808 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 



These expressions may be interpreted as follows: In case (I), flow is by 

 diffusion and the propagation factors A'l and A'2 take the form 

 ±(a/L){iyY'^. For this case attenuation and delay terms in the ex- 

 ponential are equal, and the largest negative term occurs in Z when the 

 phase angle is 57r/4 (as may be verified by differentiation.) This leads to 



^ = 2(-l + i)2~"^exp (-57r/4) = 0.028(-l + i), (3.24) 



which gives 



Z = (l/coC2)(- 0.028 -i 1.028), (3.25) 



the impedance of a condenser \\\i\i a negative Q of 37. In order to make 

 an oscillator by coupling this to an inductance, an inductance with a Q 

 of more than 37 must be used. It is obviously advantageous to reduce 

 the magnitude of the negative Q. 



For case (II) in its ideal form, the ac current simply drifts through the 

 n-layer without attenuation. This produces a phase lag of co times the 

 transit time L/u. If this were the only effect involved, a capacitor with 

 a negative Q of less than unity could be produced. In addition, however, 

 there is attenuation due to spreading by diffusion. This effect is depend- 

 ent upon the ratio of the spread by diffusion (DL/uY''^ to the separation 

 of planes of equal phase in the drifting hole current. This latter separa- 

 tion is 2Tru/o). The square of this ratio appears in the attenuation term 

 in the second form of (3. 



We shall estimate the effect of the attenuation term by taking 



ay/2 = 37r/2, (3.26) 



so that the desired phase shift is obtained. The attenuation term is 

 7/4 smaller than this so that if 7/4 is considerably less than one, the 

 attenuation in (3 will be small while the phase shift is correct. If we take 

 37r/2 for the value of ay/2, then the value of 7 becomes 



7 = 4:a>D/u = 6TkT/qAV. (3.27) 



Thus the approximation on which (II) is based fails unless qAV/kT 

 > 18, a value which implies an enormous range of concentration in the 

 n-layer. We must, therefore, investigate the case of gradients in the 

 n-layer by more complete algebraic procedures. 



We shall denote by — di the phase shift in jS due to the exponential in 

 equation (3.21). The total phase shift 6 is somewhat less since the alge- 

 braic expressions give a small positive phase shift of at most about 15°, 

 which vanishes for large and small values of 7. Similarly the attenuation 

 of ;S arises chiefly from the real part of the exponential since the absolute 



