1048 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 



Using (15), one gets: 



~ (dXs/dY)B 



= i(X-' + X) th(-Y + fn\+ (n x) + hO^~' - X) 



(18) 



This procedure, inaccurate as it is, has the advantage that no particu- 

 lar assumption need be made concerning the functional dependence of 

 X on V, it being understood that x in (18) has the value holding for v = 



— Y -\- (n X. In particular, if }^o is that ^•alue of Y at which the ratio 

 -(a2s/a5)y/(a2,/a}')5 changes sign, 



/"wxo = To - Cn X + t}r\{\ - X"')/(X + X~')] (19) 



From the experimental data, one linds, for the /^-tj-pe sample, In xo '^ 

 2.4 (at V = —3.5); for the p-type sample, (n xn ^' 1.0 (at v = 1.9). 



In \-iew of the approximations made, these estimates would not be 

 expected to be more precise than ± 1 to 2 units. Notice that both \alues 

 are positive, and that the difference between them is small in compari- 

 son with the difference in v. This suggests that we start afresh with the 

 assumption that x is independent of v, and woi'k out the surface photo- 

 voltage integral exactly. This is done in the next section. 



IV. EXACT TREATMENT FOR THE CASE N{v) = A ch {qv + B) , AVITH CON- 

 STANT CROSS-SECTIONS 



The results of the previous section suggest the procedure of assuming 

 that N{v) is of the functional form given by (16) and (17), and evaluat- 

 ing the integrals (9), (11) and (12) exactly. The integral for {dliJdY), 

 (11), depends only on the form of N{v) and ma}^ be eA'aluated at once. 

 To get idfijdb), (12), one must know how x depends on v. On the 

 basis of the work of the previous section, we shall suppose that x is in- 

 dependent of V. (Properly, we need only assume that x varies with v 

 more slowly than e^ Since the function th[\{v — Y) -f ^Cn X + (n x] 

 has one of the values ±1 everywhere except close to j' = Y — (n X 



— 2Cn X, and since the denominator of (12) has a sharp minimum at 

 V = — Y -\- (n X, it follows that the region in which (3Ss/d5)y changes 

 sign will be governed mainly by the value of x at ^ = — (n x) To get s [(9), 

 using (7)], one must also assume something about the geometric mean cross- 

 section, {an(T,^ ". In the absence of any information on this score, we 

 shall assume that (o-„a-p)' " also is independent of v, and see how the com- 

 puted variation of s with Y compares with the experimental results. 



