UNIVERSITY OF CALIFORNIA College of Engineering Department of Electrical Engineering and Computer Sciences. Midterm I
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1 UNIVERSITY OF CALIFORNIA College of Engineering Department of Electrical Engineering and Computer Sciences EECS 130 Fall 2006 Professor Ali Javey Midterm I Name: 1. Closed book. One sheet of notes is allowed. 2. Show all your work in order to receive partial credit. 3. Include correct units when appropriate. 4. Make sure everything is on the exam papers. Work on additional papers will NOT be accepted. 5. There are a total of 10 pages of this exam including this page. Make sure you have them all. Problem 1 30 Problem 2 15 & 5 extra credit Problem 3 28 Problem 4 27 Total 100 1
2 Physical Constants Electronic charge q 1.602xlO-19 C Pennittivity of vacuum co xlO-14 F cm-1 Relative pennittivity of silicon cs/ co 11.8 Boltzmann's constant k x 10-5eV/ K or 1.38xlO-23J K1 Thennal voltage at T =300K kt/q V Effective density of states Nc 2.8 x 1019cm-3 Effective density of states Ny 1.04 x 1019cm-3 Silicon Band Gap Eo 1.12 ev Intrinsic Carrier Concentration nj 1010cm-3 in Si at 300K 2
3 1. Carriers Concentrations [30 pts] This problem concerns a specimen of gallium arsenide, GaAs, which has 2x1017 cm-3donors and an unknown number of acceptors. A measurement is made on the specimen and it is found that it is p-type with an equilibrium hole concentration, p o, of 5 x 1017 cm At room temperature in GaAs, the intrinsic carrier concentration, n., I is 10 cm, the hole 2 2 mobility, J.Io,is 300 cm N-s, and the electron mobility, Ile' is 4000 cm N-s. The minority -9 carrier lifetime, t., is 10 s. mid a) [6 pts] What is the net acceptor concentration, N A(=Na -Nd),in this sample, and what is the total acceptor concentration, Na? 0:: NI1.- NrJ. + ( ~ NfA - ~rj) 2 ~. '" :1 ) Vl - P d-. ') N =,1.. " A (tv~- NJ ) ~ n?- :::) D.:: NfA-NJ.. jj" - NJ. AI _ AI -\ G-- (0 --;- T = I"q lye{ "/ Net; NA + NJ.:. 7'Xlo'7Ctry,_1 b) [6 pts] What is the equilibrium electron concentration, no' in this sample at room temperature? -- (IO~W'- 3) $ ~XI0 (1M c) [6 pts] Calculate EF- Ej in this sample at room temperature. f i..f % ~ b ~ VI i ~ KI p., - ;. e \.1: - (E ~- E,'KT ev -. P,~J I J ~10 CP"_J OJ /07 /" - ~O t\!(v ( ) S ) ~ - ~4).',M}./ 3
4 d) [6 pts] What is the electrical conductivity, <10,of this sample in thermal equilibrium at room temperature? ~O u.: = (1fJf 1. -! ~ h f. ) ~ t4 I.b110 C ~4 ~~ <10= S/cm e) [6 pts] This sample is illuminated by a steady state light which generates hole-electron pairs uniformly throughout its bulk, and the conductivity of the sample is found to increase by 1% (that is, to 1.01 (10).What are the excess hole and electron concentrations, ~p and ~n, in the illuminated sample, assuming that the illumination has been on for a long time? /Jh=~ "--- WI (fo' ~ pp (\1.1~\'1) + ~h ( ot~p)" ~.j. ~)A".A\'I+ ~ }A f ~f 0.01([0 : Cf, p" 6 f~ ff li~ ~n= ~p= - '-bx(o-(~l (h,ci+ ~ vooo(~ v.) \) 4
