1、1.2. Assuming dopant atoms are uniformly distributed in a silicon crystal, how far apart are these atoms when the doping concentration is a). 1015 cm-3, b). 1018 cm-3, c). 5x1020 cm-3.Answer:The average distance between the dopant atoms would just be one over the cube root of the dopant concentratio

2、n:a) b) c) 1.3. Consider a piece of pure silicon 100 m long with a cross-sectional area of 1 m2. How much current would flow through this “resistor” at room temperature in response to an applied voltage of 1 volt?Answer:If the silicon is pure, then the carrier concentration will be simply ni. At roo

3、m temperature, ni 1.45 x 1010 cm-3. Under an applied field, the current will be due to drift and hence, 1.10. A state-of-the-art NMOS transistor might have a drain junction area of 0.5 x 0.5 m. Calculate the junction capacitance associated with this junction at an applied reverse bias of 2 volts. As

4、sume the drain region is very heavily doped and the substrate doping is 1 x 1016 cm-3. Answer:The capacitance of the junction is given by Eqn. 1.25.The junction built-in voltage is given by Eqn. 1.24. ND is not specified except that it is very large, so we take it to be 1020 cm-3 (roughly solid solu

5、bility). The exact choice for ND doesnt make much difference in the answer.Since ND NA in this structure, the capacitance expression simplifies toGiven the area of the junction (0.25 x 10-8 cm2, the junction capacitance is thus 4.2 x 10-17 Farads.3.2. A boron-doped crystal pulled by the Czochralski

6、technique is required to have a resistivity of 10 cm when half the crystal is grown. Assuming that a 100 gm pure silicon charge is used, how much 0.01 cm boron doped silicon must be added to the melt? For this crystal, plot resistivity as a function of the fraction of the melt solidified. Assume k0

7、= 0.8 and the hole mobility p = 550 cm2 volt-1 sec-1. Answer:Using the mobility value given, and we have:10 cm NA = 1.14 x 1015 cm-3 and 0.01 cm NA = 1.14 x 1018 cm-3From Eqn. 3.38, and we want CS = 1.14 x 1015 cm-3 when f = 0.5. Thus, solving for C0 the initial doping concentration in the melt, we

8、have:The resistivity as a function of distance is plotted below and is given by 3.3. A Czochralski crystal is pulled from a melt containing 1015 cm-3 boron and 2x1014 cm-3 phosphorus. Initially the crystal will be P type but as it is pulled, more and more phosphorus will build up in the liquid becau

9、se of segregation. At some point the crystal will become N type. Assuming kO = 0.32 for phosphorus and 0.8 for boron, calculate the distance along the pulled crystal at which the transition from P to N type takes place.Answer:We can calculate the point at which the crystal becomes N type from Eqn. 3

10、.38 as follows:At the point where the cross-over occurs to N type, these two concentrations will be equal. Solving for f, we findThus only the last 0.5% of the crystal is N type.3.6. Suppose your company was in the business of producing silicon wafers for the semiconductor industry by the CZ growth

11、process. Suppose you had to produce the maximum number of wafers per boule that met a fairly tight resistivity specification.a). Would you prefer to grow N type or P type crystals? Why?b). What dopant would you use in growing N-type crystals? What dopant would you use in growing P type crystals? Exp

12、lainAnswer:a). Boron has the segregation coefficient closest to unity of all the dopants. Thus it produces the most uniform doping along the length of a CZ crystal. Thus P type would be the natural choice.b). For P type, the obvious (and only real choice) is boron as explained in part a). For N type

13、 crystals Fig. 3-18 shows that either P or As would be a reasonable choice since their segregation coefficients are quite close and are better than Sb. Table 3-2 indicates that P might be slightly preferred over As because its kO value is slightly closer to 1. 4.1. An IC manufacturing plant produces

14、 1000 wafers per week. Assume that each wafer contains 100 die, each of which can be sold for $50 if it works. The yield on these chips is currently running at 50%. If the yield can be increased, the incremental income is almost pure profit because all 100 chips on each wafer are manufactured whethe

15、r they work or not. How much would the yield have to be increased to produce an annual profit increase of $10,000,000? Answer:At 1000 wafers per week, the plant produces 52,000 wafers per year. If each wafer has 50 good die each of which sells for $50, the plant gross income is simplyIncome = (52,00

16、0)(50)($50) = $130,000,000 per year.To increase this income by $10,000,000 requires that the yield increase by4.3. As MOS devices are scaled to smaller dimensions, gate oxides must be reduced in thickness. a. As the gate oxide thickness decreases, do MOS devices become more or less sensitive to sodi

17、um contamination? Explain. b. As the gate oxide thickness decreases, what must be done to the substrate doping (or alternatively the channel VTH implant, to maintain the same VTH? Explain.Answer:a). From the text, Na+ contamination causes threshold voltage instabilities in MOS devices. Also from Eqn

18、. 4.1, the threshold voltage is given byAs the gate oxide thickness decreases, COX increases, so the same amount of mobile charge QM will have less effect on VTH as oxides get thinner. Therefore MOS devices are less sensitive to sodium contamination.b). Using the same expression for VTH as in part a

