Space-charge layers

This is specifically a model that is applied to ionic crystals, and to outline the model it is necessary to begin with a brief description of basic defect thermodynamics and diffusion theory. 

All real crystals above 0 K contain point defects which are thermodynamically inherent [21,22]. In a monatomic crystal, the simplest defects are the vacancy, a lattice site that is empty, and the interstitial atom, an atom on an interstitial site in the lattice. The equilibrium concentration of these defects is thermally controlled and has an exponential dependence on temperature. For example, the site fraction of vacancies, c in a pure monatomic crystal is given by  

Here gp is the Gibbs free energy to form a vacancy, k is the Boltzmann constant, and T is the temperature. Diffusion in a crystal lattice occurs by motion of atoms via jumps between these defects. For example, vacancy diffusion - the most common mechanism in close-packed lattices such as face-centered cubic fee) metals, occurs by the atom jumping into a neighboring vacancy. The diffusion coefficient, D, therefore will depend upon the probability that an atom is adjacent to a vacancy, and the probability that it has sufficient energy to make the jump over the energy barrier into the vacancy. The first of these probabilities is directly proportional to c,. and the 

The same defect thermodynamics and diffusion theory can be applied to ionic crystals with one important proviso, which is the need to account for the charges on the ions (and hence effective charges on the defects), and that the crystal must remain electrically neutral overall. This means that the defects will occur as multiplets to satisfy this later condition. For example, in a MX crystal they will occur as pairs the Schottky pair- a cation vacancy and an anion vacancy the cation-Prenkel pair- a cation vacancy and an interstitial cation and the anion-Frenkel pair - an anion vacancy and an interstitial anion. The concentrations of the defects in the pair are related by a solubility product equation, which for Schottky pairs in an MX equation takes the form  

c+ and c are the site fractions of the cation and anion vacancies, respectively, Ks is the equilibrium constant for the formation of the Schottky pair, and gs (= g+ + g, the sum of the individual defect formation energies) is the Gibbs free energy to form the pair. In the bulk of a pure crystal, the condition of electrical neutrality demands that the concentrations of each defect in the pair are equal that is  

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Figure Bl.28.9. Energetic sitiration for an n-type semiconductor (a) before and (b) after contact with an electrolyte solution. The electrochemical potentials of the two systems reach equilibrium by electron exchange at the interface. Transfer of electrons from the semiconductor to the electrolyte leads to a positive space charge layer, W. is the potential drop in the space-charge layer.

Although the observations for PPV photodiodes of different groups are quite similar, there are still discussions on the nature of the polymer-metal contacts and especially on the formation of space charge layers on the Al interface. According to Nguyen et al. [70, 711 band bending in melal/PPV interfaces is either caused by surface states or by chemical reactions between the polymer and the metal and... 

Dispersing a dielectric substance such as A1203 in Lil [34] enhances the ionic conductivity of Lil about two orders of magnitude. The smaller the particle size of the dielectrics, the larger is the effect. This phenomenon is explained on the basis that the space-charge layer consists of or Li, generated at the interface between the ionic conductor (Lil) and the dielectric material (A1203) [35],... 

Before constructing an electrode for microwave electrochemical studies, the question of microwave penetration in relation to the geometry of the sample has to be evaluated carefully. Typically only moderately doped semiconductors can be well investigated by microwave electrochemical techniques. On the other hand, if the microwaves are interacting with thin layers of materials or liquids also highly doped or even metallic films can be used, provided an appropriate geometry is selected to allow interaction of the microwaves with a thin oxide-, Helmholtz-, or space-charge layer of the materials. 

Figure 4a. Electrochemical cells for microwave conductivity measurements. Cell above microwave conduit (1) electrochemical cell (plastic tube, placed on working electrode material), (2) counter-electrode, (3) reference electrode, (4) electrolyte, (5) space charge layer, (6) diffusion layer, (7) contact to working electrode, (8) waveguide.

Another technique consists of MC measurements during potential modulation. In this case the MC change is measured synchronously with the potential change at an electrode/electrolyte interface and recorded. To a first approximation this information is equivalent to a first derivative of the just-explained MC-potential curve. However, the signals obtained will depend on the frequency of modulation, since it will influence the charge carrier profiles in the space charge layer of the semiconductor. 

Relation (18) for the potential-dependent PMC signal is a reasonably good approximation only for the depletion region, where the space charge layer is controlled by the presence of fixed electron donors (Afo). It would become even more complicated if bimolecular or even more complicated kinetic reaction steps were considered. 

The Gartner model simulates charge collection by a potential-dependent space charge layer and considers diffusion into the space charge layer of charge carriers generated deep inside the semiconductor. The well-known Gartner formula for the photocurrent /ph is... 

The decrease of the PMC signal toward increasing depletion therefore reflects the increasing dynamics of minority carriers passing the space charge layer. No classical electrochemical technique has up to now permitted observation of this phenomenon with such clarity. 

Figure 28. Semiconductor interfaces with increasing electric fields in the space charge layer (from top to bottom) compared with tubes of different diameters through which an equivalent amount of water is pressed per unit time (equivalent to limiting current).

Interesting results have also been obtained with light-induced oscillations of silicon in contact with ammonium fluoride solutions. The quantum efficiency was found to oscillate complementarity with the PMC signal. The calculated surface recombination rate also oscillated comple-mentarily with the charge transfer rate.27,28 The explanation was a periodically oscillating silicon oxide surface layer. Because of a periodically changing space charge layer, the situation turned out to be nevertheless relatively complicated. 

The schemes in Figs. 44 and 45 may serve to summarize the main results on photoinduced microwave conductivity in a semiconductor electrode (an n-type material is used as an example). Before a limiting photocurrent at positive potentials is reached, minority carriers tend to accumulate in the space charge layer [Fig. 44(a)], producing a PMC peak [Fig. 45(a)], the shape and height of which are controlled by interfacial rate constants. Near the flatband potential, where surface recombination... 

Between the 2 and the magnitude of this potential depends on and the ionic strength of the solution. It will be apparent that 2 will determine the concentrations of charged electroactive species, while will determine the rate of the electron transfer step if... 

Fig. 3.4 Schematic representation of the cauliflower structure, showing the space charge layer in relation to the electrolyte and the semiconductor, and the pinching of cauliflowers , which is believed to be responsible for the disorder-dominated impedance. (Reprinted with permission from [71], Copyright 2009, The Electrochemical Society)...

Lanz and Com [51] proposed a 20-nm thick space charge layer on the Ti02 surface. When the fourth-order response with our TMA-covered surface is generated in the space charge layer, the broad width of the 826-cm band is understood as a depth-dependent wavenumber of the lattice vibration. 

The simplest model for the ionic distribution at liquid-liquid interfaces is the Verwey-Niessen model [10], which consists of two Gouy-Chapman space-charge layers back to... 

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