IMPEDANCE FOR TUNNELING OF ELECTRONS 199 right-hand side is Cs so that the variation of charge in the oxide can be treated (5) as a function of the surface potential Cs [viz., Eq. [46], etc., of (5)] or, expressed differently, of the free carrier density at the surface of the semiconductor, which is a well-defined function of r Thus, in a good approximation, V(5) ~ Cs, independent of the tunnel distance 5. In other words, the variation of an energy level at the distance 5 vs. an energy level at 5 = 0, induced by an a-c field applied at the oxide, can be neglected as a second order effect. On the other hand, this second order effect is the principal effect in our case of a metal-insulator structure, where Ss --Cs : 0 and V is in proportion to b [1]. Unlike the case of a semiconductor, the surface potential and the free carrier density at the metal surface are not noticeably varied by the applied field. The mathematical consequences of this difference between the MOS structure of (5) and the MOM structure are as follows: Our equivalent parallel conductance Eq. [15] with [16] and [17] corresponds to iwCox dVss/dCs in (5) with dVss/dCs given by their Eq. [64] (see also ref. (13), Eq. [1] and [2]). Several factors in the integrands of Eq. [15] of this paper and of Eq. [64] of (5) can easily be identified with each other, e.g., nr with K 9 g; fT(1 --JT)/kT with Of/O~T; and (1 ~ iW~T)-1 with [1-p jwTc] -1. However, our [15] with [16] contains the factor (5/d) 2, which is missing in [64] of (5). This has the following reason: In our case the "driving force" for the tunneling is the a-c potential variation between surface and recipient state, which accounts for one factor b/d in [16]. The second factor 5/d results from the "dipole distance" between the trapped electron and the positive charge left behind on the metal, and the corresponding change in V'~ at a given E as expressed by [12]. The first factor 5/d is absent in [64] of (5), since their driving force ~s is independent of 5: In the MOS structures, the tunneling charge Qss changes the voltage distribution between depletion layer voltage Cs and oxide voltage by the shift to the bulk side of the depletion layer of the positive charge left behind by a tunneling electron at the semiconductor surface. Since the charge shift across the depletion layer (which, in general, does not occur by tunneling, but by diffusion and field drift) occurs over a larger distance than the tunnel distance, the charge displacement (dipole distance) is much larger in the case of the MOS structure and rather independent of 5. As a consequence, much larger charges must tunnel in the case of MOM structures than in MOS structures to obtain impedance effects of comparable magnitude.In summary, the case of metal-insulator tunneling differs from that of semiconductor-insulator tunneling physically as to the dominant effect, and mathematically by a factor (5/d) 2 << 1, occurring in the integrand of the expression for the equivalent parallel conductance.ABSTRACT Semieonducting n-type TiO_~ evolve...
