interface, 0SC - 0S = Af0, we find for the surface charge of the semiconductor in .NET

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interface, 0SC - 0S = Af0, we find for the surface charge of the semiconductor
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(4.5.5)
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The overall Galvani potential difference between the bulk of the semiconductor and the bulk of the electrolyte solution can be separated into three parts: Ar< = Af 0 + A|0 + Al2(j) (4.5.6) where Af$ is the potential difference between the bulk of the semiconductor and the semiconductor-electrolyte interface, A|0 is the potential difference in the compact layer (i.e. between the interface and the outer Helmholtz plane) and A 0 is the potential difference between the outer Helmholtz plane and the bulk of the electrolyte solution. Obviously,
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(4.5.7) where sc , Ex and Ex are the electric field strengths in the semiconductor, in the compact layer and in the diffuse layer in the electrolyte, respectively, Lx is the Debye length in the solution, dx is the thickness of the compact layer and sc, H a n d i are the permittivities of the semiconductor, compact layer and electrolyte, respectively. At sufficiently large electrolyte concentrations, the second and third terms in parentheses in Eq. (4.5.7) can be neglected. Under these conditions, the Galvani potential difference between the semiconductor and the electrolyte solution is given by the potential difference in the space charge region (diffuse layer) in the semiconductor, which also holds for larger potential differences than RTIF. The distribution of charge carriers in the space charge region of an ft-type semiconductor is shown in Fig. 4.1. It is assumed that the semiconductor contains not only electrons (majority charge carriers) but also much lower concentrations of holes (minority charge carriers). If the negative potential of an ft-type semiconductor electrode increases, then the negative charge of the electrode increases through an increase in the concentration of electrons in the interphase (accumulation layer). As the electron energy in the space charge region e(p(x) depends on the distance from the interface, the energy bands bend downwards in the direction towards the interface in the negatively charged space charge region (Fig. 4.12A). When the surface charge decreases to zero, the energy bands become horizontal. The corresponding flat-band potential A * 0 ^ is an analogy of the zero-charge potential Epzc (Fig. 4.12B). When the difference A^C0 - A* 0 ^ becomes positive, the interphase is depleted of majority charge carriers (electrons in this case) forming a
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Fig. 4.12 Dependence of concentrations of negative charge carriers ( e ) and positive charge carriers ( p ) on distance from the interface between the semiconductor (sc) and the electrolyte solution (1) in an -type semiconductor. These concentration distributions markedly differ if the semiconductor/electrolyte potential difference AfQ is (A) smaller than the flat-band potential Ar0 f b , (B) equal to the flat-band potential, (C) larger and (D) much larger than the flat-band potential. nD denotes the donor concentration. (According to Yu. V. Pleskov)
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239 depletion or Mott-Schottky layer (Fig. 4.12C). However, Eq. (4.5.5) is a simplification, as it neglects the presence of minority charge carriers, which was originally very low. The following relationship holds between the concentration of electrons and the concentration of holes np: npnc = nl (4.5.8)
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where n0 is the concentration of electrons in the same semiconductor in the absence of impurities and of an electric field inside the semiconductor, i.e. in the corresponding intrinsic semiconductor. According to the quantum theory of semiconductors, the quantity n0 is given by the relationship n0 = -3 exp
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where m e is the electron mass, h is the Planck constant and eg is the width of the forbidden band. For a sufficiently large potential increase, the charge in the interphase finally corresponds to the minority charge carriers (Fig. 4.12D). The greater the width of the forbidden band eg, the broader is the potential range A\c<j> in which the space charge region has the character of a depletion layer, i.e. is formed by ionized impurity atoms. It should be noted that the study of the properties of an electrical double layer at semiconductor electrodes corresponds to study of the cell (in the simplest formulation) M | S | electrolyte solution | M (4.5.10)
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where M denotes the metal phase and S is the semiconductor. The electrolyte solution has a composition such that the interface with the metal is unpolarizable. An electrical double layer is formed at the interface between the metal and the semiconductor (called the Schottky barrier), but this interface is unpolarized, so that Am0 = [Me(s) - jUe(m)]/e (4.5.11)
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The discussion of Eq. (4.5.7) has shown that the potential of a polarized semiconductor electrode at sufficiently high electrolyte concentrations is Ep Af 0 + constant (4.5.12)
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In Eq. (4.5.5), describing an rc-type semiconductor strongly doped with electron donors, the first and third terms in brackets can be neglected for the depletion layer (Alc(f) kT/e). Thus, the Mott-Schottky equation is obtained for the depletion layer, asc = Vlenle Af<j> (4.5.13)
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