470 BELL SYSTEM TECHNICAL JOURNAL 



F. Currents Limited by Space Charge 



Thus far we have only considered the effect of the surface and ap- 

 pUed fields on the number of electrons that escape from a cathode and 

 reach the anode at a particular temperature. However, an electron 

 traveling from the cathode to the anode is also subjected to a field 

 due to all of the electrons in the space between the electrodes. If the 

 electron density in this space is large enough, the current that reaches 

 the anode will be determined by these charges rather than by the 

 work function and temperature of the cathode. The current is then 

 said to be limited by space charge. If, on the other hand, the applied 

 potential is raised to a sufficiently high value, the current is no longer 

 limited by the charges in the space but is then determined by the work 

 function and temperature of the cathode. The current is then said 

 to be saturated or limited by emission. The space charge and satu- 

 rated emission regions are illustrated by curve 2 in Fig. 24 which is a 

 plot of log i vs. log V. In the region to the left of point A, the current 

 is limited by space charge and increases rapidly with the applied poten- 

 tial. To the right of A, the current is limited by emission. Curve 2 

 has been calculated from equations that will be discussed later. A 

 sharp break point is indicated at A, whereas experimental curves 

 usually show a gradual transition. This gradual transition is due 

 to non-uniformities in work function. 



When the current is limited by space charge, the charges in the 

 space increase the height of the potential barrier which electrons must 

 cross in traveling from cathode to anode. The current is determined 

 primarily by the applied potential and electrode geometry and second- 

 arily by the temperature of the cathode and magnitude of the saturated 

 emission. The problem of relating the current to these quantities 

 is very difficult but has been solved on the basis of certain simplifying 

 assumptions for several forms of electrodes by Child, ''* Schottky ,'''•* 

 Epstein,'*" Fry,^^ Langmuir *-"~*^ and others. These assumptions 

 together with the solutions will be summarized in this section. 



In all of these solutions it is assumed that the maximum of the 

 potential hill which is due to the surface forces and the applied po- 

 tential, occurs right at the cathode surface. Actually, in the absence 

 of space charge, the maximum in the work distance curve occurs at a 

 small but finite distance from the surface, about 3 X lO"*^ cm. for 

 the image equation with moderate applied fields. The space within 

 this distance has a much larger density of electrons than if the potential 

 had its maximum value at the surface. The above assumption, there- 

 fore, neglects the infiuence of these excess charges on the space charge. 



