Project 3 Report
CHE/MAE 294
Fall 2002
Defect Structures
Dr. J. J. Weimer
Due: September 20, 2002
Submitted: September 24, 2002
This report uses a line spacing of 12pt rather than the requested 18pt in order to fit it into fewer pages.
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Calculating Defect Concentration, Number Density, and Radius in Materials - Prof. Jeffrey , Study Guides, Projects, Research of Chemistry
The formulas and calculations for determining the concentration, number density, and effective radius of substitutional defects in metals, ceramics, and doped semiconductors. It includes expressions for weight fraction, molar defect density, and effective defect radius, as well as examples for ni/pd, mgo-nio, and various doped semiconductors.
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Project 3 Report
CHE/MAE 294
Fall 2002
Defect Structures
Dr. J. J. Weimer
Due: September 20, 2002
Submitted: September 24, 2002
This report uses a line spacing of 12pt rather than the requested 18pt in order to fit it into fewer pages.
CHE/MAE 294 Project 3 2
1 Introduction
The goal of this assignment was to learn how concentration, number density, and effective radius
of substitutional defects in materials are related. The objectives were to obtain the exact formula-
tions relating number density and effective radius to concentration for metal, ceramic, and doped
semiconductor systems and to calculate typical values using these formulations.
This assignment had numerous errors in the equations given. This mistakes were found using
unit analysis. Formulations are derived using the expressions given in the assignment. Correct
expressions are also derived.
Two quantitative factors were of concern, the concentration of defects as wt%, mol%, or ppX and
the number density of defects (number per unit volume as m
3
). The wt fraction of a component
B in binary metal or ceramic mixture or compound of A and B is given as ω
B
= m
B
=( m
A
- m
B
where m
i
(kg) is the mass of the specific component. Dopant concentration in semiconductors is
expressed as pp X , where X is m for million (
6
), b for billion (
9
), or t for trillion (
12
). The
expression given in the assignment was ppX = m
d
=( Xm
o
m
d
) where ppX is the concentration,
m
d
is the mass of dopant and Xm
o
is the mass of the pure material in the corresponding decade
factor for ppm, ppb, or ppt. For ppm concentrations, the expression would evaluate as ppm =
m
d
6
m
d
The introduction must give the above equation for ppX or ppm with explainations and units for all symbols.
Unit analysis shows the expression given in the assignment for ppXr is really the ratio of the mass
of dopant to the mass of pure material. The correct expression relating a ppX concentration to
dopant and pure material masses is
m
m
o
ppX
X ppX
where m is the mass of dopant, m o
is the mass of pure semiconductor, ppX is the dopant concen-
tration, and X is the decade multipler for the ppX scale. For example, a concentration of 5 ppm
P/Si would have ppX = 5, X = 10
6
, and m = m
o
6
g P per 1 g Si.
Award 5% extra credit if the above equation is given or used in further derivations.
The number density of substitutional defects ρ
sd
(number/m
3
) in a material was formulated in the
assignment sheet as
ρ
sd
m
sd
ρ
o
m
o
M
sd
where m
sd
is the mass of defects (kg), ρ
o
is the density of the pure material (kg/m
3
), m
o
is the mass
of pure material (kg), and M
sd
is the molar mass of the substituted atom or molecule (kg/mol).
The introduction must give the above equation for ρ
sd
with explainations and units for all symbols.
The above expression yields units of mol/m
3
. To obtain units of number/m
3
, it must be corrected
with Avagadro’s number N o
23
mol
1
) as
ρ
sd
m
sd
ρ
o
N
o
m o
M
sd
Award 5% extra credit if the above equation using N o
is given or used in further derivations.
CHE/MAE 294 Project 3 4
The MOLAR expression relating defect density to ppX is
ρ
sd
X ρ
o
ppX
( 1 + ppX ) M
sd
where ¯ X is the decade multiplier. For 100 ppm, ppX = 100 and ¯ X = 10
6
). The MOLAR expression
relating effective defect radius to weight fraction is
r
e ; sd
4 π
X ρ
o
ppX
3 ( 1 + ppX ) M
sd
1 = 3
These expressions may also be given using ppm and without ¯ X as
ρ
sd
6
ρ
o
ppm
( 1 + ppm ) M
sd
r
e ; sd
4 π 10
6
ρ
o
ppm
3 ( 1 + ppm ) M
sd
1 = 3
The multiplier 10
6
must be shown in the ppm form.
