Schottky barrier emission tunnelling

Because the spacing between pores is always less than the width of the depletion layer and PS has a very high resistivity, Beale et al. proposed that the material in the PS is depleted of carriers and the presence of a depletion layer is responsible for current localization at pore tips where the field is intensified. This intensification of field is attributed to the small radius of curvature at the pore tips. For lowly doped p-Si the charge transfer is by thermionic emission and the small radius of curvature reduces the height of the Schottky barrier and thus increases the current density at the pore tips. For heavily doped materials the current flow inside the semiconductor is by a tunneling process and depends on the width of the depletion layer. In this case the small radius of curvature results in a decrease of the width of the depletion layer and increases the current density at pore tips. The initiation was considered to be associated with the surface inhomogeneities, which provide the initial localized high current density at small surface depressions. 

Several possible current transport mechanisms are illustrated in Fig. 3. The schematic represents a Schottky barrier on an undoped sample under forward bias. The three arrows for electron transport are drawn for comparison with crystalline semiconductors in which thermionic emission, tunneling via thermionic field emission, or field emission represent the usual mechanisms. 

Field emission and thermionic field emission tunneling through a Schottky barrier on an n-type semiconductor (a) forward bias and (b) reverse bias. 

Tunneling through a Schottky barrier has been analyzed theoretically by Padovani and Stratton, and by Crowell and Rideout. The main results of their study are described below. Field emission in the forward direction occurs only in degenerate semiconductors and, except for very low forward biases, the I-V characteristic in the presence of tunneling can be described by the relation ... 

Under reverse bias, the FN tunnelling mechanism did not apply. On the other hand, the current-density was found to follow Richardson-Schottky thermionic emission model, where tunnelling through the barrier is ignored and field-induced barrier lowering is taken into consideration. The current density at a temperature T is given by [13] ... 

Field emission is characterized by its temperature independence. Here meff is the effective mass of the carrier in the dielectric. The essential assumption of the Schottky model is that a carrier can gain sufficient thermal energy to cross the barrier that results from superposition of the external field and image charge potential. Neither tunnelling nor inelastic carrier scattering is taken into account. The following current characteristic is predicted for the Schottky junction ... 

Field emission potential diagram. Large electric fields induce barrier narrowing, which increases the number of electrons tunnelling from the Fermi level in the metallic, electron-rich, surface into the vacuum. Variations in the density of occupied states, N E), and current density, J[E), as a function of electron energy. Surface contamination and charging effects (Schottky rounding) can be seen to drastically alter the potential profile. 

Simmons treated tunneling and Schottky emission through very thin films with a double-image force barrier. With thin enough films the current depended on the work function of the positive electrode, as in Standley and Maissel s work. 

We now turn attention to conditions at the electrodes. These play vital roles in establishing the pre-breakdown conditions in the liquid under high electric stress and in triggering the breakdown itself. It has been natural to invoke electron injection at the cathode as an important component since high fields will lower the potential barrier to electron transfer across the interface whether it occurs by a thermally activated or tunnelling process. However, employment of the Schottky formula for field-assisted thermionic emission or the Fowler-Nordheim one for tunnel emission which are appropriately applicable only for electron transfer to a vacuum is a much too simplified solution to the problem. 

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