19、), we observe that as the oxide thickness decreases, (COX increases), to maintain the same VTH, NA will have to increase. NA will actually have to increase by the square root of the oxide thickness decrease to keep VTH constant. 4.4. A new cleaning procedure has been proposed which is based on H2O s

20、aturated with O2 as an oxidant. This has been suggested as a replacement for the H2O2 oxidizing solution used in the RCA clean. Suppose a Si wafer, contaminated with trace amounts of Au, Fe and Cu is cleaned in the new H2O/O2 solution. Will this clean the wafer effectively? Why or why not? Explain.A

21、nswer:As described in the text, cleaning metal ions off of silicon wafers involves the following chemistry:The cleaning solution must be chosen so that the reaction is driven to the right because this puts the metal ions in solution where they can be rinsed off. Since driving the reaction to the rig

22、ht corresponds to oxidation, we need an oxidizing solution to clean the wafer. H2O/O2 is certainly an oxidizing solution. But whether it cleans effectively or not depends on the standard oxidation potential of the various possible reactions. From Table 4-3 in the text, we have:Oxidant/ReductantStand

23、ard OxidationPotential (volts)Oxidation-Reduction ReactionSiO2/Si0.84/Fe0.17/Cu-0.34/H2O-1.23/Au-1.42The stronger reactions (dominating) are at the bottom. Thus the H2O/O2 reaction will clean Fe and Cu, but it will not clean Au off the wafer. 4.5. Explain why it is important that the generation life

24、time measurement illustrated in Figure 4-19 is done in the dark. Answer:The measurement depends on measuring carriers generated thermally in the silicon substrate (or at the surface). If light is shining on the sample, then absorbed photons can also generate the required carriers. As a result, the e

25、xtracted generation lifetime with the light on would really be measuring the intensity of the incident light and not a basic property of the silicon material. 5.1. Calculate and plot versus exposure wavelength the theoretical resolution and depth of focus for a projection exposure system with a NA o

26、f 0.6 (about the best that can be done today). Assume k1 = 0.6 and k2 = 0.5 (both typical values). Consider wavelengths between 100 nm and 1000 nm (DUV and visible light). ). Indicate the common exposure wavelengths being used or considered today on your plot (g-line, i-line, KrF and ArF). Will an A

27、rF source be adequate for the 0.13 m and 0.1 m technology generations according to these simple calculations?Answer:The relevant equations are simplyThese equations are plotted below. Note that the ArF (193 nm) will not reach 0.13 m or 0.1 m resolution according to these simple calculations. In fact

28、, with more sophisticated techniques such as phase shift masks, off axis illumination etc., ArF is expected to reach 0.13 m and perhaps the 0.1 m generations.5.3. An X-ray exposure system uses photons with an energy of 1 keV. If the separation between the mask and wafer is 20 m, estimate the diffrac

29、tion limited resolution that is achievable by this system.Answer:The equivalent wavelength of 1 keV x-rays is given byX-ray systems operate in the proximity printing mode, so that the theoretical resolution is given by Eqn. 5.12:5.8. As described in this chapter, there are no clear choices for litho

30、graphy systems beyond optical projection tools based on 193-nm ArF eximer lasers. One possibility is an optical projection system using a 157-nm F2 excimer laser. a. Assuming a numerical aperture of 0.8 and k1 = 0.75, what is the expected resolution of such a system using a first order estimate of r

31、esolution? b. Actual projections for such systems suggest that they might be capable of resolving features suitable for the 2009 0.07 m generation. Suggest three approaches to actually achieving this resolution with these systems.Answer a). The simple formula for resolution is b). The calculated res

32、olution in part a is a factor of two larger than required for the 0.07 m generation. Therefore some “tricks” will have to be used to actually achieve such resolution. There are a number of possibilities:1. Use of phase-shift masks. This technique, discussed in this chapter, has the potential for sig

33、nificant resolution improvements. It works by designing a more sophisticated mask. Simple masks are digital - black or white. Phase shifting adds a second material to the mask features, usually at the edges which shifts the optical phase and sharpens up the aerial image. Sophisticated computer progr

34、ams are required to design such masks. 2. Use of optical proximity correction in the mask design. This is another approach to designing a better mask and as discussed in class, can also improve resolution significantly. The approach involves adding extra features to the mask, usually at corners wher

35、e features are sharp, to compensate for the high frequency information lost to diffraction effects.3. Off-axis illumination. This allows the optical system to capture some of the higher order diffracted light and hence can improve resolution. 5.9. Current optical projection lithography tools produce

36、 diffraction limited aerial images. A typical aerial image produced by such a system is shown in the simulation below where a square and rectangular mask regions produce the image shown. (The mask features are the black outlines, the calculated aerial image is the grayscale inside the black rectangl

37、es.) The major feature of the aerial image is its rounded corners compared to the sharp square corners of the desired pattern. Explain physically why these features look the way they do, using diffraction theory and the physical properties of modern projection optical lithography tools.Answer:Modern

38、 optical projection lithography systems are limited in the resolution they can achieve by diffraction effects. The finite size of the focusing lens means that the high order diffraction components are “l(fā)ost” and are therefore not available to help in printing a replica of the mask image. But the hig

