HOLE CONCENTRATION AND POINT CONTACTS 491 



has the dimensions of a length. For 5 = 1500 cm/sec and Mp = 1700 

 cm-/volt sec, corresponding to germanium at room temperature, X is 

 about 35 cm~^. 



The boundary condition on the disk is similar to (3A) except that s is 

 replaced by vj^ (cf. Eq. (3)). Thus for r < p, 



dp/dz = Xcp z = 0,r < p, (5 A) 



where 



X, = Vae/4:tJLpkT. (6A) 



Evaluated for germanium at room temperature, X^ is about 6 X 10^. 



In order to have a dependent variable which vanishes at infinity, we 

 replace p by: 



y = pa- p + y^paz, (7A) 



so that /» -^ />a for s = as r ^ oc . The variable y satisfies Laplace's 

 equation subject to the boundary conditions: 



dy/dz = \y z =- 0, r > p (8A) 



dy/dz = \c {y - pa) z = 0, r < p (9A) 



y = r ,z^ <».' (lOA) 



An exact solution of the problem is difficult. We shall obtain an approxi- 

 mate solution which satisfies (8A) but not (9A) and which applies when 



Xp « 1 « Kp. (11 A) 



This approximation is valid for a germanium point contact, since, for p '^ 

 10-3 cm, 



Xp ^ .035, \cP ~ 60. (12A) 



We shall first discuss the limiting case for which X — ^ and X,. — ^ =o . 

 The former implies neglect of surface recombination and the latter 



y = pa for z = 0, r < p. (13 A) 



The problem is the same as that of finding the potential due to a conduct- 

 ing circular disk. The solution of this problem, which is well known, is: 



y = ilpjr) r e-''Mrt) '-^^ dt. (14A) 



Jo t 



The current flowing to the disk is obtained from integrating: 



i, - kTp.p{dy/dz), (15 A) 



