MATERIALS OF IMPORTANCE Aluminum for Integrated Circuit Interconnects
5.7 OTHER DIFFUSION PATHS
Atomic migration may also occur along dislocations, grain boundaries, and exter- nal surfaces. These are sometimes called “short-circuit” diffusion paths inasmuch as rates are much faster than for bulk diffusion. However, in most situations short-circuit contributions to the overall diffusion flux are insignificant because the cross- sectional areas of these paths are extremely small.
S U M M A R Y
Introduction
• Solid-state diffusion is a means of mass transport within solid materials by step- wise atomic motion.
• The term interdiffusion refers to the migration of impurity atoms; for host atoms, the term self-diffusion is used.
Diffusion Mechanisms
• Two mechanisms for diffusion are possible: vacancy and interstitial.
Vacancy diffusion occurs via the exchange of an atom residing on a normal lattice site with an adjacent vacancy.
For interstitial diffusion, an atom migrates from one interstitial position to an empty adjacent one.
• For a given host metal, interstitial atomic species generally diffuse more rapidly.
Steady-State Diffusion
• Diffusion flux is defined in terms of mass of diffusing species, cross-sectional area, and time according to Equation 5.1a.
• Concentration profile is represented as a plot of concentration versus distance into the solid material.
• Concentration gradient is the slope of the concentration profile curve at some specific point.
• The diffusion condition for which the flux is independent of time is known as steady state.
• For steady-state diffusion, diffusion flux is proportional to the negative of the concentration gradient according to Fick’s first law, Equation 5.3.
• The driving force for steady-state diffusion is the concentration gradient (dC/dx).
Nonsteady-State Diffusion
• For nonsteady-state diffusion, there is a net accumulation or depletion of diffusing species and the flux is dependent on time.
• The mathematics for nonsteady state in a single (x) direction (and when the diffusion coefficient is independent of concentration) are described by Fick’s second law, Equation 5.4b.
• For a constant surface composition boundary condition, the solution to Fick’s second law (Equation 5.4b) is Equation 5.5, which involves the Gaussian error function (erf).
Factors That Influence Diffusion
• The magnitude of the diffusion coefficient is indicative of the rate of atomic mo- tion and depends on both host and diffusing species as well as on temperature.
• The diffusion coefficient is a function of temperature according to Equation 5.8.
Diffusion in Semiconducting Materials
• The two heat treatments that are used to diffuse impurities into silicon during integrated circuit fabrication are predeposition and drive-in.
During predeposition, impurity atoms are diffused into the silicon, often from a gas phase, the partial pressure of which is maintained constant.
For the drive-in step, impurity atoms are transported deeper into the silicon so as to provide a more suitable concentration distribution without increas- ing the overall impurity content.
• Integrated circuit interconnects are normally made of aluminum—instead of met- als such as copper, silver, and gold that have higher electrical conductivities—on the basis of diffusion considerations. During high-temperature heat treatments, interconnect metal atoms diffuse into the silicon; appreciable concentrations will compromise the chip’s functionality.
