DEAN AND NUESE: STEP-RECOVERY TECHNIQUE FOR MEASURING PARAMETERS IN ASYMMETRIC p-n JUNCTION DIODES theoretically for three different idealized junction im- The triangle which approximates the actual initial dis- purity distributions In a graded p-n junction [9]and in tribution has an area which is exactly one half the area a p-i-n diode [10]the relation is under the corresponding exponential T/T=loge(1+IF/IR (1) For a graded impurity distribution(L/Lp<<1) with IF/IR>>(L/LD)?, the actual minority carrier n an ideal step junction [7] the relation is density at t= Ts can be expressed as [9] eri(T/r)=(1+I/I)-1 nt = x(Ir/qAD)exp(-x/L) We will compare the results of our approximate theory We will approximate this distribution with a triangle writing the equation for the extraction of the total in- fusion exactly supplies the reverse current I R. This slope jected charge Q is 2+=-1 Ir/QAD=(/IF)(L/Lp)(no/L) where the reverse current IR is considered to be con- where d is the minority carrier diffusion coefficient, LD stant for this derivation. The solution of this equation is the diffusion length, and the other variables have been defined previously. Solving for the intersection of e()=[(0)+ IrT]exp(t/)-IRt the two lines yields At t=T, the charge has decayed to that indicated by n;=n/[1+(In/)(LD/D)] the shaded area of Fig. 1, and ha as a value e(T)=le(0)+iRT]exp(Ts/T)-Iy x;=L/[1+(IR/1)(L/LD)2] Now we will define a new parameter a to be the frac- The area under our triangle is n L/2, which is exactl of initially injected carriers that remain att= Ta half the area under the more rigorously derived [9] a=Q(T)/Q(0 graded-junction distribution indicated above. For an abrupt junction (L/LD=1) with IF/IR<<l, the tri- This fraction a can vary from zero for a graded junction angular area is again half the area under the initial or large IF/IR to unity for a step junction or small exponential. This is as it should be since this limit cor- Ip/IR. Setting (3)equal to Q(T)=aQ(o), noting the responds to the case in which almost none of the in- elation Q(0)=IFT [9], and rearranging yields jected carriers are removed during the storage period T/r=loge [(1+Ip/IR)/(1+alp/Ir)]. (4) Since for the cases of graded and ideally abrupt p-n junctions, the areas of our triangular density approxi quation(4)is not yet a useful solution, since a is a mations are one half those under the more rigorous function of both impurity grading and Ip/IR. For an derived curves, we postulate that the ratio of the ti estimate of a we will use straight line approximations to ngular areas (at t=0 and t= T) provides a reasonable he actual density distributions, as shown by the dotted general approximation to the area ratio for actual minority tributions. Then, fro The initial density varies approximately exponentially etry a=Q(T /Q(0)an/no, and from(5) with distance a according to the expression [7] a s[1+(P/IE)(LD/L)2-1 Substituting(7)for a in(4)yie With an initial total charge given by T loge( [1 +IpIr][1+(IP/IR)(Lp/L)? (0)=q4|adx≈g4n0L /[1+(p/1)(L/D)2+Ip/l.(8) one obtains For a graded junction, a retarding field crowds the n0≈Ip/qAL condition L<<LD. Thus for L/LD=0,( 8)reduces to where q is the electronic charge and A is the junction (1), as expected. For an ideally abrupt junction,we area. We will approximate this distribution with a line have L=LD, and( 8)reduces to exponential ([1+Ip/l2/1+2r/l]l.