432 BELL SYSTEM TECHNICAL JOURNAL 



The potential V is determined from the charge density, q, by Poisson's 

 equation 



d'^V/dx^ = -4Trq/K. (2) 



Since the charge density may be expressed in terms of n{x,l) and the 

 density of fixed charge, these two equations may be used to determine n 

 and V when /(/) is specified. Spenke eliminates the potential V between 

 (1) and (2) and gets a rather complicated equation for n. We prefer to deal 

 with Eq. (1) directly, to treat the potential V{x,l) as a known function, and 

 to solve for the concentration, n{x,l). 



The plane x = is taken at the interface betw-een metal and semiconduc- 

 tor and the plane .v — xi just beyond the barrier layer in the semiconductor. 

 It is assumed that F = at .r = .vi. Under thermal equilibrium conditions, 

 with no current flowing, the hole concentration in the barrier layer varies 

 as exp (—eV/kT), taking the values: 



ft = Ho at X = xi (3a) 



n = iim = iio exp (—eVm/kT) at x = 0, (3b) 



where no is the equilibrium concentration in the body of the semiconductor 

 and Vm is the height of the potential barrier. We suppose that the boundary 

 conditions (3a) and (3b) also hold when a current is flowing and when there 

 is an additional voltage, Va, across the barrier layer. Our procedure is to 

 solve Eq. (1) for n{x,l), with V{x,l) assumed known, and then to determine 

 /(/) in such a way that the boundary conditions are satisfied. The solution 

 of Eq. (1) which satisfies (3b) is: 



„(,, ,) . ,, exp[-.(F - K)/m --L I' (/ + ^1 g^ 



expHV - V)/kT] dx' (4) 



The prime indicates that the variable is .v' rather than x. At .v = 0, V is 

 the sum of Vm and the applied potential, Va'. 



V = Va-h Vmat X = (5) 



The current /(/) is determined in such a way that (3a) is satisfied. Setting 

 X — xi, using (3a), and solving the resulting equation for /(/), we get: 



nkT[noexp(eVa/kT) - noexp{eV/kT)] - [ '^ ^ exp{eV'/kT) dx' 

 ^/ N _ Jo 47r ax ot 



I \xp{eV'/kJ)dx' 



(6) 



