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Chapter 7 - CHAPTER 7 E1(a The in and out movement of the...

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Unformatted text preview: CHAPTER 7 E1' (a) The in and out movement of the majority carriers about the steady-state depletion width in response to the applied a.c. signal. (1)) N D—N A (c) Quasistatically is an adverb used to describe a situation where carriers or a device subject to non-steady—state conditions responds as if steady-state conditions applied at each instant of time. . (d) Vamctor—a contraction of variable reactor. A commercial device, such as a reverse- biased pn junction diode, where the reactance = llij varies as a function of the applied voltage. (e) Profiling—the process of determining the doping concentration inside a device as a function of position. (i) The low-frequency conductance of an ideal diode was noted to be (Eq. 7.15), G = .21 I + I 0 Id“ ( 0) Go o: I when the diode is forward biased and vanishes for reverse biases greater than a few kT/q. Also note that G0 = qlolkT when VA = 0. We conclude Go 7-1 (g) The diffusion admittance an'Ses from fluctuations in the number and position of minority carriers stored in the quasineutral regions adjacent to the depletion region. (h) At signal frequencies where mp 3 1, the minority carriers have trouble following the a.c. signal and the resulting out-of-phase oscillations enhance the diffusion conductance at the expense of the diffusion capacitance. 1L2, Given N300 = ND(x) = bxm x > 0 the application of the depletion approximation yields p E qND = qum ...0$x5xn§W Next substituting into Poisson's equation gives 'ECZ=_E_= 4” xm momsw Separating variables and solving for the electric field, we find 0 w dg‘ .1] .K_‘l_b__ mm» 80) :1: SEC _ = iv: = Lb Wm” 8(x) dx K580 m+1 0! W = L(wm+1w xm+l) x (m+l)Ks£0 Again separating variables and this time integrating across the entire depletion region, we obtain ' Vm—VA W I dv = 4'” [tvm+‘—xm+1] dx 0 (m+1)K580 0 or Vbi—VA = ————qb [W"’+'x— xm+2] W (m+l)KS£{} m+2 0 Note that the second term on the right hand side of the Vbi—VA expression blows up when evaluated at the lower limit if m < —2. The solution likewise blows up at the upper limit if m = -2. It is for this reason that we must restrict m to values m > ~2. With m > —2, we conclude 01' 2; (a)l(b) Script of a MATLAB program yielding fully-dimensioned reverse—bias C-V curves, and a sample output to be compared with Fig. 7.3. are reproduced below. Using a computed Vbl consistent with the specified doping yields capacitance values that are too low. This is especially true at small applied voltages where IVAI is comparable to Vbi. For example, at VA = 0, the computed C1 is 106 pF while the observed value is approximately 1 123 pF. The noted discrepancy is indeed related to the result in Exercise 7.2 where a lower m, a Vbi not consistent with the doping concentration. was deduced from the Fig. 7.3 experimental data. Not surprisingly, if one employs the Vbi deduced in Exercise 7.2 instead of the computed value (which is possible with the supplied m—file), one obtains excellent agreement with the Fig. 7.3 data. (It should be noted that even better agreement is obtained if 2 pF are added to the computed values to account for stray capacitance.) (c) Because the depletion width at a given reverse bias shrinks with increased doping, the capacitance, which is proportional to l/W, increases with increased doping on the lightly doped side of the junction. This is readily verified by simply running the P_07_03.m program with different ND inputs. MATLAB program script... % Fully—dimensioned Reverse—bias C—V curves % appropriate fer p+-n step junction diodes %Initialization Clear; close %Constants and Parameters q=1.6e-19: e0=8.85e—14: EG=1.12: kT=0.0259: ni=1.0e10; KS=11.8: 7-»3 s=menu('CHOOSE Vbi APPROACH','Compute','Input'): A=input('Input the diode area in cmAZ, A = '): ND=input('Input the n-side (of p+~n) doping, ND = '): VAmax=input('Input reverse-bias IVAImax, IVAimax = '); if s==1, Vbi=EG/2+kT*log(ND/ni); else Vbi=input('Input Vbi, Vbi = '); end %C‘V Computation VA=1inspace(0,—Vnmax): CJO=(KS*e0*A)/sqrt(2*KS*e0*Vbi/(q*ND)); CJ=CJO./sqrt(1-VA/Vbi); %Plot result ymax=1.2*max(CJ): plot(VA,CJ): axis([—VAmax,5,0,ymax]}; grid xlabelt'VA (volts)'): ylabel('CJ (pF)') 3: 104° -25 —2o .15 -10 -5 o 5 VA (volts) 7-4 LA For an abrupt p+-n junction, we know in general from Eq. (7. I 1) that 1 2 —- = mos-vii) c} czirvoKseoA2 1 After reducing all capacitance values in Table PTA by 3pF to account for the stray capacitance shunting the encapsulated diode’f, a least squares fit to the corrected data employing the MATLAB polyfil function yields J; = (8.254 x 1020) - (1.123 x 1021)VA C; in Farads C: We therefore conclude qKseoAZIsIopeI (1.6X10'19)(11.8)(8.85X10'14)(6x10'3)2(1.123X1021) = 2.96 X 1014fcm3 and 20 Va = @19— = 0.135 V 1.123x1021 Referring to Fig. E5.1, one finds the Vbi result here is reasonably close to the theoretically computed Vbi = 0.83V associated with an ND 5 3 X 1014/cm3 p+-n step junction. i It was incorrectly stated in the first printing of the text that the data listed in Table P14 had already been canceled to account for the cited stray capacitance. 