39、h frequency spatial components are exactly the components that contain information about “sharp” features, i.e. corners etc. Thus the projected aerial image loses this information and corners become rounded. The only ways to improve the image are by using shorter wavelength light, or a higher NA len

40、s.5.10. Future optical lithography systems will likely use shorter exposure wavelengths to achieve higher resolution and they will also likely use planarization techniques to provide “flat” substrates on which to expose the resist layers. Explain why “flat” substrates will be more important in the f

41、uture than they have been in the past. Answer:As the wavelength of the exposure system decreases, the depth of focus of the exposure system also decreases. Thus it will be necessary to make sure that the resist in which the image is to be exposed, is flat and does not require much depth of focus. Pl

42、anarization techniques will be required to accomplish this. This could mean CMP to planarize the substrate before the resist is applied, or it could mean using a spun on resist which planarizes the substrate and which is then covered with a thin, uniform imaging resist layer.6.4. Construct a HF CV p

43、lot for a P-type silicon sample, analogous to Fig. 6-9. Explain your plot based on the behavior of holes and electrons in the semiconductor in a similar manner to the discussion in the text for Fig. 6.9.Answer:The C-V plot looks basically the same as the N substrate example in the text, that we disc

44、ussed in class, except that the horizontal axis is flipped. For negative applied gate voltages, the majority carrier holes in the substrate are attracted to the surface. This is the accumulation region a) above. We measure just COX for the capacitance since there is no depletion in the substrate. Fo

45、r + VG, the holes are driven away from the surface creating first a depletion region as in b) and finally an inversion layer of electrons as in c). The measured capacitance drops as we move into depletion and finally reaches a minimum value after an inversion layer forms. The C-V curves shown are hi

46、gh frequency curves. As discussed in the text, the capacitance remains at its minimum value for + VG values greater than VTH because the inversion layer electrons cannot be created or destroyed as fast as the signal is changing. Hence the small AC signal must “wiggle” the bottom of the depletion reg

47、ion to balance VG. 6.6. In a small MOS device, there may be a statistical variation in VT due to differences in QF from one device to another. In a 0.13 m technology minimum device (gate oxide area = 0.1m x 0.1m) with a 2.5nm gate oxide, what would the difference in threshold voltage be for devices

48、with 0 or 1 fixed charge in the gate oxide?Answer:The oxide capacitance is The change in threshold voltage is given by This shows that a single electron trap in a gate oxide will have a negligible effect on the threshold voltage at this technology generation.6.12 A silicon wafer is covered by an SiO

49、2 film 0.3 mm thick. a. What is the time required to increase the thickness by 0.5 mm by oxidation in H2O at 1200C?b. Repeat for oxidation in dry O2 at 1200C.Answer:We will perform the calculation for silicon wafers. For wafers, the linear rate constant should be divided by 1.68.a. At 1200C, in H2OT

50、he initial oxide, if grown at 1200C would have taken this long to growThe time required to grow at 1200C isThus, the time required to add to an existing film is or 41.7 minutes.b. At 1200C, in dry oxygenThe initial oxide would have taken 2.206 hours to grow in dry oxygen, it would require 14.217 hou

51、rs to grow , thus would require an additional 12 hours to add to an existing film.6.13. Suppose an oxidation process is used in which (100) wafers are oxidized in O2 for three hrs. at 1100C, followed by two hrs. in H2O at 900C, followed by two hrs in O2 at 1200C. Use Figs. 6-19 and 6-20 in the text

52、to estimate the resulting final oxide thickness. Explain how you use these figures to calculate the results of a multi-step oxidation like this.Answer:We can use these figures to estimate the oxide thickness as follows. First, we use Fig. 6-19 for the first dry oxidation cycleA three hour oxidation

53、at 1100C produces an oxide thickness of about 0.21 m. We next use Fig. 6-20 for the wet oxidation as shown below. The oxidation is 2 hrs in H2O at 90 C. We start by finding the point on the 900C curve that corresponds to 0.21 m since this is the starting oxide thickness. This is point A. We then mov

54、e along the 900C curve by two hours to point B. This corresponds to a thickness of about 0.4 m which is the thickness at the end of the wet oxidation.We now go back to Fig. 6-19 for the final dry O2 cycle. This process is 2 hrs at 1200C. We start by finding the point on the 1200C curve that correspo

55、nds to a starting oxide thickness of 0.4 m. This is point A below. We then increment the time by 2 hrs along the 1200C curve, to arrive at a final oxide thickness of about 0.5 m.6.18. Silicon on Insulator or SOI is a new substrate material that is being considered for future integrated circuits. The

56、 structure, shown below, consists of a thin single crystal silicon layer on an insulating (SiO2) substrate. The silicon below the SiO2 provides mechanical support for the structure. One of the reasons this type of material is being considered, is because junctions can be diffused completely through the thin silicon layer to the underlying SiO2. This reduces junction capacitances and produces faster circuits. Isolation is also easy to achieve in this material, because the thin Si layer can be completely oxidized, resulting in devices completely surrounded by SiO2. A LOCOS process