Equation Summar y
Summary • 143
Equation Page
Number Equation Solving for Number
5.1a Diffusion flux 126
5.3 Fick’s first law—diffusion flux for steady-state diffusion 127
5.4b Fick’s second law—for nonsteady-state diffusion 128
5.5 Solution to Fick’s second law—for constant surface composition 129
5.8 DD0expa Qd Temperature dependence of diffusion coefficient 133 RTb
CxC0
CsC0 1erfa x 22Dtb 0C
0t D 02C 0x2 J D dC dx J M
At
List of Symbols
Symbol Meaning
A Cross-sectional area perpendicular to direction of diffusion C Concentration of diffusing species
C0 Initial concentration of diffusing species prior to the onset of the diffusion process Cs Surface concentration of diffusing species
Cx Concentration at position x after diffusion time t D Diffusion coefficient
D0 Temperature-independent constant M Mass of material diffusing
Qd Activation energy for diffusion R Gas constant (8.31 J/mol K)
t Elapsed diffusion time
x Position coordinate (or distance) measured in the direction of diffusion, normally from a solid surface
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Processing/Structure/Properties/Performance Summar y
Diffusion in semiconducting materials was discussed in Section 5.6. For both pre- deposition and drive-in treatments, diffusion is nonsteady-state—solutions to Fick’s second law were provided for both. Nonsteady-state diffusion and these treatments are two of the processing components for silicon, as noted in the following diagram:
Important Terms and Concepts activation energy
carburizing
concentration gradient concentration profile diffusion
diffusion coefficient diffusion flux driving force
Fick’s first and second laws interdiffusion (impurity diffusion)
interstitial diffusion nonsteady-state diffusion self-diffusion
steady-state diffusion vacancy diffusion
R E F E R E N C E S
Carslaw, H. S., and J. C. Jaeger, Conduction of Heat in Solids, 2nd edition, Oxford University Press, Oxford, 1986.
Crank, J., The Mathematics of Diffusion, Oxford University Press, Oxford, 1980.
Gale, W. F., and T. C. Totemeier (Editors), Smithells Metals Reference Book, 8th edition, Butterworth- Heinemann, Woburn, UK, 2003.
Glicksman, M., Diffusion in Solids, Wiley- Interscience, New York, 2000.
Shewmon, P. G., Diffusion in Solids, 2nd edition, The Minerals, Metals and Materials Society, Warrendale, PA, 1989.
Nonsteady-state diffusion Diffusion in semiconductors
(Chapter 5) (Chapter 5)
Silicon (Processing)
Temperature dependence Diffusion in semiconductors of diffusion coefficient (impurity doping)
(Chapter 5) (Chapters 5 and 18)
Silicon (Processing)
Temperature dependence of diffusion (Chapter 5) Iron–Carbon
Alloys (Steels) (Processing)
In the design of heat treatments to be used for introducing impurities into semiconductors (i.e., doping, Chapter 18), and, in addition, in the production of steel alloys (Chapter 10), an understanding of the temperature dependence of the diffusion coefficient (i.e., Equation 5.8) is essential. The following diagrams illus- trate the preceding relationships for these two materials.
Isothermal transformation diagrams
(Chapter 10)
Tempering (tempered martensite) (Chapter 10)
Q U E S T I O N S A N D P R O B L E M S
Questions and Problems • 145
and a decarburizing atmosphere on the other side at 725C. After reaching steady state, the iron was quickly cooled to room temperature.
The carbon concentrations at the two surfaces of the sheet were determined to be 0.012 and 0.0075 wt%. Compute the diffusion coefficient if the diffusion flux is 1.4 108 kg/m2 Hint: Use Equation 4.9 to convert the con- centrations from weight percent to kilograms of carbon per cubic meter of iron.
5.9 When ␣-iron is subjected to an atmosphere of hydrogen gas, the concentration of hydrogen in the iron, CH(in weight percent), is a func- tion of hydrogen pressure, (in MPa), and absolute temperature (T ) according to
(5.14) Furthermore, the values of D0and Qdfor this diffusion system are 1.4 107 m2/s and 13,400 J/mol, respectively. Consider a thin iron membrane 1 mm thick that is at 250C.
Compute the diffusion flux through this mem- brane if the hydrogen pressure on one side of the membrane is 0.15 MPa (1.48 atm), and on the other side 7.5 MPa (74 atm).
Nonsteady-State Diffusion 5.10 Show that
is also a solution to Equation 5.4b. The parameter B is a constant, being independent of both x and t.
5.11 Determine the carburizing time necessary to achieve a carbon concentration of 0.45 wt% at a position 2 mm into an iron–carbon alloy that initially contains 0.20 wt% C. The surface con- centration is to be maintained at 1.30 wt% C, and the treatment is to be conducted at 1000C.
Use the diffusion data for -Fe in Table 5.2.
