468 c ( x , t ) = 10 21 × 0.8468 c ( x , t ) = 8.468 × 10 20 atoms / cm 3 Thus, concentration at a depth of 0.75 μm is c ( x , t ) = 8.468 × 10 20 atoms /cm 3 . Concentration with respect to depth and time is calculated as follows:. Considering depth as x = 1 μm = 1 × 10 − 4 cm Value of diffusion coefficient (D) is 2.5 × 10 − 4 cm 2 / sec . Total time for carrying out the process is 8 hr = 8 hr × 60 sec hr = 480 sec . Substituting the given value in the above relation, c ( x , t ) = C s × e r f c ( x 2 D t ) c ( x , t ) = C s × e r f c ( x 2 D t ) c ( x , t ) = 10 21 × e r f c ( 1 × 10 − 4 2 2.5 × 10 − 14 × 480 ) c ( x , t ) = 10 21 × e r f c ( 14.433 ) When the value of error function exceeds the range of limit 4. The formula used for calculation of error function is, e r f ( x ) = 1 − e − x 2 π x x = 14.433 e r f ( 14.433 ) = 1 − e − 14.433 2 π × 14.433 e r f ( 14.433 ) = 0.9999 e r f ( 14.433 ) = 0.838 c ( x , t ) = 10 21 × 0.838 c ( x , t ) = 8.38 × 10 20 atoms / cm 3 Thus, concentration at a depth of 1 μm is c ( x , t ) = 8.38 × 10 20 atoms / cm 3 . Concentration with respect to depth and time is calculated as follows:. Considering depth as x = 1.25 μm = 0.000125 cm . Value of diffusion coefficient (D) is 2.5 × 10 − 4 cm 2 / sec . Total time for carrying out the process is 8 hr = 8 hr × 60 sec hr = 480 sec . Substituting the given value in the above relation, c ( x , t ) = C s × e r f c ( x 2 D t ) c ( x , t ) = C s × e r f c ( x 2 D t ) c ( x , t ) = 10 21 × e r f c ( 0.000125 2 2.5 × 10 − 14 × 480 ) c ( x , t ) = 10 21 × e r f c ( 18.042 ) When the value of error function exceeds the range of limit 4

The Science and Engineering of Materials (MindTap Course List)

7th Edition

ISBN: 9781305076761

Author: Donald R. Askeland, Wendelin J. Wright

Publisher: Cengage Learning

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Chapter 5, Problem 5.85DP

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