Essentials Of Materials Science And Engineering
Essentials Of Materials Science And Engineering
4th Edition
ISBN: 9781337385497
Author: WRIGHT, Wendelin J.
Publisher: Cengage,
Question
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Chapter 5, Problem 5.85DP
Interpretation Introduction

Interpretation:

Considering the depth of material, the concentration profile should be determined. Also, the graph for the profile P for pre-deposition process should be plotted.

Concept introduction:

Constant source diffusion is the relationship of diffusion source with depth, diffusion coefficient and time. The relationship is given by using the below equation,

  ξ=x2Dt

Where,

ξ is the constant source diffusion.

D is the diffusion coefficient.

t is the time.

x is the depth.

Constant source diffusion is of two types:

  1. Pre deposition step: It is the first step for constant source diffusion in which a small quantity of impurity is doped using the constant source.
  2. Drive in deposition: It is the second step for constant source diffusion in which the material is maintaining the higher value of temperature for a certain period of time.

Equation used for calculating the diffusion is,

Fick's Law of diffusion: This law states that molar flux is directly proportional to concentration gradient. The law is stated as:

  J=DdCAdx

Where,

J is the molar flux defined as the number of atoms passing per unit area per unit time.

D is the diffusion coefficient in cm2/sec.

  dCAdx is the concentration gradient in atomscm3.cm.

Factors affection diffusion are as follows:

  1. Temperature
  2. Diffusion coefficient

The following equation is stated as:

  D=D0exp(-QRT)

Where,

Q is the activation energy in calorie/ mole.

R is universal gas constant in calmoleK.

T is the absolute temperature in kelvin.

  D0 is constant for given diffusion system and its value is defined as 1T=0 or T=.

Expert Solution & Answer
Check Mark

Explanation of Solution

Given Information:

The given equation for basis is:

  c(x,t)=Cs[1erf(x4Dt)]

Where,

c is the concentration considering pre deposition function.

D is the diffusion coefficient.

X is the depth.

T is the time.

erf is the error function.

The given boundary conditions are:

Depth = 0.25μm.

Concentration of P at depth of 0.25μm is 1013atoms/cm3.

The temperature at which material is conducted is 1000°C.

Value of diffusion coefficient is 2.5×104cm2/sec.

Total time for carrying out the process is 8hr.

Equation considered for calculation is,

  c(x,t)=Cs[1erf(x4Dt)]

On integrating the above equation within limits,

  c(x,t)=Cs[12 π0 x 2 Dt e y 2 dy]c(x,t)=Cs×erfc(x2 Dt)

Thus, the equation considered for calculation is,

  c(x,t)=Cs×erfc(x2 Dt)

Assuming the various values of depth required for calculation. Assumed values of depth are,

  x=0.25μm=2.5×105cmx=0.50μm=5×105cmx=0.75μm=7.5×105cmx=1μm=1×104cmx=1.25μm=0.000125cmx=1.5μm=0.00015cmx=1.75μm=0.000175cmx=2μm=0.0002cmx=2.25μm=0.000225cmx=2.5μm=0.00025cm

Calculation of concentration with respect to time and depth. Considering depth as

  x=0.25μm=2.5×105cm

Value of diffusion coefficient (D) is 2.5×104cm2/sec.

Total time for carrying out the process is 8hr = 8hr×60sechr=480sec.

Substituting the given value in the above relation,

  c(x,t)=Cs×erfc(x2 Dt)

  c(x,t)=Cs×erfc(x 2 Dt )c(x,t)=1021×erfc( 2.5× 10 5 2 2.5× 10 14 ×480 )c(x,t)=1021×erfc(3.60)

Using table for calculation of error function,

  erfc(3.60)=0.99c(x,t)=1021×0.99c(x,t)=9.9×1020atoms/cm3

Thus, concentration at a depth of 0.25μm is c(x,t)=9.9×1020atoms/cm3.

Calculation of concentration with respect to time and depth. Considering depth as x=0.50μm=5×105cm

Value of diffusion coefficient (D) is 2.5×104cm2/sec.