(9) Values of T/T calculated from (9) are only about 25 L percent higher than those accurately calculated fromtheoretically for three different idealized junction im￾purity distributions. In a graded p-n junction 193 and in a p-i-n diode [ 101 the relation is T,/T = log, (1 i- IF/IR). (1) In an ideal step junction [7] the relation is erf (T,/T) = (1 + IR/Ip)-’. (2) We will compare the results of our approximate theory with these formulas in the appropriate limits. Following Moll, Krakauer, and Shen [9], we start by writing the equation for the extraction of the total in￾jected charge Q: dQ Q -+-= -rR at T where the reverse current IR is considered to be con￾stant for this derivation. The solution of this equation is @(t) = [Q(O) + IRT] exp (-t/r) - IRT. At t = T,, the charge has decayed to that indicated by the shaded area of Fig. 1 , and has a value Q(TJ = [Q(o) + 1x71 exp (-T,/T) - IRT. (3) Now we will define a new parameter a to be the frac￾tion of initially injected carriers that remain at t = T, : = Q(Ta)/Q(O). This fraction CY can vary from zero for a graded junction or large IF/IR to unity for a step junction or small IF/IR. Setting (3) equal to Q(T,) =crQ(O), noting the relation Q(0) = Ipr [9], and rearranging yields Ts/T = loge [(I + IF/IR)/(l aIF/IR)]* (4) Equation (4) is not yet a useful solution, since 01 is a function of both impurity grading and I~/IR. For an estimate of a! we will use straight line approximations to the actual density distributions, as shown by the dotted lines in Fig. 1. The initial density varies approximately exponentially with distance x according to the expression [7] n = lzo exp (-x/L). With an initial total charge given by 187 = @(o) = qA ndx z qAnoL, SoW one obtains no S IFT/qAL, where q is the electronic charge and A is the junction area. We will approximate this distribution with a line through no having a slope equal to the initial slope of the exponential : An Ax -_ - - no/L. The triangle which approximates the actual initial dis￾tribution has an area which is exactly one half the area under the corresponding exponential. For a graded impurity distribution (L/LD < < 1) with IF/IR > > (L/LD)2, the actual minority carrier density at t = T, can be expressed as [9] n x(TR/qAD) exp (-x/L). We will approximate this distribution with a triangle formed by the original straight line and another one passing through the origin with a slope such that dif￾fusion exactly supplies the reverse current IR. This slope is A 1z Ax _- - IR/~AD = (IR/IF)(L/LD)’(%O/L) where D is the minority carrier diffusion coefficient, LD is the diffusion length, and the other variables have been defined previously. Solving for the intersection of the two lines yields ni = nO/[l + (I~/I&) (LD/L) ‘1 (5) at xi == L/[1 4- (IR/IF) (L/LD)’]. (6) The area under our triangle is niL/2, which is exactly half the area under the more rigorously derived [9] graded-junction distribution indicated above. For an abrupt junction (L/LD = 1) with IF/IR< <1, the tri￾angular area is again half the area under the initial exponential. This is as it should be since this limit cor￾responds to the case in which almost none of the in￾jected carriers are removed during the storage period. Since for the cases of graded and ideally abrupt p-n junctions, the areas of our triangular density approxi￾mations are one half those under the more rigorously derived curves, we postulate that the ratio of the tri￾angular areas (at t = 0 and t = T,) provides a reasonable general approximation to the area ratio for actual minority carrier distributions. Then, from simple geom￾etry, a= Q(T,)/Q(O) =ni/no, and from (j), O( = [I + (WI~ (wLPJ-’. (7) Substituting (7) for CY in (4) yields - loge { [I f IF/IR] [1 + (IF/IR)(LD/L)*] Ts T /[1 + (IF/IIJ (LD/L)’ + IF/IR] 1. (8) For a graded junction, a retarding field crowds the injected carriers close to the junction, which leads to the condition L < <LO. Thus for L/LD =O, (8) reduces to (l), as expected. For an ideally abrupt junction, we have L =LD, and (8) reduces to T,/T = log, { [1 + IP/’TR]~/[~ + ~IF!JR]). (9) Values of T,/r calculated from (9) are only about 25 percent higher than those accurately calculated from (2) over a wide range of IF/IR. For most purposes, such