Department of Electrical and Computer Engineering: Final Examination

University of Waterloo Department of Electrical &038 Computer Engineering E&038CE 231 Final Examination Spring 2000 Aids principle Sheets (attached), scientific Calculator Time Allowed 3 hours Exam Type Closed script Instructor C. R. Selvakumar Date August 10, 2000 Max Marks light speed operating instructions Answer all questions in PART-A and any two questions in replete(p) from PART-B. State your assumptions clearly. Be concise, precise and clear in your answers General assumptions to be made when not specified in a question (a) bust that the semiconductor is te. (b) relieve that the temperature T = three hundredK c) Use the data given in the principle sheets where needed. (d) Use the following expressions for the Effective Density of States in the conduction Band (NC) and in the Valence Band (NV) respectively 3 2 3 3 3 ? m ? ? T ? 2 ? 3 ?? N C = 2. 5 ? 1019 ? ? cm ? m 0 ? ? ccc ? * n ? m* ? 2 ? T ? 2 p ?3 19 N V = 2. 5 ? 10 ? ? m ? ? 300? cm ?? ? ? 0? PART -A 1a) Consid er a Silicon p+-n diode with the following doping densities NA = 1019 cm-3 and ND is 1016 cm-3. The diode has an area of 100 m by 20 m. (i) Without doing any calculations, skeleton the capability versus mouse quintupletage (VR) starting from VR = 0. (4 marks) (ii)Calculate the voltage at which you will confine the minimal capacitance and also determine (calculate) the minimum capacitance at that voltage. (10 marks) (iii) go down the mathematical relations you use in calculating the quantities in (ii) above. (16 marks) 1b) anticipate that the p+ region and the n-region of the diode described in 1a) above are long compared to the minority bearer scattering lengths in those regions, doom how you would obtain the complete Current-Voltage (I-V) Characteristic of the diode. You can assume that there is no recombination in the space-charge layer and you need not solve the continuity equation.Sketch the negatron and repair current distributions in the entire device. (10 marks ) Page 1 PART B 2a) Draw a clearly labelled surround diagram of an n-p-n electronic transistor under thermic equilibrium and superimpose on it a band diagram of the same transistor when it is under normal forward nimble path of operations. (8 marks) 2b) Derive an expression for the common emitter current gain $ ($ = IC/IB), in terms of the doping densities in the different regions, thickness and flattop diffusivities and diffusion lengths. arrogate that there is no recombination in the neutral subject or in the space-charge layers.Also, assume that the conventional reverse saturation current of the reverse-biased diode, IC0, is minimal. Assume that short-region approximation is valid in the base and that the bandgap narrowing in the emitter is important. No need to solve continuity equations and you can assume the expected carrier distributions. (12 marks) 2c) Obtain the modified Ebers-Moll (EM) equations from the original EM equations given in the formula sheet. Sketch Common -Base output characteristics based on the modified EM equations and show the Forward Active Region of operation, Saturation Region and Cut-off Region. 