The NUMBER expressions are
ρ
sd
X ppX ρ
o
N
o
( 1 + ppX ) M
sd
r
e ; sd
4 π
X ppX ρ
o
N
o
3 ( 1 + ppX ) M
sd
1 = 3
ρ
sd
6
ppm ρ
o
N
o
( 1 + ppm ) M
sd
r
e ; sd
4 π 10
6
ppm ρ
o
N
o
3 ( 1 + ppm ) M
sd
1 = 3
Award 5% bonus points for giving the above expressions.
The correct expression relating defect number density to ppX concentration should be
ρ
sd
ppX ρ
o
N
o
X ppX ) M
sd
where ppX is the concentration and ¯ X is the decade multiplier. The corresponding effective defect
radius is
r
e ; sd
4 π ppX ρ
o
N
o
X ppX ) M
sd
1 = 3
Award 10% bonus points for giving the above expressions. All other expressions are to be marked wrong.
Results will be calculated using the MOLAR and NUMBER forms of the expressions as well as
the last two expressions for defect number density and effective radius.
CHE/MAE 294 Project 3 5
For the Ni/Pd system, the molar mass of Ni is M
sd
3
kg/mol, the density of Pd is
ρ o
3
kg/m
3
, and the empirical atomic radius of a Ni atom is r
sd
= 0 :135 nm [3].
For Ni/Pd, the molar mass of Ni, density of Pd, and atomic or covalent radius of Ni must be reported.
References must be given for the densities and radii. Web page references must include a copy of the Web
page in the Appendix.
For the MgO/NiO system, the molar mass of MgO is M
sd
3
3
kg/mol and the density of NiO is ρ o
3
kg/m
3
[2]. The radius of the MgO is taken as the
bond length of the Mg
2 +
O
2
compound as r
sd
= 0 : 078 + 0 : 132 = 0 :210 nm [1].
For MgO/NiO, the molar mass of MgO, density of NiO, and radius of the MgO compound must be reported.
References must be given for the densities and radii. Web page references must include a copy of the Web
page in the Appendix.
Molar masses M
sd
(kg/mol) [1], densities ρ
o
(Mg/m
3
) [1], and empirical atomic radii r
sd
(nm) [3]
for the doped semiconductors (dSC) are given in Table 3.A below. In each case for the element
pair A/B, the substitutional defect (for molar mass and radius) is element A and the doped material
(for density) is element B.
Table 4.A Materials parameters used for calculations (see text description).
ID 1-3 4-6 7-9 10-12 13-15 16-18 19-21 22-24 25-27 28 +
dSC Al/Si P/Si As/Si Sb/Si Ga/Si Al/Ge P/Ge As/Ge Sb/Ge Ga/Ge
M
sd
r
sd
ρ
o
Molar masses, densities, and atomic or covalent radii as above must be reported for doped semiconductors.
References must be given for the densities and radii. Web page references must include a copy of the Web
page in the Appendix.
3 Discussion (not required)
The effective radius of the Ni substitutional defect in the Ni/Pd alloy approaches 1 as the weight
fraction approaches 50%. In the MgO/NiO ceramic, the effective radius of the MgO substitution
is smaller than one because a 50 wt% mixture has a higher mole fraction of MgO than NiO. In
the doped semiconductor systems, the dopant has an effective radius of between about 20 to 200
atoms depending on ppm concentration.
References
- Introduction to Materials Science for Engineers, 5th Ed., J. F. Shackelford (Prentice Hall, New
Jersey) 2000.
- CRC Handbook of Chemistry and Physics, 73rd Ed., Ed. D. R. Lide (CRC Press, Ann Arbor)
- http://www.webelements.com/
CHE/MAE 294 F02 Project 3
Table 4.1:
Results for the Ni/Pd alloy system. Values for MOLAR and NUMBER expressions are given.
MOLAR EXPRESSIONS
NUMBER EXPRESSIONS
weight fraction
Ni
molar defect density
(mol/m3)
reNi/rNi
(mol^(-1/3))
weight fraction
Ni
number defect density (#/m3)
reNi/rNi
(unitless)
2.065E+
3.61E+
1.243E+
2.271E+
1.62E+
1.368E+
8.760E+
1.03E+
5.275E+
2.044E+
7.80E+
1.231E+
The table MUST include correct units for the values in the column for defect density.