5.12 An FCC iron–carbon alloy initially containing 0.35 wt% C is exposed to an oxygen-rich and virtually carbon-free atmosphere at 1400 K (1127C). Under these circumstances the carbon diffuses from the alloy and reacts at the surface
g Cx B
1Dt exp a x2 4Dtb
CH1.341021pH2exp a27.2 kJ/mol
RT b
pH2
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s.Introduction
5.1 Briefly explain the difference between self- diffusion and interdiffusion.
5.2 Self-diffusion involves the motion of atoms that are all of the same type; therefore, it is not subject to observation by compositional changes, as with interdiffusion. Suggest one way in which self-diffusion may be monitored.
Diffusion Mechanisms
5.3 (a) Compare interstitial and vacancy atomic mechanisms for diffusion.
(b) Cite two reasons why interstitial diffusion is normally more rapid than vacancy diffusion.
Steady-State Diffusion
5.4 Briefly explain the concept of steady state as it applies to diffusion.
5.5 (a) Briefly explain the concept of a driving force.
(b) What is the driving force for steady-state diffusion?
5.6 The purification of hydrogen gas by diffusion through a palladium sheet was discussed in Section 5.3. Compute the number of kilo- grams of hydrogen that pass per hour through a 5-mm-thick sheet of palladium having an area of 0.20 m2at 500C. Assume a diffusion coefficient of 1.0 108 m2/s, that the con- centrations at the high- and low-pressure sides of the plate are 2.4 and 0.6 kg of hydrogen per cubic meter of palladium, and that steady-state conditions have been attained.
5.7 A sheet of steel 1.5 mm thick has nitrogen atmospheres on both sides at 1200C and is per- mitted to achieve a steady-state diffusion con- dition. The diffusion coefficient for nitrogen in steel at this temperature is 6 1011m2/s, and the diffusion flux is found to be 1.2 107kg/m2 s. Also, it is known that the con- centration of nitrogen in the steel at the high- pressure surface is 4 kg/m3. How far into the sheet from this high-pressure side will the concentration be 2.0 kg/m3? Assume a linear concentration profile.
5.8 A sheet of BCC iron 1 mm thick was exposed to a carburizing gas atmosphere on one side
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with the oxygen in the atmosphere; that is, the carbon concentration at the surface position is maintained essentially at 0 wt% C. (This process of carbon depletion is termed decarburization.) At what position will the carbon concentration be 0.15 wt% after a 10-h treatment? The value of D at 1400 K is 6.9 1011m2/s.
5.13 Nitrogen from a gaseous phase is to be dif- fused into pure iron at 700C. If the surface concentration is maintained at 0.1 wt% N, what will be the concentration 1 mm from the surface after 10 h? The diffusion coefficient for nitrogen in iron at 700C is 2.5 1011m2/s.
5.14 Consider a diffusion couple composed of two semi-infinite solids of the same metal, and that each side of the diffusion couple has a different concentration of the same elemen- tal impurity; furthermore, assume each impu- rity level is constant throughout its side of the diffusion couple. For this situation, the solu- tion to Fick’s second law (assuming that the diffusion coefficient for the impurity is inde- pendent of concentration) is as follows:
(5.15) In this expression, when the x 0 position is taken as the initial diffusion couple interface, then C1is the impurity concentration for x0;
likewise, C2is the impurity content for x0.
A diffusion couple composed of two silver–gold alloys is formed; these alloys have compositions of 98 wt% Ag–2 wt% Au and 95 wt% Ag–5 wt% Au. Determine the time this diffusion couple must be heated at 750C (1023 K) in order for the composition to be 2.5 wt% Au at the 50 m position into the 2 wt% Au side of the diffusion couple. Pre- exponential and activation energy values for Au diffusion in Ag are 8.5 105 m2/s and 202,100 J/mol, respectively.
