EXCESS SEMICONDUCTOR HOLE TRANSPORT 4()3 



holes and electrons occurs only at the electrodes, and that the disappearance 

 of mobile holes and electrons is caused only by mechanisms which cause 

 holes and eleclrons to disai)i)ear in equal numbers at essentially the same 

 time and i)hice. If this assumi)tion is valid, the charge density due to im- 

 purity centers will never differ from its equilibrium value by an amount 

 comparable with the density due to free electrons. This assumption can be 

 expected to be reasonably good for an n-iype impurity semiconductor in 

 which the number of donor levels is very much greater than the number of 

 acceptor levels and for which, at the operating temperature, practically all 

 the donor levels have been thermally ionized, while thermal excitation of 

 electrons from the normally full band has not yet become appreciable. 



As has just been mentioned, the differential equations for the behavior 

 of the electron and hole densities involve migration under the influence of 

 the local electric field E{x, t). This field is in turn influenced by the space 

 charge due to any inequality between the hole density iih and the electron 

 excess {rie - »o), where wo is the normal electron density. If the difference 

 {nh - He + no) were comparable with Uh or fie , the problem would be very 

 complicated. Fortunately, however, this difference cannot have an appreci- 

 able value over an appreciable range of x, on the scale of typical experiments. 

 For example, if (»/, - iie + m) were IQ-^ of Wo for a range Ax of 1/x, and if 

 no is 10^^ cm-3, then the difference in field strength on the two sides of Ax- 

 would be about 2000v/cm, a field which would outweigh all other fields in 

 the problem and rapidly neutralize the space charge. Moreover, the time 

 required for the evening out of any such abnormally high space charge would 

 be very short, of the order of magnitude of the resistivity of the specimen 

 expressed in absolute electrostatic units (1 sec. = 9 X 10' ^ 12 cm). Thus it 

 will be quite legitimate to assume {nu - th + wo) = in all equations of 

 the problem except Poisson's equation which determines the field E, and 

 so fie can be eliminated from the conduction-diffusion equations for holes 

 and electrons. These two equations can then be used, as is shown below, to 

 determine the two unknown functions fih and E, Poisson's equation being 

 discarded as unnecessary. 



The boundary conditions for these differential equations consist of two 

 parts, the conditions at / = and those at and to the left oi x = 0. In most 

 of the applications to be considered, the injection current j^ will be assumed 

 to commence at / = 0. Thus, initially, the specimen will be free of holes and, 

 at / = 0+, will have a field Ea = jjco in the region -a < .t < 0, and a 

 field Eo = jb/ao in the region < .r < b, where ao is the normal conductivity 

 of the specimen and jb = ja + je is the total current density to the right 

 of :c = 0. The boundary condition at .t = is determined by the magnitudes 

 of the electronic and hole contributions to the injection current je . If no 

 electrons are withdrawn by the electrode at x = 0, then the electron cur- 

 rents just to the left and just to the right of x = must be equal, and the 