10 marks) 3a) A silicon n-p-n transistor has an emitter doping NDE = 1020 cm-3 and a base doping NAB = 1016 cm-3. The emitter is 1 m thick and assume that the hole diffusion length in the emitter is 0. 1 m. The base is 0. 35 m thick and you can use the value of mobilities and lifetimes given in the tables in the formula sheet to determine the electron diffusion length in the base. Verify that the short-region approximation is applicable to the base. Assume that the carrier recombinations in the neutral base an in the emitter-base depletion layer are zero. When this transistor is operating in the normal forward active mode with 0. volts forward bias across the emitter-base junction and a 2 volt reverse bias across the collector-base junction, what is the collector current density (JC) and the base current density (JB) ? You can assume that the depletion layer thicknesses are negligible at both junctions. Assume that bandgap narrowing for the emitter doping is 100 meV and the room temperature is 300K. (15 marks) 3b) What is the emitter efficacy of the transistor in 3a)? (5 marks) 3c) What do you understand by diffusion capacitance of a diode? Show (derive) that the diffusion capacitance of a p+ n diode is approximately given by C scattering ?Qp Vt where Qp is the total injected minority hole charge on the n-side quasi-neutral=region and Vt is the thermal voltage (kT/q). Prove that the quantity Q p ? qAL p pn 0 e V Vt (10 marks) Page 2 4a) Consider an n- have a bun in the oven MOSFET and explain how the MOSFET operates using list band diagrams (along source, channel and drain and vertically along the metal gate, oxide and the channel region) and cross-sectional diagrams. State clearly wherefrom the channel electrons come and explain how this is controlled by the gate voltage. (10 marks) 4b) With reference to an n-p-n transistor, explain what is Early Effect and how it arises.Using an approximate sketch show the Early Voltage. Clearly illustrate your answer with the aid of carrier profiles and common-emitter output characteristics. (10 marks) 4c) Contrast the Temperature-dependence of Avalanche Breakdown Mechanism and Zener breakdown Mechanism. expand your answer with sketches of Reverse bias I-V characteristics giving physical reasons. (10 marks) Page 3 E&038CE 231 1/4 Formula Sheet C. R. Selvakumar E&038CE 231 Formula Sheet 3 1 4? *2 g c (E) = 3 (2m n ) ( E ? E C )) 2 (E ? E c ) h 3 1 4? *2 2 g V (E) = 3 2m p ( E V ? E)) (E ? E V ) h 1 f FD (E) = (E-E F )/kT 1+ e p 0 = N V e (E V ? E F )/kT = n i e (Ei ?E F )/kT () n 0 p0 = n 2 i 3/2 ? 2? m* kT ? p N V = 2? ? 2 ? ? ?h ? n = q? c,n m* n and p = q? c,p m* p ? max = ? qN A x p0 ? 0? r qN + x n0 D = ?0? r 1/2 x n0 ? 2? r ? 0 V0 ? NA =? ? q N D (N A + N D ) ? ? ? 2? r ? 0 V0 ? ND =? ? q N A (N A + N D ) ? ? 1/2 3/2 ? p 0 + N + = n0 + N A D + ? ?2 ? N D ? NA ?? N + ? NA ? D ? + n2 ? + ?? n0 = i 2 2 ? ? ? ?? ? + ? N D x n0 = N A x p0 x p0 n 0 = N C e (E F ? EC )/ kT = n i e (E F ? E i )/kT ? 2? m* kT ? n N C = 2? ? 2 ?h ? ? kT ? n no p po ? kT ? N + N A ? D V0 = ln? ?= ln? ? q ? n2 ? q ? n2 ? i i p( x n0 ) = pn e qV / kT and ? pn = pn ( e qV / kT ? 