MOLAR EXPRESSIONS
units are mol/m
NUMBER EXPRESSIONS
units are numberl/m3 or just 1/m
The number of significant digits for values in either defect density column cannot exceed four (as shown).The number of significant digits for values in either radius ratio column cannot exceed three (as shown).All calculations should yield values that are within 10% of those in the above table.Values Used in Calculations
Pd density (g/m3)
1.20E+
MOLAR EXPRESSION
MNi (g/mol)
rho = (omega * rho_o) / ((1 - omega) * M)
rNi (m)
1.35E-
No (#/mol)
6.022E+
NUMBER EXPRESSIONrho = (omega * rho_o * N_o) / ((1 - omega) * M)
Ni-Pd
CHE/MAE 294 F02 Project 3
Table 4.2:
Results for the NiO-MgO ceramic. Values for MOLAR and NUMBER expressions are given.
MOLAR EXPRESSIONS
NUMBER EXPRESSIONS
weight fraction
MgO
molar defect density
(mol/m3)
reMgO/rMgO
(mol^(-1/3))
weight fraction
NiO
number defect density (#/m3)
reMgO/rMgO(
unitless)
1.67E+
2.49E+
1.01E+
1.84E+
1.12E+
1.11E+
7.09E+
7.14E+
4.27E+
1.65E+
5.38E+
9.96E+
The table MUST include units for the values in the column of defect density.
MOLAR EXPRESSIONS
units are mol/m
NUMBER EXPRESSIONS
units are numberl/m3 or just 1/m
The number of significant digits for values in either defect density column cannot exceed three (as shown).The number of significant digits for values in either radius ratio column cannot exceed three (as shown).All calculations should yield values that are within 10% of those in the above table.Values Used in Calculations
NiO den(g/m3)
6.67E+
MOLAR EXPRESSION
MMgO(g/mol)
rho = (omega * rho_o) / ((1 - omega) * M)
rMgO(m)
2.10E-
No (#/mol)
6.022E+
NUMBER EXPRESSIONrho = (omega * rho_o * N_o) / ((1 - omega) * M)
MgO-NiO
CHE/MAE 294 F02 Project 3
Table 4.2:
Results for P/Si doped semiconductor. Values for MOLAR and NUMBER expressions are given.
MOLAR EXPRESSIONS
NUMBER EXPRESSIONS
CORRECT NUMBER EXPRESSIONS
ppm
molar defect
density (mol/m3)
reP/rP
(mol^(-1/3))
ppm
number defect density (#/m3)
reP/rP
(unitless)
ppm
number defect density (#/m3)
reP/rP
(unitless)
3.76E+
1.85E+
2.27E+
4.53E+
5.64E+
1.62E+
3.40E+
1.36E+
6.84E+
1.52E+
4.12E+
4.53E+
7.28E+
1.49E+
4.38E+
1.36E+
7.45E+
1.47E+
4.49E+
4.53E+
The table MUST include units for the values in the column of defect density.
MOLAR EXPRESSIONS
units are mol/m
NUMBER EXPRESSIONS
units are numberl/m3 or just 1/m
The number of significant digits for values in either defect density column cannot exceed three (as shown).The number of significant digits for values in either radius ratio column cannot exceed three (as shown).All calculations should yield values that are within 10% of those in the above table.Values Used in Calculations
Si density (g/m3)
2.33E+
MP (g/mol)
rP (m)
1.00E-
No (#/mol)
6.022E+
P-Si
CHE/MAE 294 F02 Project 3
Table 4.2:
Results for As/Si doped semiconductor. Values for MOLAR and NUMBER expressions are given.
MOLAR EXPRESSIONS
NUMBER EXPRESSIONS
CORRECT NUMBER EXPRESSIONS
ppm
molar defect
density (mol/m3)
reAs/rAs
(mol^(-1/3))
ppm
number defect density (#/m3)
reAs/rAs (unitless)
ppm
number defect density (#/m3)
reAs/rAs (unitless)
1.55E+
2.16E+
9.36E+
1.87E+
2.33E+
1.89E+
1.40E+
5.62E+
2.83E+
1.77E+
1.70E+
1.87E+
3.01E+
1.73E+
1.81E+
5.62E+
3.08E+
1.72E+
1.85E+
1.87E+
The table MUST include units for the values in the column of defect density.