Total time for carrying out the process is 8hr = 8hr×60sechr=480sec.

Substituting the given value in the above relation,

  c(x,t)=Cs×erfc(x2 Dt)

  c(x,t)=Cs×erfc(x 2 Dt )c(x,t)=1021×erfc( 5× 10 5 2 2.5× 10 14 ×480 )c(x,t)=1021×erfc(7.216)

When the value of error function exceeds the range of limit 4. The formula used for calculation of error function is,

  erf(x)=1e x 2 πxx=7.216erf(7.216)=1e 7.216 2 π×7.216erf(7.216)=12.43× 10 2312.790erf(7.216)=1erf(7.216)=0.8427c(x,t)=1021×0.8427c(x,t)=8.427×1021atoms/cm3

Thus, concentration at a depth of 0.50μm is c(x,t)=8.427×1021atoms/cm3.

Calculation of concentration with respect to time and depth. Considering depth as x=0.75μm=7.5×105cm

Value of diffusion coefficient (D) is 2.5×104cm2/sec.

Total time for carrying out the process is 8hr = 8hr×60sechr=480sec.

Substituting the given value in the above relation,

  c(x,t)=Cs×erfc(x2 Dt)

  c(x,t)=Cs×erfc(x 2 Dt )c(x,t)=1021×erfc( 7.5× 10 5 2 2.5× 10 14 ×480 )c(x,t)=1021×erfc(10.825)

When the value of error function exceeds the range of limit 4. The formula used for calculation of error function is,

  erf(x)=1e x 2 πxx=10.825erf(10.825)=1e 10.825 2 π×10.825erf(10.825)=11.2855× 10 5119.186erf(10.825)=1erf(10.825)=0.8468c(x,t)=1021×0.8468c(x,t)=8.468×1020atoms/cm3

Thus, concentration at a depth of 0.75μm is c(x,t)=8.468×1020atoms/cm3.

Calculation of concentration with respect to time and depth. Considering depth as x=1μm=1×104cm

Value of diffusion coefficient (D) is 2.5×104cm2/sec.

Total time for carrying out the process is 8hr = 8hr×60sechr=480sec.

Substituting the given value in the above relation,

  c(x,t)=Cs×erfc(x2 Dt)

  c(x,t)=Cs×erfc(x 2 Dt )c(x,t)=1021×erfc( 1× 10 4 2 2.5× 10 14 ×480 )c(x,t)=1021×erfc(14.433)

When the value of error function exceeds the range of limit 4. The formula used for calculation of error function is,

  erf(x)=1e x 2 πxx=14.433erf(14.433)=1e 14.433 2 π×14.433

  erf(14.433)=0.9999erf(14.433)=0.838c(x,t)=1021×0.838c(x,t)=8.38×1020atoms/cm3

Thus, concentration at a depth of 1μm is c(x,t)=8.38×1020atoms/cm3.

Calculation of concentration with respect to time and depth. Considering depth as x=1.25μm=0.000125cm.

Value of diffusion coefficient (D) is 2.5×104cm2/sec.

Total time for carrying out the process is 8hr = 8hr×60sechr=480sec.

Substituting the given value in the above relation,

  c(x,t)=Cs×erfc(x2 Dt)

  c(x,t)=Cs×erfc(x 2 Dt )c(x,t)=1021×erfc( 0.000125 2 2.5× 10 14 ×480 )c(x,t)=1021×erfc(18.042)

When the value of error function exceeds the range of limit 4. The formula used for calculation of error function is,

  erf(x)=1e x 2 πxx=18.042erf(18.042)=1e 18.042 2 π×18.042

  erf(18.042)=1erf(18.042)=0.842c(x,t)=1021×0.842c(x,t)=8.42×1020atoms/cm3

Thus, concentration at a depth of 1.25μm is c(x,t)=8.42×1020atoms/cm3.

Calculation of concentration with respect to time and depth. Considering depth as x=1.50μm=0.00015cm.

Value of diffusion coefficient (D) is 2.5×104cm2/sec.