7—5 A plot of the corrected 1/C 12 versus VA data, which may be used for obtaining a result by “eyeballing,” is displayed below. (Also see m-file P__07__04.m available on the instructor's disk.) x to“ ”cm (1iF‘2) 7.5 For concreteness, we take the device under test to be a ptn junction diode, with ND(x) the arbitraty nondegenerate donor doping on the lightly doped side of the junction. Based on the depletion approximation, the total charge in the depletion region on the n-side of the junction will be InEW ' W QN = A I p(x)dx -—- CIA] ND(x)dx 0 0 Assuming the diode follows the applied a.c. signal quasistatically, W C = §Q_P = _€QN.. = _qA _d_ ND(x)dx = —4AND(W)fl/' dV 0 WA 7-6 However, C] = KSEOA W 29. = HMM dVA W2 dVA andtherefore dW =__W2 3’6; _ _KsEoA dCJ dVA KSEOA dVA C32 dVA Substituting back into the generalized capacitance expression then yields K 2N dVA . C} (WA and solving for ND(W) gives — ___L_____ ND(W) — 3 ' qKS£0A2[(dCJ/dVA)/CJ] Finally, noting _ A Q _d_(i] = 3 WA c; c] M and realizing W is synOnymous with the distance x from the junction being probed, we obtain - NDUT) = —"——'2—*'"— qKssoAzlda/nydml where x = W = 53% ‘ ...(from CJ=K580A/m J l._6_ (Solution not supplied.) 7-7 L7. As deduced by combining Eqs. (7.29) and (7.30), 60/60 = $441+ w213+1)ll2 (I'D/Cm = %(41 + (021% — 1W2 Computations based on the above relationships and implemented using the program listed below yield an almost perfect reproduction of the text plot. MAnAB program script... % Frequency variation of the normalized diffusion % conductance (GD/GO) and capacitance (CD/C00) % (reproduction of Fig. 7.10) %Initialization clear; close- %Computation x=logspace(—2,2); Gratio=sqrt(sqrt(l+x.‘2)+1)./sqrt(2): Cratio=sqrt(sqrt(l+x.*2)—1).*(sqrt(2)./x); %Plot loglog(x,Gratio,x,Cratio): axis([0.01,200,0.1,20]); grid xlabel('angular frequency * lifetime') text(2.4,2.2,'GD/GO') text(2.2,0.45,'CD/CDO') 1 o“ 1 0° 1 o 10 angular frequency ' lifelimo 7-8 1L8 , As deduced by combining Eqs. (7.30a) and (7.30b), (BUD/GD —> corp/2 mp << 1 As deduced from Eqs. (7.293) and (7.2%), Go CD Aa—fl— ”Mp ... CUT]; >>1 GD aggianp ...(nrp>>1 and ' wCD/GD —> 1 corp >> 1 In general, again referring to Eqs. (7.29), 2 01:0 _ 1+wzrg—1Yl GD V1+w2173+11 A plot of aJCD/GD versus amp that is consistent with the limiting-case solutions and the script of the generating MATLAB program are disPIayed below. The result here provides some food for thought. Even though GD increases and CD decreases with increased frequency above 0311, = 1, the relative size of the real and imaginary components of the diffusion admittance approach the same value and increase at the same rate if an}, >> 1. Also, the result emphasizes that the diffusion conductance is the larger admittance component at low frequencies. MATLAB program script... % Relative size of the capacitive and conductive % components of the diffusion admittance (wCD/GD) %Initialization Clear: close %Computation x=logSpace(—2,2); ratio=sqrt((sqrt(1+x.“2)—l)./(sqrt(1+x.‘2)+l)); %Plot loglog(x,ratio); axis([0.01,200,0.001,2]); grid x1abe1('angular frequency * lifetime') ylabel('wCD/GD...w=angular frequency') _'7-9 wCDIGD...w=angular frequency 1 0° 1 o' 2 angular frequency ' lifetime 10' A table listing the computational variables and the deduced Values of In is presented below. Capacitance entries in this table were established as follows: (1) The CTOTAL = C; + CD data spanning the voltage range from 0.5V to 0.58V were extracted from line 20 of the MATLAB program script in Exercise 7.4. (2) C] was Computed using C] = de‘il—VA/Vbi = 120/114 VA/OJ (pF) The values of C10 and 'Vbi were noted from entries in the Exercise 7.4 program script. (3) CD = CTOTAL— CJ ZVA(V01tS) CTOTAL(PF) CJ(PF)' CD (PF) 013(3) Tn=2CD/GD(SCC) 0.5 276 224 52 2.00x10-4 5.20x10-7 0.52 346 237 109 3.90x10-4 5.59x10-7 0.54 440 251 189 7.15x10-4 5.29x10-7 0.56 654 268 386 1.33x10-3 5.81x10-7 0.58 . 938 290 648 2.28X10'3 5.68x10-7 :7. = 5.51 x 10-7sec 7—10 ...
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