MOLAR EXPRESSIONS
units are mol/m
NUMBER EXPRESSIONS
units are numberl/m3 or just 1/m
The number of significant digits for values in either defect density column cannot exceed three (as shown).The number of significant digits for values in either radius ratio column cannot exceed three (as shown).All calculations should yield values that are within 10% of those in the above table.Values Used in Calculations
Si density (g/m3)
2.33E+
MAs (g/mol)
rAs (m)
1.15E-
No (#/mol)
6.022E+
As-Si
CHE/MAE 294 F02 Project 3
Table 4.2:
Results for Ga/Si doped semiconductor. Values for MOLAR and NUMBER expressions are given.
MOLAR EXPRESSIONS
NUMBER EXPRESSIONS
CORRECT NUMBER EXPRESSIONS
ppm
molar defect
density (mol/m3)
reGa/rGa
(mol^(-1/3))
ppm
number defect density (#/m3)
reGa/rGa(unitless)
ppm
number defect density (#/m3)
reGa/rGa(unitless)
1.67E+
1.87E+
1.01E+
2.01E+
2.51E+
1.63E+
1.51E+
6.04E+
3.04E+
1.53E+
1.83E+
2.01E+
3.23E+
1.50E+
1.95E+
6.04E+
3.31E+
1.49E+
1.99E+
2.01E+
The table MUST include units for the values in the column of defect density.
MOLAR EXPRESSIONS
units are mol/m
NUMBER EXPRESSIONS
units are numberl/m3 or just 1/m
The number of significant digits for values in either defect density column cannot exceed three (as shown).The number of significant digits for values in either radius ratio column cannot exceed three (as shown).All calculations should yield values that are within 10% of those in the above table.Values Used in Calculations
Si density (g/m3)
2.33E+
MGa (g/mol)
rGa (m)
1.30E-
No (#/mol)
6.022E+
Ga-Si
CHE/MAE 294 F02 Project 3
Table 4.2:
Results for Al/Ge doped semiconductor. Values for MOLAR and NUMBER expressions are given.
MOLAR EXPRESSIONS
NUMBER EXPRESSIONS
CORRECT NUMBER EXPRESSIONS
ppm
molar defect
density (mol/m3)
reAl/rAl
(mol^(-1/3))
ppm
number defect density (#/m3)
reAl/rAl
(unitless)
ppm
number defect density (#/m3)
reAl/rAl
(unitless)
9.86E+
1.07E+
5.94E+
1.19E+
1.48E+
9.38E+
8.91E+
3.56E+
1.79E+
8.80E+
1.08E+
1.19E+
1.91E+
8.62E+
1.15E+
3.56E+
1.95E+
8.55E+
1.18E+
1.19E+
The table MUST include units for the values in the column of defect density.
MOLAR EXPRESSIONS
units are mol/m
NUMBER EXPRESSIONS
units are numberl/m3 or just 1/m
The number of significant digits for values in either defect density column cannot exceed three (as shown).The number of significant digits for values in either radius ratio column cannot exceed three (as shown).All calculations should yield values that are within 10% of those in the above table.Values Used in Calculations
Ge density (g/m3)
5.32E+
MAl (g/mol)
rAl (m)
1.25E-
No (#/mol)
6.022E+
Al-Ge
CHE/MAE 294 F02 Project 3
Table 4.2:
Results for As/Ge doped semiconductor. Values for MOLAR and NUMBER expressions are given.
MOLAR EXPRESSIONS
NUMBER EXPRESSIONS
CORRECT NUMBER EXPRESSIONS
ppm
molar defect
density (mol/m3)
reAs/rAs
(mol^(-1/3))
ppm
number defect density (#/m3)
reAs/rAs (unitless)
ppm
number defect density (#/m3)
reAs/rAs (unitless)
3.55E+
1.64E+
2.14E+
4.28E+
5.33E+
1.43E+
3.21E+
1.28E+
6.46E+
1.34E+
3.89E+
4.28E+
6.87E+
1.32E+
4.14E+
1.28E+
7.03E+
1.31E+
4.23E+
4.28E+
The table MUST include units for the values in the column of defect density.
