# Department of Electrical and Computer Engineering: Final Examination

University of Waterloo Department of Electrical & Computer Engineering E&CE 231 Final Examination – Spring 2000 Aids: Formula Sheets (attached), Scientific Calculator Time Allowed: 3 hours Exam Type: Closed Book Instructor: C. R. Selvakumar Date: August 10, 2000 Max Marks: 100 Instructions: Answer all questions in PART-A and any two questions in full 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) Assume that the semiconductor is Silicon. (b) Assume that the temperature T = 300K c) Use the data given in the formula 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 ? ? 300 ? * n ? m* ? 2 ? T ? 2 p ?3 19 N V = 2. 5 ? 10 ? ? m ? ? 300? cm ?? ? ? 0? PART -A 1a) Consider 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 µm by 20 µm. (i) Without doing any calculations, sketch the capacitance versus reverse voltage (VR) starting from VR = 0. (4 marks) (ii)

Calculate the voltage at which you will obtain the minimum capacitance and also determine (calculate) the minimum capacitance at that voltage. (10 marks) (iii) Derive the mathematical relations you use in calculating the quantities in (ii) above. (16 marks) 1b) Assuming that the p+ region and the n-region of the diode described in 1a) above are ‘long’ compared to the minority carrier diffusion lengths in those regions, show 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 electron and hole current distributions in the entire device. (10 marks) Page 1 PART B 2a) Draw a clearly labelled band diagram of an n-p-n transistor under thermal equilibrium and superimpose on it a band diagram of the same transistor when it is under normal forward active mode 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 carrier diffusivities and diffusion lengths. Assume that there is no recombination in the neutral base or in the space-charge layers.

Also, assume that the conventional reverse saturation current of the reverse-biased diode, IC0, is negligible. 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 µm 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 values 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 efficiency 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 Diffusion ?

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-channel MOSFET and explain how the MOSFET operates using key 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. Illustrate your answer with sketches of Reverse bias I-V characteristics giving physical reasons. (10 marks) Page 3 E&CE 231 1/4 Formula Sheet C. R. Selvakumar E&CE 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 ? µn = q? c,n m* n and µ 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 ? type , where ? c,n and ? ,p are mean time between collisions ? = qmn n + qm p p and r = 1/s dn ? dp ? ? ? J n = q? nµn ? + Dn ? ; J p = q ? pµ p ? ? D p ? ? ? dx ? dx ? D p Dn kT = = = 0. 0259 V at 300K µ p µn 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 short 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 = µ n Ci ? ? ? (VG ? VT )V D ? VD ? ? L? ? 2 ? µ n Ci ? Z ? 2 I DSat = ? ? (V ? VT ) V Dsat = VG ? VT 2 ? L? G E&CE 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&CE 231 3/4 Formula Sheet C. R. Selvakumar Mobilities in Silicon N = doping density (cm ? 3 ) µ (N) = µ min + Carrier type µ0 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 Lifetimes (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&CE 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 weight 28. 08 144. 63 density g/cm3 2. 33 5. 32 breakdown field 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 Physical Constants ?1. 38×10 ? 23 J / K ? k ? ?8. 62×10 ? 5 eV / K ? ? 31 m0 9. 11×10 kg ?0 8. 85×10 ? 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. 62×10 c 3×10 q 1. 6×10 4. 07 energy gap, eV 1. 12 1. 43 intrinsic carrier conc. , ni cm-3 at T = 300K 1. 5 x 1010 1. 8 x 106 effective mass 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