5.15 For a steel alloy it has been determined that a carburizing heat treatment of 10-h duration will raise the carbon concentration to 0.45 wt% at a point 2.5 mm from the surface. Estimate the time necessary to achieve the same concentra- tion at a 5.0-mm position for an identical steel and at the same carburizing temperature.
Cx aC1C2
2 baC1C2
2 b erf a x 21Dtb
Factors That Influence Diffusion
5.16 Cite the values of the diffusion coefficients for the interdiffusion of carbon in both -iron (BCC) and -iron (FCC) at 900C. Which is larger? Explain why this is the case.
5.17 Using the data in Table 5.2, compute the value of D for the diffusion of zinc in copper at 650C.
5.18 At what temperature will the diffusion coef- ficient for the diffusion of copper in nickel have a value of 6.5 1017m2/s? Use the dif- fusion data in Table 5.2.
5.19 The preexponential and activation energy for the diffusion of iron in cobalt are 1.1 105 m2/s and 253,300 J/mol, respectively. At what temperature will the diffusion coefficient have a value of 2.1 1014m2/s?
5.20 The activation energy for the diffusion of car- bon in chromium is 111,000 J/mol. Calculate the diffusion coefficient at 1100 K (827C), given that D at 1400 K (1127C) is 6.25 1011m2/s.
5.21 The diffusion coefficients for iron in nickel are given at two temperatures:
T (K) D (m2/s)
1273 9.4 1016
1473 2.4 1014
(a) Determine the values of D0 and the activation energy Qd.
(b) What is the magnitude of D at 1100C (1373 K)?
5.22 The diffusion coefficients for silver in copper are given at two temperatures:
T( C) D(m2/s)
650 5.5 1016
900 1.3 1013
(a) Determine the values of D0and Qd. (b) What is the magnitude of D at 875C?
5.23 The following figure shows a plot of the logarithm (to the base 10) of the diffusion coefficient versus reciprocal of the absolute temperature, for the diffusion of iron in chromium. Determine values for the activation energy and preexponential.
ⴗ
5.24 Carbon is allowed to diffuse through a steel plate 15 mm thick. The concentrations of car- bon at the two faces are 0.65 and 0.30 kg C/m3 Fe, which are maintained constant. If the pre- exponential and activation energy are 6.2 107m2/s and 80,000 J/mol, respectively, com- pute the temperature at which the diffusion flux is 1.43 109kg/m2 s.
5.25 The steady-state diffusion flux through a metal plate is 5.4 1010kg/m2 s at a temperature of 727C (1000 K) and when the concentration gradient is 350 kg/m4. Calculate the diffusion flux at 1027C (1300 K) for the same concen- tration gradient and assuming an activation energy for diffusion of 125,000 J/mol.
5.26 At approximately what temperature would a specimen of -iron have to be carburized for 2 h to produce the same diffusion result as at 900C for 15 h?
5.27 (a) Calculate the diffusion coefficient for copper in aluminum at 500C.
(b) What time will be required at 600C to produce the same diffusion result (in terms of concentration at a specific point) as for 10 h at 500C?
5.28 A copper–nickel diffusion couple similar to that shown in Figure 5.1a is fashioned. After a 700-h heat treatment at 1100C (1373 K), the concentration of Cu is 2.5 wt% at the 3.0-mm position within the nickel. At what tempera- ture must the diffusion couple be heated to produce this same concentration (i.e., 2.5 wt%
Cu) at a 2.0-mm position after 700 h? The pre- exponential and activation energy for the diffusion of Cu in Ni are given in Table 5.2.
5.29 A diffusion couple similar to that shown in Figure 5.1a is prepared using two hypothetical
g
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Reciprocal temperature (1000/K)
0.55 0.60 0.65 0.70
Diffusion coefficient (m2/s) 10–16 10–15
Questions and Problems • 147 metals A and B. After a 30-h heat treatment at 1000 K (and subsequently cooling to room tem- perature) the concentration of A in B is 3.2 wt%
at the 15.5-mm position within metal B. If an- other heat treatment is conducted on an iden- tical diffusion couple, only at 800 K for 30 h, at what position will the composition be 3.2 wt%
A? Assume that the preexponential and acti- vation energy for the diffusion coefficient are 1.8 105m2/s and 152,000 J/mol, respectively.
