Depletion layers equilibrium

Figure 12.17. (a) Diode laser band structure. (1) In thermal equilibrium. (2) Under forward bias and high carrier injection. Ec, v, and f are the conduction band, valence band, and Fermi energies respectively, (b) Fabry-Perot cavity configuration fora GaAs diode laser. Typical cavity length is 300//m and width 10/tm. d is the depletion layer. 

Fig. 6-48. Differential capacity of a space charge layer of an n-type semiconductor electrode as a function of electrode potential solid cunre = electronic equilibrium established in the semiconductor electrode dashed curve = electronic equilibrium prevented to be established in the semiconductor electrode AL = accumulation layer DL = depletion layer IL = inversion layer, DDL - deep depletion layer.

Fig. 8-41. Electron transfer reaction of hydrated redox particles in equilibrium on a metal electrode covered with a thick film (F, solid curve) and with a thin film (F, dashed curve) >cs = electron transfer current via the conduction band >scl = tunneling electron current through a depletion layer in the conduction band >vb = hole transfer current via the valence band.

Figure 4, Semiconductor-electrolyte solution interface in the dark. An n-type semiconductor with a depletion layer at the surface is illustrated. E is electron energy, Ej and pEj . are the equal Fermi leveh for electrons and holes at equilibrium, other symbols as in Figure 3 (13).

Mechanisms 1 and 2 are included in the model that is used here for comparison with experimental data. Interface recombination and dark current effects are not included however, the experimental data have been adjusted to exclude the effects of dark current. To include the additional bulk and depletion layer recombination losses, the diffusion equation for minority carriers is solved using boundary conditions relevant to the S-E junction (i.e., the photocurrent is linearly related to the concentration of minority carriers at the interface). Using this boundary condition and assuming quasi-equilibrium conditions (flat quasi-Fermi levels) ( 4 ) in the depletion region, the following current-voltage relationship is obtained. 

A fifth type of space charge layer, the deep depletion layer, may be formed under non-equilibrium conditions at the semiconductor surface when a high voltage is applied such that an inversion layer should form, but either (a) minority carriers are not available to accumulate at the surface in the time allotted or (b) the minority carriers are consumed in an electrochemical reaction as soon as they reach the surface. Such a space charge layer is unlikely to form within semiconductor electrodes at open circuit and is included here solely for completeness. 

Fig. 9.4. Comparison of the band bending, space charge layer formation and Fermi levels (E,r) for a large particle when r = r throughout the depletion layer and equation (9.18) applies, and for a small particle when r = tv and equation (9.19) applies. The semiconductor particles are considered to be in thermodynamic equilibrium with a redox pair of Nernst...

Fig. 16. Principles of the photovoltage measurement, (a) Semiconductor in the dark, with the bulk Fermi level E in equilibrium with P ,dol( (b) under illumination, when the open-circuit potential adjusts so that, in principle, there is no potential drop in the depletion layer.

Type I. Equilibrium is established between the surface states and the solution redox couple but not between the semiconductor majority carriers and either the surface states or the solution couple. Under these circumstances, the flat-band potential will apparently change with redox couple, but any change in the potential is dropped across the depletion layer and the semiconductor will appear otherwise to behaving classically. 

Type II. Equilibrium is established between the surface states and the majority carriers within the semiconductor. Under these circumstances, a fraction of the potential change will be dropped across the depletion layer and a fraction across the Helmholtz layer. If a redox couple is present in solution, and the kinetics of electron transfer between this and the surface states are also rapid, then a large dark current will be found. 

The complexity afforded by these traps arises from the fact that the frequency range normally used in a.c. studies, 1-105 Hz, is likely to contain the relaxation frequency appropriate to such trap sites. To explore this concept, we may consider Fig. 32 [76]. We identify a deep trap level of energy E0 relative to the conduction band, lying below the Fermi level in the bulk of the semicondutor and therefore ionised only up to a distance l into the depletion layer. For x > l, only the shallow traps are ionised at equilibrium. Within the spirit of the Mott-Schottky approximation, we obtain (provided V 3= V0)... 

In his treatment of the photoresponse in the semiconductor, Wilson assumed, as had Memming, that quasi-equilibrium conditions obtained across the depletion layer, i.e. the product np is a constant. 

To obtain relations between carrier density at the interface and at the inner edge of the depletion layer (the thickness of the space charge layer dsc is defined by Eq. (22)), we assume Boltzmann equilibrium for the carriers across the space charge layer. Using Eqs. (3a) and (3b), we have... 

Fermi level is a kind of measiue of equilibrium electronic energy of a solid material. It is thought that Fermi level is located just below the CB bottom and above the VB top for n-type and p-type semiconducting materials 13), respectively. Most metal oxides are categorized as n-type semiconductors with Fermi levels more cathodic (higher) than the standard electrode potential of electrolyte in contact with the metal oxide and thereby electrons in donor levels a little below the CB are injected into the electrol5rte to form a space charge (depletion) layer with an electric field, that is, Schottky barrier. In the... 

An n-type semiconductor at room temperature has free electrons in the conduction band compensated by positively charged immobile ionized donors (Section 4.3.6). The concentration of free electrons (donors), buik, is usually between lO and lO cm", hence 10 — 10 smaller than in a metal. A common situation is that the electrons are transferred from the semiconductor to a redox system, so that equilibrium is achieved. The surface region of the solid is then depleted of free electrons (Figure 19) a depletion layer of width rfsc is formed. The concentration of free electrons in the depletion layer, w(x), is much smaller than the concentration of the immobile donors buik- Thus, the charge density in the depletion layer per unit volume is e buik- Therefore, the surface charge density at the solid side is ... 

It is interesting to consider in some detail the exponential region I in the p-type case. An important remark is that this region is situated negatively with respect to flat-band potential I fb. which implies that holes are being captured from a depletion layer. Assuming that the holes are at quasi-equilibrium in this layer, their con-... 

Water dissociation occurs in the transition or depletion layer of the membrane and the dissociated ions removed from this region are replenished by the following water dissociation equilibrium ... 

Three types of space charge layers, namely, depletion layer, accumulation layer, and inversion layer, may occur in a semiconductor depending on the bias and equilibrium conditions as shown in Fig. 1.7. 

In ideal semiconductor/electrolyte junctions, the presence of an energetic barrier in the semiconductor phase, originated from the equilibrium of the Fermi level in the solid and liquid phases, can be approached by the semiconductor depletion layer capacitance or space charge capacitance, Cgc- Measurement of capacitance versus... 

Both electrodes take the equilibrium potential of this redox couple, so that the Fermi levels of the metal and CdS, and the level redox = Fs -/sl e solution, become equal. Good separation of light-generated electrons and holes requires that a depletion layer... 

Fig. 4. Energy diagram for a bulk monocrystalline n-type semiconductor electrode in the dark, in equilibrium with a redox system that has an equilibrium potential U. The Fermi-level (Ep) and the energy of the band edges are shown as a function of distance, x, perpendicular to the surface. The electrode is depleted of majority caniers at its surface, dsc is the width of the depletion layer and is the potential drop over the depletion layer.

In the absence of electronic equilibrium established by a redox electrolyte, the potential difference across the semiconductor/electrolyte jimction can be controlled in a three electrode cell with a reference electrode. The variation of the depletion layer capacitance with potential, U, is described by the Mott-Schottky equation [1] ... 

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