1) 1/2 for n ? shell , where ? c,n and ? ,p are symbolize time between collisions ? = qmn n + qm p p and r = 1/s dn ? dp ? ? ? J n = q? nn ? + Dn ? J p = q ? p p ? ? D p ? ? ? dx ? dx ? D p Dn kT = = = 0. 0259 V at 300K p n q n( ? x p0 ) = n p e qV / kT and ? n p = n p (e qV / kT ? 1) ? p( x n ) = ? pn e or ? p( x n ) = ? pn ( 0) e ? x p / Ln or ? n( x p ) = ? n p ( 0) e ?n( x p ) = ? n p e ? xn / L p ? x p / Ln ? Dn ? Dp ? I = qA? n p0 + p n0 ? (e qV/ kT ? 1) ? Lp ? Ln ? ? ? qN ? C j = A? Si d ? ? 2(V0 ? V ) ? 1/ 2 for p + ? n diffusion capacitance C s = q 2 AL p kT p n0 e qV/kT for p + ? n n ? type regions of width, W long base diode approx I p = qAD p ? pn ( 0 )Lp sh ort base diode approx I p = qAD p ?p 1 dJ p ?n 1 dJ n =? + G ? Rp = ? + G ? Rn ?t q dx ?t q dx Wm = L p = D p ? p and Ln = Dn ? n VT = d 2V d? ? ? 2= = where ? = q ( p ? n + N d ? N a ) dx ? 0 ? r dx dV 1 dE c 1 dE v 1 dE t ?= ? = = = dx q dx q dx q dx ? xn / L p 2? Si ( 2? F ) qN a for VG > Vth ? pn ( 0 ) W ? Si = ? 0 ? r ? Qd Qi + 2? F + ? ms ? , Ci Ci Q d = Q B = ? qN a x dm ,x dm = Wm ?? ? Ci = Cox = 0 ox = i t ox d 1 2? ? Z? ? I D = n Ci ? ? ? (VG ? VT )V D ? VD ? ? L? ? 2 ? n Ci ? Z ? 2 I DSat = ? ? (V ? VT ) V Dsat = VG ? VT 2 ? L? G E&038CE 231 2/4 Formula Sheet C. R. Selvakumar Eber-Moll Model (n-p-n transistor)I EBO (e VBE / Vt ? 1) RIC I CBO (e VBC /Vt ? 1) FIE ? VBE ? ? VBC ? I E = ? I ES ? e Vt ? 1? + ? R I CS ? e Vt ? 1? ? ? ? ? ? ? ? ? ? VBE ? ? VBC ? Vt ?e ? + I CS ? e Vt ? 1? I C = ? R I ES ? ? 1? ? ? ? ? ? ? E&038CE 231 3/4 Formula Sheet C. R. Selvakumar Mobilities in Silicon N = doping density (cm ? 3 ) (N) = min + Carrier type 0 N 1+ N ref min 0 cm2 / (v. s ) Nref cm-3 electron 88 1251. 8 1. 26 x 1017 hole 54. 3 406. 97 2. 35 x 1017 Doping density Mobilities life-times (J) as function of doping density N n p 1 1 = + cA N2 ? ? SRH 1015 1016 1017 1018 1019 1020 1322. 3 1218. 2 777. 3 262. 1 114. 1 91. 5 457. 96 437. 87 330. 87 43. 23 68. 77 56. 28 cm 2 v. sec cm 2 v. sec cm ? 3 Doping density N cm-3 Lifetime J sec For both electrons and holes 1015 1016 1017 1018 1019 1020 9. 8 x 10-6 8. 3 x 10-6 3. 3 x 10-6 4. 5 x 10-7 3. 3 x 10-8 8. 3 x 10-10 Obtained using the above formula for lifetime using JSRH = 10-5/(1 + 5 x 1016/N) and CA = 10-31 cm6s-1 E&038CE 231 4/4 Formula Sheet C. R. Selvakumar Properties of Silicon and Gallium Arsenide PROPERTY Si GaAs atoms or molecules/ cm3 5. 0 x 1022 4. 42 x 1022 atomic or molecular burthen 28. 08 144. 63 density g/cm3 2. 33 5. 32 breakdown field of operations V/cm 3 x 105 4 x 105 dielectric constant, gr 11. 8 13. 1 effective density of tates Nc cm-3 Nv cm-3 natural Constants ?1. 38&21510 ? 23 J / K ? k ? ?8. 62&21510 ? 5 eV / K ? ? 31 m0 9. 11&21510 kg ?0 8. 85&21510 ? 14 ? r (Si) 2. 8 x 1019 1. 04 x 1019 4. 7 x 1017 7. 0 x 1018 11. 8 ? r (SiO 2 ) 3. 9 h electron affinity, eV 4. 05 6. 62&21510 c 3&21510 q 1. 6&21510 4. 07 cipher gap, eV 1. 12 1. 43 intrinsic carrier conc. , ni cm-3 at T = 300K 1. 5 x 1010 1. 8 x 106 effective potful electrons holes m*n = 1. 1 m0 m*p = 0. 56 m0 m*n = 0. 067 m0 m*p = 0. 48 m0 intrinsic mobility 300K electrons cm2/Vs holes cm2/Vs 1350 480 8500 400 diffusivity 300K electrons cm2/s holes cm2/s 35 12. 5 220 10 F / cm 10 ? 34 J ? s cm / s ? 19 C