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Fig. 2.3 Schematic structure of a silicon semiconductor of (A) n type and (B) / type together with the energy level array. (According to C. Kittel)
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If, on the other hand, the impurity is an electron acceptor (here, elements of group III), then the new acceptor band lies closely above the valence band (see Fig. 2.3B). The electrons from the valence band pass readily into this new band and leave holes behind. These holes are the main charge carriers in p-type semiconductors. Compared with metals, semiconductors have quite high resistivity, as conduction of current requires a supply of activation energy. The conductivity of semiconductors increases with increasing temperature. Germanium provides a good illustration of conductivity conditions in semiconductors. In pure germanium, the concentration of charge carriers is rte = 2 x 1013 cm"3 (in metals, the concentration of free electrons is of the order of 1022cm~3). The intrinsic conductivity of germanium is about the same as that of pure (conductivity) water, K = 5.5 x 108 Q~l cm"1 at 25 C. In strongly doped semiconductors, the concentration of charge carriers can increase by up to four orders of magnitude, but is nonetheless still comparable with the concentration of dilute electrolytes. Ionic (electrolytic) conduction of electric current is exhibited by electrolyte solutions, melts, solid electrolytes, colloidal systems and ionized gases. Their conductivity is small compared to that of metal conductors and increases with increasing temperature, as the resistance of a viscous medium acts against ion movement and decreases with increasing temperature.
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90 A special class of conductors are ionically and electronically conducting polymers (Sections 2.6.4 and 5.5.5). 2.4.2 Conductivity of electrolytes This part will be concerned with the properties of electrolytes (liquid or solid) under ordinary laboratory conditions (i.e. in the absence of strong external electric fields). The electroneutrality condition (Eq. 1.1.1) holds with sufficient accuracy for current flow under these conditions:
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2^, = 0
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(2.4.1)
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where ct are the concentrations of ions / and z, are their charge numbers. The basic equations for the current density (Faraday's law), electrolytic mobility and conductivity are (2.3.14), (2.3.15) and (2.3.17). The conductivity K = ^ z2F2uiCi = 2 N FUiCi
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(2.4.2)
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in dilute solutions is thus a linear function of the concentrations of the components and the proportionality constants are termed the (individual) ionic conductivities: li = zJP2ui = \zi\FUi (2.4.3)
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Consider a solution in which a single strong electrolyte of a concentration c is dissolved; this electrolyte consists of v+ cations B 2+ in concentration c+ and v_ anions Az~ in concentration c__. Obviously v+z+ = -v_z_ = v_ |z_| and
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c. c_ c= = (2.4.5)
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(2.4.4)
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Substitution into Eq. (2.4.2) yields K = z2+F2u+v+c + Z2-F2u_v_c = (U+ + U_)z+v+Fc = (U+ + UJ) |z_| v_Fc The quantity (U+ + U_)z+v+F = (U+ + I/_) |z_| v_F = - = A (2.4.7) c is called the molar conductivity, which is as shown below, a concentrationdependent quantity except in an ideal solution (in practice at high dilution). (2.4.6)
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91 For first class conductors, the conductivity is a constant characterizing the ability of a given material at a given temperature to conduct electric current. However, for electrolyte solutions, it depends on the concentration and is not a material constant. Thus the fraction A = K/C is introduced; however, it will be seen below that the constant characterizing the ability of a given electrolyte to conduct electric current in solution is given by the limiting value of the molar conductivity at zero concentration. The main unit of molar conductivity is Q" 1 m2 mol"1, corresponding to K in Q" 1 m"1 and c in mol m~3. However, units of Q" 1 cm2 mol"1 are often used. If units of Q" 1 cm"1 are simultaneously used for K and the usual units of mol dm"3 for the concentration, then Eq. (2.4.7) becomes
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When reporting the molar conductivity data, the species whose amount is given in moles should be indicated. Often, a fractional molar conductivity corresponding to one mole of chemical equivalents (called a val) is reported. For example, for sulphuric acid, the concentration c can be expressed as the 'normality', i.e. the species ^H2SO4 is considered. Obviously, A(H2SO4) = 2A(^H2SO4). Consequently, the concept of the 'equivalent conductivity' is often used, defined by the relationship A* = - ^ - = ^ = (U+ + UJ)F = At + A* (2.4.9)
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where A* = A+/z+ and A* = A_/|z_| (cf. Eq. 2.4.3). It follows that, for our example of sulphuric acid, A*(H2SO4) = A(^H2SO4) = ^A(H2SO4). Combination of Eqs (2.3.15) and (2.3.17) yields j^-^gradtf) (2.4.10)
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where tt is the transport (transference) number; giving the contribution of the ith ion to the total conductivity K, that is
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for a single (binary) electrolyte,
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U+ + U. t/_
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At + A! (2.4.12) A*