1258 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 



The first term has a maximum value of 0.35 (L,£/L^)^'^ at 2 = 0.6 

 and the second term a maximum value of 0.18 at 2 = 0.5 and x = L. 



The dashed curve in Fig. 2 for ^ = ,01 is the corrected cubic. For 

 the other curves in Fig. 2, the correction is smaller. For the curves in 

 Figs. 3 and 4 the correction is too small to show. 



Limits on the Solution 



We now show that 8E as derived above is not only a first approxima- 

 tion but also upper limit on the correction necessary to take account of 

 charge diffusion. That is, an exact solution to (3.8) lies between the cubic 

 and the corrected cubic. 



Consider the region where the second derivative of E is positive so 

 that the perturbed curve lies above the cubic as in Fig. 2. On the cubic 

 we have s£' — 7 = 0. As we move upward from the cubic and toward the 

 dashed curve, sE — I increases. The value oi sE — I on the dashed curve 

 just equals the value of £' d'E/dx' on the cubic. However, the dashed 

 curve has a smaller second derivative than the cubic. Thus in moving 

 upward from the cubic toward the dashed curve sE — I increases from 

 zero and £' d E/dx , which is positive, decreases; on the dashed curve 

 sE — I is actually greater. Therefore there is a curve lying just under 

 the dashed curve where the two sides of (3.8) are equal. The same argu- 

 ment applied to the region where the second derivative is negative shows 

 that the equation is satisfied by a curve lying just above the first per- 

 turbation of the cubic. Where the curvature changes sign, the cubic is 

 correct. 



It should be emphasized again that the neglect of the diffusion term 

 in the current is justified only for the ideal case of equal hole and electron 

 mobilities. For unequal mobilities both drift and diffusion will be im- 

 portant in current flow. However, as we will discuss in section 5, we can 

 simplify the problem of unequal mobilities by defining a fictitious current 

 that has the same form as / in (2.6) and (3.8). 



IV. RECOMBINATION 



As discussed in Section III, when the voltage for a given current is re- 

 duced, s increases and near x = becomes comparable to unity. Then 

 recombination becomes important and the cubic solution breaks down, 

 or rather joins onto a solution that takes account of recombination. 

 When recombination is important the center Xi of the intrinsic region is 

 no longer at the .r = point on the cubic but to the left of it. That is, if 

 we want the same current with continually decreasing voltage, we even- 



