Helmholtz potential

For an interface described by a constant Helmholtz potential electron exchange between the semiconductor and redox electrolyte solution. The result is that dV = d(psc, and for a non-equilibrium system one can obtain the current-voltage relation ... 

Moderately doped diamond demonstrates almost ideal semiconductor behavior in inert background electrolytes (linear Mott -Schottky plots, photoelectrochemical properties (see below), etc.), which provides evidence for band edge pinning at the semiconductor surface. By comparison in redox electrolytes, a metal-like behavior is observed with the band edges unpinned at the surface. This phenomenon, although not yet fully understood, has been observed with numerous semiconductor electrodes (e.g. silicon, gallium arsenide, and others) [113], It must be associated with chemical interaction between semiconductor material and redox system, which results in a large and variable Helmholtz potential drop. 

The formula for the adiabatic Gibbs potential (Equation 6.51) is exact within the framework of the mean held description. As with the adiabatic Helmholtz potential, we examine it numerically following the Carlson theory of elliptic integrals [15-21], In order to make clear comparisons between the two types of free energy, the width D of the container is set to be 200 nm throughout (except for Figure 6.21). 

From this equation, it can be seen that the Helmholtz potential is expected to increase with the square root of the acceptor concentration. This result, shown in Fig. 15, has been confirmed by Gaspard et al. [65]. 

The band structure at equilibrium for the interface region is illustrated in Fig. 1.5. The band bending K is the potential drop in the space charge layer. The potential drop on the solution side is measured by A. the Helmholtz potential. As and As are the conduction band edge and the valence band edge at the surface, respectively. Vm is the measured potential. Normally, the band edge positions at the surface are equal for p-and n-type materials because the same Helmholtz potential is expected for the two... 

This equation indicates that potential change occurs mainly in the Helmholtz layer at a current significantly less than the limiting current whereas it occurs mainly in the space charge layer at a current near or equal to the limiting current and the Helmholtz potential is almost fixed. 

It can be expected that the Helmholtz potential of a silicon electrode in an aqueous electrolyte is a strong function of pH since hydrogen adsorption is a dominant process on the surface of silicon. Table 2.15 shows the dependence of flatband potential on pH... 

In this equation, the term 4.4 corresponds to the position of the SCE with respect to the vacuum level and AEp represents the difference between the Fermi level and the valence band edge (AEp = Ep- Eyp a parameter that can be estimated from thermoelectric power measurements. is the Helmholtz potential between the electrode, and the solution originated from the preferential adsorption of either OH or H+ ions from the solution on the metal oxide surface. Vjj depends on the pH according to... 

The anodic and cathodic currents are given in a semilogarithmic plot vs. overvoltage in Fig. 7.14. It should be emphasized that the Tafel curves have a slope of 60 mV/decade which corresponds to a transition factor of a = 1, whereas with metal electrodes an a value of about 0.5 is usually found (see Section 7.1). Semiconductor and metal electrodes behave differently because any overvoltage occurs across the space charge region of the semiconductor, whereas it leads to change of the Helmholtz potential in the case of metal electrodes. 

Mott-Schottky theory can be used to determine the flat band potential (see Sect. 2.1.3.1). This then allows one to calculate the band-bending for any value of the applied potential, if the Helmholtz potential remains constant (i.e. the band edges are pinned). The band-bending determines the concentration of majority carriers at the surface (see Eqs. 25 and 26). In a p-type electrode these are holes, which are essential for the dissolution reaction. In an n-type electrode, the band-bending determines the surface electron concentration and thus, the rate of recombination... 

The assumption of pinned band edges is very often not valid. Lincot and Vedel [50] in an early study of the photoanodic dissolution of -type CdTe used EIS to show that the Fermi level becomes pinned in a wide potential range positive with respect to Vfb. This means that in this range the band-bending within the semiconductor remains constant while the potential changes across the Helmholtz layer. In their analysis, Lincot and Vedel consider the rate constants for change transfer to be exponentially dependent on the Helmholtz potential. 

This important result shows that the Helmholtz potential changes with -59 mV (23kTle) per pH unit at 25°C. 

Both the above are satisfied by a thermodynamic potential, when the thermodynamic process happens imder conditions where its natural variables remain constant. For example, for a closed system N = constant) undergoing a process under controlled temperature and volume (T, V are also constant), the Helmholtz potential F = F(N, V, T) is the corresponding free energy similarly, for the same system undergoing a process where NPT are constant, the Gibbs potential G = G(N, P, T) is now the free energy in this case and so on. 

One can derive the critical phase equation in a much simpler form in many specific cases using Equations 1.1.2--30... 34 as the stability conditions rather than Equation 1.1.2-12. The subsequent line of argument should remain unchanged. The molar Helmholtz potential F is convenient to use for a one-component system with Equations 1.1.2-31 and 1,1.2-32 to express the thermal and mechanical stability conditions, respectively. As the second inequality is stronger, one can confine oneself with it. In this case, the determinant D/r degenerates into one term... 

The scattered light intensity is presumed to be equal to the squared amplitude of this qlh wave (see Elquation 9), i.e. Equation 1.4-18 which is calculated with the density-fluctuations-related Helmholtz potential increment as a distribution function... 

The Helmholtz potential, [7h, which depends, of course, on the composition of the electrolyte, can be determined quantitatively by flatband potential measurements using Eq. (16). A strong pH dependence has been found for most semiconductors. Especially in the case of oxide semiconductors, the pH dependence has been measured quantitatively. Such experiments have yielded a shift of 0.059 V/pH as shown by various authors. These results indicate that the adsorption or bonding of hydroxyl groups determines the behavior of the Helmholtz layer. A pH dependence has also been observed for some III-V compounds " and II-IV compounds. In these cases, however, hydroxyl groups are not necessarily the potential-determining species. For instance, Ginley and Butler identified such species to be HS and H at CdS electrodes. ... 

U +500 mV) and hydride surfaces U +30 mV) at low pH. Since these have different positions, AtpH differs by more than 0.5 V for the two cases. This behavior can only be explained by a different electrical moment of the hydroxide and hydride surfaces. Accordingly, the Helmholtz potential at a Ge electrode is caused by a dissociation double layer and a dipole within the surface layer. 

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