MOLAR EXPRESSIONS
units are mol/m
NUMBER EXPRESSIONS
units are numberl/m3 or just 1/m
The number of significant digits for values in either defect density column cannot exceed three (as shown).The number of significant digits for values in either radius ratio column cannot exceed three (as shown).All calculations should yield values that are within 10% of those in the above table.Values Used in Calculations
Ge density (g/m3)
5.32E+
MAs (g/mol)
rAs (m)
1.15E-
No (#/mol)
6.022E+
As-Ge
CHE/MAE 294 F02 Project 3
Table 4.2:
Results for Sb/Ge doped semiconductor. Values for MOLAR and NUMBER expressions are given.
MOLAR EXPRESSIONS
NUMBER EXPRESSIONS
CORRECT NUMBER EXPRESSIONS
ppm
molar defect
density (mol/m3)
reSb/rSb
(mol^(-1/3))
ppm
number defect density (#/m3)
reSb/rSb (unitless)
ppm
number defect density (#/m3)
reSb/rSb
(unitless)
2.18E+
1.53E+
1.32E+
2.63E+
3.28E+
1.34E+
1.97E+
7.89E+
3.97E+
1.25E+
2.39E+
2.63E+
4.23E+
1.23E+
2.55E+
7.89E+
4.33E+
1.22E+
2.61E+
2.63E+
The table MUST include units for the values in the column of defect density.
MOLAR EXPRESSIONS
units are mol/m
NUMBER EXPRESSIONS
units are numberl/m3 or just 1/m
The number of significant digits for values in either defect density column cannot exceed three (as shown).The number of significant digits for values in either radius ratio column cannot exceed three (as shown).All calculations should yield values that are within 10% of those in the above table.Values Used in Calculations
Ge density (g/m3)
5.32E+
MSb (g/mol)
rSb (m)
1.45E-
No (#/mol)
6.022E+
Sb-Ge
CHE/MAE 294 Project 3 Solution
Derivation of Expressions
Dr. J. J. Weimer
23.September.
restart:
Metals and Ceramics
Formulations
Weight Fraction
mo - mass of pure material m - mass of substitutional defect
eqnwtf := omega = m/(m + mo);
eqnwtf :=ω =
m
m + mo
Defect Number Density
rho[o] - density of pure material M - molar mass of substitutional defect
mo - mass of pure material m - mass of substitutional defect
eqnnd := rho = mrho[o]No/(mo*M);
eqnnd :=ρ =
m ρ
o
No
mo M
Solutions
Solve the weight fraction expression for the mass of the defect and subsitute into the defect number
density expression.
ms := solve(eqnwtf,m);
ms :=−
ω mo
ω − 1
Defect Number Density
eqnrhos := subs(m=ms,eqnnd);
eqnrhos :=ρ =−
ω ρ
o
No
( ω − 1 ) M
eqndr := 1/rho = (4/3)Pir^3;
eqndr := =
ρ
π r
3
Defect Effective Radius
eqnrsd := subs(rho=rhs(eqnrhos),eqndr);
eqnrsd :=− =
( ω − 1 ) M
ω ρ
o
No
π r
3
Doped Semiconductors
Formulations
Concentration as Given
X is the decade multiplier for the total mass of material.
eqnppX := ppX = m/(X*mo - m);
eqnppX := ppX =
m
X mo − m
Solutions
Solve the ppX expression for the mass of the defect. Substitute into the expression for defect density.
mXs := solve(eqnppX,m);
mXs :=
ppX X mo
ppX + 1
Defect Number Density as Given
eqnrhoX := subs(m=mXs,eqnnd);
eqnrhoX :=ρ =
ppX X ρ
o
No
( ppX + 1 ) M
Actual Concentration
The actual expression for ppX concentration is given below, where X is the decade multiplier.
eqnppXa := (m/mo) = (ppX/(X - ppX));
eqnppXa := =
m
mo
ppX
X − ppX
mXsa := solve(eqnppXa,m);
mXsa :=−
ppX mo
X ppX
eqnrhoXa := subs(m=mXsa,eqnnd);
eqnrhoXa :=ρ =−
ppX ρ
o
No
( − X + ppX ) M
The effective defect radius can be found in the same manner as for metals and ceramics.