On the Theory of the A-C. Impedance of a Contact Rectifier 



By J. BARDEEN 



THE a-c. impedance of the rectifying contact between a metal and a 

 semiconductor is measured by superimposing a small a-c. current on 

 a d-c. bias current. It is generally recognized^ that an equivalent circuit 

 consists of a parallel resistance and capacitance in series with a resistance 

 as shown in Fig. 1. The parallel components represent the impedance of 

 the barrier layer itself and depend on the d-c. bias current flowing. The 

 series resistance is that of the body of the semiconductor. It has been shown 

 theoretically by Spenke- that under quite general conditions the parallel 

 capacitance and resistance are independent of frequency. Unfortunately 

 Spenke's proof is highly mathematical and is also not readily available. 

 The derivation of the impedance relations which is presented here is in 

 some ways more general and gives more physical insight into the problem. 



The method of analysis which is used is similar to that emjDloyed by Miss 

 C. C. Dilworth'* for the d-c. case. Except for some obvious differences in 

 sign, the theory is the same for ii- and p-type semiconductors.^ We give 

 the theory for the latter because the signs are a little simpler for positively 

 charged holes than for negatively charged conduction electrons. Before 

 the discusssion of the theory of the a-c. impedance, a brief outline of Schott- 

 ky's theory of the barrier layer will be given. 



A rough schematic energy level diagram, based on Schottky's theory of 

 the barrier layer at a contact between a metal and a ^-type semiconductor, 

 is illustrated in Fig. 2. The diagram is plotted upside down from the usual 

 one in order to show the energy of holes increasing upward. The energy of 

 electrons increases downwartl. In a defect or p-type semiconductor, such 

 as CU2O, electrons are thermally excited to acceptor levels, charging the 

 acceptors negatively, and le.iving missing electrons or holes in the filled 

 band. The holes are mobile and provide the conductivity. Electron states 

 with energies lying above the Fermi level in the diagram, corresponding to 

 lower energies for electrons, have a probalMlity of more than one-half of 



' For an outline of ihc iheorv of conlucl rectiliers togflher willi references to the earlier 

 literature, see H. C. Torrey and C. .\. Whitmer, "Crystal Rectifiers," McGraw-Hill Book 

 Company, Inc., New York, New York (1948). 



2 Eherhard Sjjenke, Wiss. Verotf. Siemen's Konzern, 20, 40 (1941). 



3 C. C. Dilworth, Froc. Plivs. So:. London. 60. 31.=^ ( 1947). A similar method was used 

 earlier by H. .\. Kramers I'hysica 1 , 284 ( 1940), in a discussion of the ditifusion of particles 

 over potential barriers. 



^ We suppose that only one t>i)e of carrier takes part in conduction. 



428 



