NEGATIVE RESlSTAiNCE IN SEMICONDUCTOR DIODES 807 



where 



Lh ^ (.To - .ri)A-i = «[1 + (1 + iyY"], (3.14) 



LA-o ^ (x, - .ri)A-2 = «[1 - (1 + iyY"] (3.15) 



where 



a = qAV/2kT, (3.16) 



7 = 4coDpAr', (3.17) 



?< = UpE = MpAF/L. (3.18) 



The current is 



j(p, X, t) = q(up - Dpdp/dx) (3.19) 



and the ratio of currents at .Ti and xo , which is /3 by definition, is 



l3 = j{p, .1-2 , t)/{j(p, Xi , t), 



^ Lh - Lk. 



Kh exp (-LA:,) - KA;2 exp {-Lh) (3.20) 



^ 2(1 + iyye" 



[1 + (1 +iyy'^expa{l+iyy'^ - [1 - (1 +*7)''']exp - cx(l-\-iyy'' ' 



The phase lag in jS must exceed 180° or tt in order to give negative re- 

 sistance. It can be seen that this phase factor must result from the first 

 exponential in the denominator by the line of reasoning suggested below: 

 The real part of the exponent is larger than the imaginary part. Hence 

 the absolute ratio of the two exponentials is at least 27r. For this condi- 

 tion the second term in the denominator is negligible compared to the 

 first. Hence the phase of (3.20) is determined largely by the first ex- 

 ponential. As a helpful approximation we may neglect the second term 

 and write 



. ^ 2(1 + n)"'exp [g - c. (1 + iyY"] . . 



^ 1 + (1 + iyy^ • ^^-^^^ 



Two limiting cases are worthy of special note: 



(I) a ^ 0, uniform n-layer, y — > so . 



^ = 2 exp -aiiyY" = 2 exp -(1 + i)(o:/2Dy" L. (3.22) 



(II) a -^ 00 , qAV/kT » 1 , 7 ^ 0. 



/3 = exp [-za7/2 - ay^/S\, 



(3 23) 



- exp [-i{c^L/u) - (DL/u)/(u/o:f]. 