Total time for carrying out the process is 8hr = 8hr×60sechr=480sec.

Substituting the given value in the above relation,

  c(x,t)=Cs×erfc(x2 Dt)

  c(x,t)=Cs×erfc(x 2 Dt )c(x,t)=1021×erfc( 0.00015 2 2.5× 10 14 ×480 )c(x,t)=1021×erfc(21.650)

When the value of error function exceeds the range of limit 4. The formula used for calculation of error function is,

  erf(x)=1e x 2 πxx=21.650erf(21.650)=1e 21.650 2 π×21.650

  erf(21.650)=1erf(21.650)=0.842c(x,t)=1021×0.85c(x,t)=8.5×1020atoms/cm3

Thus, concentration at a depth of 1.50μm is c(x,t)=8.5×1020atoms/cm3.

Calculation of concentration with respect to time and depth. Considering depth as x=1.75μm=0.000175cm.

Value of diffusion coefficient (D) is 2.5×104cm2/sec.

Total time for carrying out the process is 8hr = 8hr×60sechr=480sec.

Substituting the given value in the above relation,

  c(x,t)=Cs×erfc(x2 Dt)

  c(x,t)=Cs×erfc(x 2 Dt )c(x,t)=1021×erfc( 0.000175 2 2.5× 10 14 ×480 )c(x,t)=1021×erfc(25.259)

When the value of error function exceeds the range of limit 4. The formula used for calculation of error function is,

  erf(x)=1e x 2 πxx=25.259erf(25.259)=1e 25.259 2 π×25.259

  erf(25.259)=1erf(25.259)=0.842c(x,t)=1021×0.85c(x,t)=8.5×1020atoms/cm3

Thus, concentration at a depth of 1.75μm is c(x,t)=8.5×1020atoms/cm3.

Calculation of concentration with respect to time and depth. Considering depth as x=2μm=0.0002cm.

Value of diffusion coefficient (D) is 2.5×104cm2/sec.

Total time for carrying out the process is 8hr = 8hr×60sechr=480sec.

Substituting the given value in the above relation,

  c(x,t)=Cs×erfc(x2 Dt)

  c(x,t)=Cs×erfc(x 2 Dt )c(x,t)=1021×erfc( 0.0002 2 2.5× 10 14 ×480 )c(x,t)=1021×erfc(28.867)

When the value of error function exceeds the range of limit 4. The formula used for calculation of error function is,

  erf(x)=1e x 2 πxx=28.867erf(28.867)=1e 28.867 2 π×28.867

  erf(28.867)=1erf(28.867)=0.89c(x,t)=1021×0.89c(x,t)=8.9×1020atoms/cm3

Thus, concentration at a depth of 2μm is c(x,t)=8.9×1020atoms/cm3.

Calculation of concentration with respect to time and depth. Considering depth as x=2.25μm=0.000225cm.

Value of diffusion coefficient (D) is 2.5×104cm2/sec.

Total time for carrying out the process is 8hr = 8hr×60sechr=480sec.

Substituting the given value in the above relation,

  c(x,t)=Cs×erfc(x2 Dt)

  c(x,t)=Cs×erfc(x 2 Dt )c(x,t)=1021×erfc( 0.000225 2 2.5× 10 14 ×480 )c(x,t)=1021×erfc(32.475)

When the value of error function exceeds the range of limit 4. The formula used for calculation of error function is,

  erf(x)=1e x 2 πxx=32.475erf(32.475)=1e 32.475 2 π×32.475erf(32.475)=1

  erf(32.475)=1erf(32.475)=0.89c(x,t)=1021×0.89c(x,t)=8.9×1020atoms/cm3

Thus, concentration at a depth of 2.5μm is c(x,t)=8.9×1020atoms/cm3.