5 2. Temperature Dependence of Carrier Concentrations and Mobility [15 pts] A silicon wafer is moderately doped with arsenic. The plots in parts (a)-(c) show the relationship between In(n) and lit for this Si wafer, where n is the electron density in the conduction band and T is the temperature. In each case, clearly mark any pertinent shift in the curve and/or the slopes of the two non-flat regions as various properties of the semiconductor is changed. a) [5 pts] Draw a second curve that would correspond to an intrinsic (undoped) Si wafer. 1. E./ \ n: ~ Ale. tj" ~p (- ~/LtTJ nu.:i JMc.Jl -t)(p (- '~~-Q '~(V\it)~ '~JM:.tJ~- E~/~T b) [5 pts] Draw a second curve that would correspond to using a Ge (Eg-O.67)wafer with the same dopant density instead of a Si (Eg-l.l) wafer. Assume the same (ED- Ed. ~ ~1: 0.' E~.s" 1",("01.) "1..[,ieJIi - ~."..,. 1" S~AU(,~ S I'>pe lit lit 5
6 lit d) [5pts extra credit] Draw a second curves that would correspond to a heavily doped Si wafer. Hint: when doping density is high, the impurity energy level splits into a band of available states due to Pauli exclusion principle. This impurity band crosses Ec. t:::. ~; 1'L Jep I"'~ ; c \,., i {c."lc", IP!/1f1, r\~ ;s. 7' (YO r.,-, 0,. ~ ~ r {~., ',' r;-. ~ C '.'. " / c-? VI (" ( O'c fc"")'; ~ Cv", :;;.c,~, " <:., r ~.- t: r.,. J 1. ' 'r I '\.?, <:::.,..., F e I;oC.lt"O\l\j,c-""., '.-/'C\Jf.rcv-.r- OCpc.-"'.::.(-f' -.\) (Ycp~, C-. CA~ ~U T ('::>0 VI0 {,,(J(?" out c - lit 6
7 3. Band Model [28 pts] A silicon device maintained at 300 K is characterized by the following band diagram. Use the cited energy band diagram in answering parts (a)-(e) EG/ r:... 0 Xl E.. 1 E p..., :. Ev Eo/3 ~ :~ hole U3 2U3 L x (a) [8 pts] Sketch the electric field inside the semiconductor. C: J cl~t.,:.2 ~h :. i.1 bf~lyl'e(,. 'b d.'j. ~ L..h \.L t.. ~L L/:1 "l/ L (b) [5 pts] Do equilibrium conditions pr~vail (yes, no, or cannot determine)? YeJ. EFj) (O~)t." i."j (o"d ;1I100V 1. :~ s. :: 0 (c) [5 pts] Is the semiconductor degenerate at any point? If so, specify one point where this is the case. YeL De3eV\eVQ, Y Th~s ;~ c leull I.>' o (c.vv.s (/JSe 7
8 .. (d) [5 pts] What is the electron CUlTentdensity (IN)flowing at x =Xl? Bj ~p~',., ~ ~, J v.,j ',. e~"; I; L.: vw\ C\.: ()J (e) [5 pts] What is the kinetic energy of the hole shown in the diagram? 8
9 4. [27 pts] Assume a Si PN junction with the following dopant density profiles for the two segments: N ND = 2 X 1016 cm-3 NA = 1 X 1016 cm-3 P NA = 1 X 1017 cm-3 ND= 1 X 1013cm-3 a) [6 pts] Find Vbi. - Vbi = k T O. 76 ~ v b) [7 pts] Draw a band diagram for the structure with a forward bias of VA=O.5V. Label VA,Vbi.Ev,Ec,and Fermi (or quasi-fermi) levels. r( f"fp I:: V 9
10 c) [4 pts] For part b, using arrows, indicate direction of In,diff,Ip,drifbIn, and Itotal (Redraw the band diagrams from b here). N e '" f' o~~ T,-il ~ - f\ olnf[ P U ovuf - I t -),j f tj~ -) 0 d) [10 pts] So far, we have been assuming that there is no series resistance (and therefore, no potential drop) in the neutral P and N regions of our diodes. However, when lightly doped (-<5e16cm-\ the resistivity of the P and N type regions are often high, leading to series resistance or potential drop in the P and N regions under an applied voltage. Draw a band diagram for this PN junction in equilibrium and then under forward bias, this time including the effect of the series resistance (qualitatively) of the N segment. Hint: assume the series resistance is constant throughout the N segment. ~J-' r= -.-._-- -,-..-' ~...<..---_.. ~ ~e.>vu..>. Re.A -==) P (Y~) J<J'"H-~r _J V(x\ /'-i 10
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