5.30 The outer surface of a steel gear is to be hard- ened by increasing its carbon content. The carbon is to be supplied from an external carbon-rich atmosphere, which is maintained at an elevated temperature. A diffusion heat treatment at 850C (1123 K) for 10 min in- creases the carbon concentration to 0.90 wt%
at a position 1.0 mm below the surface. Estimate the diffusion time required at 650C (923 K) to achieve this same concentration also at a 1.0-mm position. Assume that the surface car- bon content is the same for both heat treat- ments, which is maintained constant. Use the diffusion data in Table 5.2 for C diffusion in
-Fe.
5.31 An FCC iron–carbon alloy initially containing 0.20 wt% C is carburized at an elevated tem- perature and in an atmosphere wherein the surface carbon concentration is maintained at 1.0 wt%. If after 49.5 h the concentration of carbon is 0.35 wt% at a position 4.0 mm below the surface, determine the temperature at which the treatment was carried out.
Diffusion in Semiconducting Materials
5.32 Phosphorus atoms are to be diffused into a silicon wafer using both predeposition and drive-in heat treatments; the background con- centration of P in this silicon material is known to be 5 1019atoms/m3. The prede- position treatment is to be conducted at 950C for 45 minutes; the surface concentration of P is to be maintained at a constant level of 1.5 1026atoms/m3. Drive-in diffusion will be carried out at 1200C for a period of 2.5 h. For the diffusion of P in Si, values of Qd and D0are 3.40 eV/atom and 1.1 104m2/s, respectively.
(a) Calculate the value of Q0.
(b) Determine the value of xjfor the drive-in diffusion treatment.
a
(c) Also for the drive-in treatment, compute the position x at which the concentration of P atoms is 1024m3.
5.33 Aluminum atoms are to be diffused into a sil- icon wafer using both predeposition and drive-in heat treatments; the background con- centration of Al in this silicon material is known to be 3 1019atoms/m3. The drive-in diffusion treatment is to be carried out at 1050C for a period of 4.0 h, which gives a junction depth xjof 3.0 m. Compute the pre- deposition diffusion time at 950C if the sur- face concentration is maintained at a constant level of 2 1025 atoms/m3. For the diffusion of Al in Si, values of Qd and D0 are 3.41 eV/atom and 1.38 104m2/s, respectively.
Spreadsheet Problems
5.1SS For a nonsteady-state diffusion situation (constant surface composition) wherein the surface and initial compositions are pro- vided, as well as the value of the diffusion coefficient, develop a spreadsheet that will allow the user to determine the diffusion
time required to achieve a given composition at some specified distance from the surface of the solid.
5.2SS For a nonsteady-state diffusion situation (constant surface composition) wherein the surface and initial compositions are provided, as well as the value of the diffusion coeffi- cient, develop a spreadsheet that will allow the user to determine the distance from the surface at which some specified composition is achieved for some specified diffusion time.
5.3SS For a nonsteady-state diffusion situation (con- stant surface composition) wherein the sur- face and initial compositions are provided, as well as the value of the diffusion coefficient, develop a spreadsheet that will allow the user to determine the composition at some specified distance from the surface for some specified diffusion time.
5.4SS Given a set of at least two diffusion coeffi- cient values and their corresponding tem- peratures, develop a spreadsheet that will allow the user to calculate (a) the activation energy and (b) the preexponential.