The table represents the data of concentration with variation in depth,

DepthConcentration
0.25μmc(x,t)=9.9×1020atoms/cm3
0.50μmc(x,t)=8.427×1021atoms/cm3
0.75μmc(x,t)=8.468×1020atoms/cm3
1μmc(x,t)=8.38×1020atoms/cm3
1.25μmc(x,t)=8.42×1020atoms/cm3
1.5μmc(x,t)=8.5×1020atoms/cm3
1.75μmc(x,t)=8.5×1020atoms/cm3
2μmc(x,t)=8.9×1020atoms/cm3
2.5μmc(x,t)=8.9×1020atoms/cm3

Based on the given values drawing graph of concentration versus depth,

  Essentials Of Materials Science And Engineering, Chapter 5, Problem 5.85DP

Thus, the required graph of concentration versus depth is shown above. As for pre deposition process concentration has dependency on time and depth, blue line is showing variation of concentration versus time and yellow line represents the variation of concentration with time.

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Chapter 5 Solutions

Essentials Of Materials Science And Engineering

Ch. 5 - Prob. 5.11PCh. 5 - Prob. 5.12PCh. 5 - Prob. 5.13PCh. 5 - Prob. 5.14PCh. 5 - Prob. 5.15PCh. 5 - Prob. 5.16PCh. 5 - Prob. 5.17PCh. 5 - Prob. 5.18PCh. 5 - Prob. 5.19PCh. 5 - Prob. 5.20PCh. 5 - Prob. 5.21PCh. 5 - Prob. 5.22PCh. 5 - Prob. 5.23PCh. 5 - Prob. 5.24PCh. 5 - Prob. 5.25PCh. 5 - Prob. 5.26PCh. 5 - Prob. 5.27PCh. 5 - Prob. 5.28PCh. 5 - Prob. 5.29PCh. 5 - Prob. 5.30PCh. 5 - Prob. 5.31PCh. 5 - Prob. 5.32PCh. 5 - Prob. 5.33PCh. 5 - Prob. 5.34PCh. 5 - Prob. 5.35PCh. 5 - Prob. 5.36PCh. 5 - Prob. 5.37PCh. 5 - Prob. 5.38PCh. 5 - Prob. 5.39PCh. 5 - Prob. 5.40PCh. 5 - Prob. 5.41PCh. 5 - Prob. 5.42PCh. 5 - Prob. 5.43PCh. 5 - Prob. 5.44PCh. 5 - Prob. 5.45PCh. 5 - Prob. 5.46PCh. 5 - Prob. 5.47PCh. 5 - Prob. 5.48PCh. 5 - Prob. 5.49PCh. 5 - Prob. 5.50PCh. 5 - Prob. 5.51PCh. 5 - Prob. 5.52PCh. 5 - Prob. 5.53PCh. 5 - Prob. 5.54PCh. 5 - Prob. 5.55PCh. 5 - Prob. 5.56PCh. 5 - Prob. 5.57PCh. 5 - Prob. 5.58PCh. 5 - Prob. 5.59PCh. 5 - Prob. 5.60PCh. 5 - Prob. 5.61PCh. 5 - Prob. 5.62PCh. 5 - Prob. 5.63PCh. 5 - Prob. 5.64PCh. 5 - Prob. 5.65PCh. 5 - Prob. 5.66PCh. 5 - Prob. 5.67PCh. 5 - Prob. 5.68PCh. 5 - Prob. 5.69PCh. 5 - Prob. 5.70PCh. 5 - Prob. 5.71PCh. 5 - Prob. 5.72PCh. 5 - Prob. 5.73PCh. 5 - Prob. 5.74PCh. 5 - Prob. 5.75PCh. 5 - Prob. 5.76PCh. 5 - Prob. 5.77PCh. 5 - Prob. 5.78PCh. 5 - Prob. 5.79PCh. 5 - Prob. 5.80PCh. 5 - Prob. 5.81DPCh. 5 - Prob. 5.82DPCh. 5 - Prob. 5.83DPCh. 5 - Prob. 5.84DPCh. 5 - Prob. 5.85DPCh. 5 - Prob. 5.86CPCh. 5 - Prob. 5.87CPCh. 5 - Prob. 5.88CPCh. 5 - Prob. K5.1KP
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