Semiconductors diffuse double layer

Although a family of OgS - Jig8 values are allowed under Equation 7 the actual equilibrium state of the oxide/solution interface will be determined by the dissociation of the surface groups and the properties of the electrolyte or the diffuse double layer near the surface. For surfaces that develop surface charges by different mechanisms such as for semiconductor, there will be an equation of state or charge-potential relationship that is analogous to Equation 7 which characterizes the electrical response of the surface. 

B , while for an n-type semiconductor the reverse is true. An analog to the SCR in the semiconductor is an extended layer of ions in the bulk of the electrolyte, which is present especially in the case of electrolytes of low concentration (typically below 0.1 rnolh1). This diffuse double layer is described by the Gouy-Chap-man model. The Stern model, a combination of the Helmholtz and the Gouy-Chapman models, was developed in order to find a realistic description of the electrolytic interface layer. 

In an attempt to rationalize the measured capacitance values, and especially the low value for the basal plane (ca. 3pF/cm2), these authors first concluded that space charge within the electrode is the dominant contribution (rather than the compact double layer with ca. 15-20 pF/cm2, or the diffuse double layer with >100 pF/cm2). They then applied the theory of semiconductor electrodes to confirm this and obtained a good agreement by assuming for SAPG a charge carrier density of 6 x 1018/cm3 and a dielectric constant of 3 for GC, they obtained 13 pF/cm2 with the same dielectric constant and 1019 carriers per cubic centimeter. 

At oxide semiconductor electrode-electrolyte interfaces, with no contribution from surface states, the electrical potential drop exhibits three components the potential drop across the space-charge region, sc, across the Helmholtz layer, diffuse double layer, d, the latter becoming negligible in concentrated electrolytes... 

The capacity of the space-charge region czm be related to the dopant concentration (or fixed charge) in a semiconductor. The space-charge region is essentially equivalent to the diffuse double layer treated in electrolytes with the exception that ionized impurities are present that, at room temperatures, are immobile. For this case, Poisson s equation becomes... 

The capacity of the electrolytic diffuse double layer is often ignored when Mott-Schottky plots are used to characterize semiconductor-electrol3ffe interfaces. Under what conditions is this assumption justified ... 

Development of a mathematical model of photoelectrochemical devices requires treatment of the diffuse double layer (or space charge region) in the semiconductor. The principles of electro-... 

A further complication arises when attention is focussed on the electron density distribution within the semiconductor solid. This, in contrast to the metal case, now is able to vary from a low to a high concentration level as electrons in a conduction band or as holes in a valence band. The electric field on the solid side of the electrical double layer now has spatial extent - a diffuse double layer character exists within the solid. The conventional electric field effects previously associated with ion motion and ion distributions in the electrolyte have a counterpart within the solid phase. 

Additional issues arise when reduction of the size of a semiconductor is considered. In bulk semiconductors the valence and conduction bands bend. Because of the low carrier concentration, the electrical double layer in a bulk semiconductor/solution system extends into the interior of the semiconductor rather than into the solution. In terms of the Gouy Chapman model, the width of a diffuse double layer is inversely proportional to the square root of the carrier concentration for a typical semiconductor carrier concentration of 10 cm the band bending is calculated to occur over several hundred nanometers (in which region there is about one carrier). 

The space-charge layer in semiconductors is comparable to the diffuse double layer in electrolyses that forms at a low concentration. Indeed, the concentration of... 

The situation is analogous to the Stem model for the combination of the Helmholtz layer with a diffuse double layer in the electrolyte described in Figure 2.12. The only difference here is that the diffuse part of the charge appears in the semiconductor. The total voltage drop between the bulk of the semiconductor and the electrolyte due to the excess electric charge in both phases is... 

Thus, the situation of a semiconductor is similar to the diffuse double layer described by the Debye-Hiickel theory. In the case of the semiconductor, the charge carriers are electrons for an n-type semiconductor and positive holes for a p-type semiconductor. The width of the space charge layer is given by the Debye length p. For the semiconductor, the expression 2Nj I of Equation 1.22b has to be replaced by the concentration of charge carriers within the semiconductor, i.e., for an n-type semiconductor by the concentration of electrons, which equals approximately that of the donors as shown in Equation 1.179. 

The capacity of the semiconductor electrode contains two parts, Cgc of fhe space charge layer of the semiconductor and Ch of fhe Helmholfz layer when the diffuse double layer within the electrolyte may be neglected due to a high electrolyte concentration (f > 1M). According to Equation 1.181, their inverse values have to be added because they are connected in series. 

The basic difference between metal-electrolyte and semiconductor-electrolyte interfaces lies primarily in the fact that the concentration of charge carriers is very low in semiconductors (see Section 2.4.1). For this reason and also because the permittivity of a semiconductor is limited, the semiconductor part of the electrical double layer at the semiconductor-electrolyte interface has a marked diffuse character with Debye lengths of the order of 10 4-10 6cm. This layer is termed the space charge region in solid-state physics. 

For semiconductor electrodes and also for the interface between two immiscible electrolyte solutions (ITIES), the greatest part of the potential difference between the two phases is represented by the potentials of the diffuse electric layers in the two phases (see Eq. 4.5.18). The rate of the charge transfer across the compact part of the double layer then depends very little on the overall potential difference. The potential dependence of the charge transfer rate is connected with the change in concentration of the transferred species at the boundary resulting from the potentials in the diffuse layers (Eq. 4.3.5), which, of course, depend on the overall potential difference between the two phases. In the case of simple ion transfer across ITIES, the process is very rapid being, in fact, a sort of diffusion accompanied with a resolvation in the recipient phase. 

At a semiconductor-electrolyte interface, if there is no specific interaction between the charge species and the surface an electrical double layer will form with a diffuse space-charge region on the semiconductor side and a plate-like counter ionic charge on the electrolyte side resulting in a potential difference (j) across the interface. The total potential difference across the interface can be given by... 

In most semiconductors the concentration of mobile carriers (electrons or holes) is very low. Therefore a diffuse part of the double layer can extend into the interior of a semiconductor electrode (2) (this point is discussed in more detail by... 

Under the assumption that there are no surface states or specific adsorption of charged species, the space charge hi a semiconductor in contact with an electrolyte is balanced by the charge in the diffuse part of the double layer thus, (]sc = cfd- Gauss s law can therefore be used to provide a boimdary condition for the electric field at the surface of the semiconductor as... 

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