D E S I G N P R O B L E M S
Steady-State Diffusion
(Factors That Influence Diffusion)
5.D1 It is desired to enrich the partial pressure of hydrogen in a hydrogen–nitrogen gas mixture for which the partial pressures of both gases are 0.1013 MPa (1 atm). It has been proposed to accomplish this by passing both gases through a thin sheet of some metal at an elevated temperature; inasmuch as hydrogen diffuses through the plate at a higher rate than does nitrogen, the partial pressure of hydrogen will be higher on the exit side of the sheet.The design calls for partial pressures of 0.0709 MPa (0.7 atm) and 0.02026 MPa (0.2 atm), respectively, for hydrogen and nitro- gen. The concentrations of hydrogen and nitrogen (CHand CN, in mol/m3) in this metal are functions of gas partial pressures (pH2and pN2, in MPa) and absolute temperature and are given by the following expressions:
(5.16a)
(5.16b) Furthermore, the diffusion coefficients for the diffusion of these gases in this metal are func- tions of the absolute temperature as follows:
(5.17a)
(5.17b) Is it possible to purify hydrogen gas in this manner? If so, specify a temperature at which DN1m2/s23.0107 expa76.15 kJ/mol
RT b
DH1m2/s21.4107 expa13.4 kJ/mol
RT b
CN2.751031pN2 exp a37.6 kJ/mol
RT b
CH2.51031pH2 exp a27.8 kJ/mol
RT b
the process may be carried out, and also the thickness of metal sheet that would be re- quired. If this procedure is not possible, then state the reason(s) why.
5.D2 A gas mixture is found to contain two di- atomic A and B species for which the partial pressures of both are 0.05065 MPa (0.5 atm).
This mixture is to be enriched in the partial pressure of the A species by passing both gases through a thin sheet of some metal at an elevated temperature. The resulting en- riched mixture is to have a partial pressure of 0.02026 MPa (0.2 atm) for gas A, and 0.01013 MPa (0.1 atm) for gas B. The con- centrations of A and B (CA and CB, in mol/m3) are functions of gas partial pressures (pA2and pB2, in MPa) and absolute temper- ature according to the following expressions:
(5.18a)
(5.18b) Furthermore, the diffusion coefficients for the diffusion of these gases in the metal are func- tions of the absolute temperature as follows:
(5.19a)
(5.19b) Is it possible to purify the A gas in this man- ner? If so, specify a temperature at which the process may be carried out, and also the DB1m2/s22.5106 exp a24.0 kJ/mol
RT b
DA1m2/s24.0107 expa15.0 kJ/mol
RT b
CB1.01031pB2 exp a30.0 kJ/mol
RT b
CA2001pA2 exp a25.0 kJ/mol
RT b
Design Problems • 149 thickness of metal sheet that would be re- quired. If this procedure is not possible, then state the reason(s) why.
Nonsteady-State Diffusion (Factors That Influence Diffusion)
5.D3 The wear resistance of a steel shaft is to be improved by hardening its surface. This is to be accomplished by increasing the nitrogen content within an outer surface layer as a result of nitrogen diffusion into the steel. The nitrogen is to be supplied from an external nitrogen-rich gas at an elevated and constant temperature. The initial nitrogen content of the steel is 0.002 wt%, whereas the surface concentration is to be maintained at 0.50 wt%.
For this treatment to be effective, a nitrogen content of 0.10 wt% must be established at a position 0.40 mm below the surface. Spec- ify appropriate heat treatments in terms of temperature and time for temperatures be- tween 475C and 625C. The preexponential and activation energy for the diffusion of ni- trogen in iron are 3 107m2/s and 76,150 J/mol, respectively, over this temperature range.
Diffusion in Semiconducting Materials
5.D4 One integrated circuit design calls for the diffusion of arsenic into silicon wafers; the background concentration of As in Si is 2.5 1020atoms/m3. The predeposition heat treatment is to be conducted at 1000C for 45 minutes, with a constant surface con- centration of 8 1026 As atoms/m3. At a drive-in treatment temperature of 1100C, determine the diffusion time required for a junction depth of 1.2 m. For this system, values of Qd and D0 are 4.10 eV/atom and 2.29 103m2/s, respectively.