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Diffusion, Static Charges, and the Conduction of Electricity in Nonmetallic Solids by a Single Charge Carrier. I. Electric Charges in Plastics and Insulating Materials
1.P. C. Woodland and E. E. Ziegler, Modern Plastics 28, 95 (May, 1951).
2.Skinner, Savage, and Rutzler, Jr., J. Appl. Phys. 24, 438 (1953);
2.Skinner, Savage, and Rutzler, Jr., 25, 1055 (1954). The present paper incorporates certain material in correspondence to Messrs. West and Morant which could not be included in the letter because of space limitations., J. Appl. Phys.
3.Woodland and Ziegler, reference 1, and personal communication from R. F. Boyer, Dow Chemical Company, Midland, Michigan. Similar difficulty is experienced in the radio parts industry.
4.For example, F. A. Vick, “Theory of contact electrification,” a paper in the symposium on Static Electrification, of the Institute of Physics in London, 25–27 March 1953, Supplement No. 2 to the British J. Appl. Phys.
5.A particular case was treated by N. E. Mott and R. W. Gurney, Electronic Processes in Ionic Crystals (Oxford University Press, New York, 1940), pp. 168–173.
6.Eley, Parfitt, Perry, and Taysum, Trans. Faraday Soc. 49, 79 (1953).
7.The letter b is used for the mobility to avoid confusion in later equations, since the symbol, μ, usually used for this quantity is also universally used for the chemical potential.
8.The general case with, however, the assumption of electrical neutrality and fixed ionic charges has been treated by a number of authors including: W. Shockley, Bell System Tech. J. 28, 435 (1949);
8.W. van Roosbroeck, Bell System Tech. J. 29, 560 (1950);
8.and R. C. Prim, III, Bell System Tech. J. 30, 1174 (1951).
8.C. Herring, Bell System Tech. J. 28, 401 (1949), has considered some more general cases without fixed ionic charges, but assuming electrical neutrality and eliminating the use of Poisson’s equation; diffusion is neglected until near the end, when an estimate of its effects is given.
9.If δ is small but not zero, and Eq. (4) is linearized to the first order in δ. i.e. and only terms in δ retained, the assumption that is small yields Therefore, a better approximation to the effect of charges thrown up from the valence level may be obtained from the solution for that for being obtained as (4.2) in which δ [Eq. (4)] may be given its local value.
10.A. Sommerfeld and H. Bethe, Handbuch der Physik (Springer‐Verlag, Berlin, 1933), second edition Vol. 24.2, p. 333.
10.The treatment is also in R. H. Fowler, Statistical Mechanics (The Cambridge University Press, New York, 1936), second edition;
10.and J. E. Mayer and M. G. Mayer, Statistical Mechanics (John Wiley and Sons, Inc., New York, 1940).
11.For example, P. T. Landsberg, Proc. Roy. Soc. (London) A213, 226 (1953) who has used the diffusion equations effectively, to investigate the Einstein “mobility‐diffusion constant” relation and the rectifying properties of semiconducting materials.
12.Transistors Teachers Summer School, Phys. Rev. 88, 1368 (1952);
12.Landsberg, reference 11.
13.M. V. Laue, Jahrb. Radioakt. u. Elektronik 15, pp. 205, 257 (1928);
13.M. V. Laue, Handb. d. Radiol. 6, 452 (1925).
13.W. Schottky, Jahrb. Radioakt. u. Elektronik 12, 14 (1915).
13.O. W. Richardson, The Emission of Electricity from Hot Bodies (Longmans Green and Company, New York, 1921), second edition.
13.R. H. Fowler, Statistical Mechanics (The Cambridge University Press, New York, 1936), second edition.
13.F. Borgnis, Z. Physik 100, 117 (1936). After this discussion was in manuscript form, a letter was also received from D.C. West of the Nova Scotia Research Foundation in which the existence of the three solutions was pointed out independently.
13.Dr. West’s letter is in J. Appl. Phys. 25, 1054 (1954).
14.W. Shockley, Electrons and Holes in Semiconductors (D. van Nostrand Company, Inc., New York, 1950), p. 350;
14.J. R. Haynes and W. Shockley, Phys. Rev. 81, 835 (1951).
15.A rough conceptual picture can also be obtained as follows: let δ be the distance a particle of energy kT could go against the electric field before being brought to rest, reversed in direction, and given again an energy kT, but now traveling in the opposite direction. Then and if which relates the change of this characteristic distance, δ, to the change in the space coordinate. Or again, since is the ratio, (by Sommerfeld and Bethe) of the transport of heat, or (by irreversible thermodynamics) of the transport of the energy distinct from heat, to the electric current under classical isothermal conditions, φ is related to the ratio of the energy due to the electric potential to that transported by the transfer of electrons.
16.S. M. Skinner (to be published).
17.W. Shockley, Bell System Tech. J. 28, 435 (1949);
17.also, Electrons and Holes in Semiconductors (D. Van Nostrand Company, Inc., New York, 1950).
18.S. R. DeGroot, The Thermodynamics of Irreversible Processes (North Holland Publishing Company, Amsterdam), and (Interscience Publishers, Inc., New York, 1951).
18.K. G. Denbigh, The Thermodynamics of the Steady State (Methuen and Company, Ltd., London), and (John Wiley and Sons, Inc., New York, 1951). The special case is diffusion and electrical conduction, U is the generalized potential, the generalized force, and the conductivity divided by the charge. The correspondence extends even further: the rate of entropy production resulting from the thermodynamic treatment is the natural generalization of the relation, i.e. Also, the free path integrals of Sommerfeld‐Bethe are consistent with and Eq. (2).
19.Earnshaw’s theorem in classical electrostatics [J. A. Stratton, Electromagnetic Theory (McGraw‐Hill Book Company, Inc., New York, 1941). p. 116;
19.G. P. Harnwell, Principles of Electricity and Magnetism (McGraw‐Hill Book Company, Inc., New York, 1949), second edition, p. 63;
19.or W. R. Smythe, Static and Dynamic Electricity (McGraw‐Hill Book Company, Inc., New York, 1939), first edition, p. 13] that a charged particle placed in an electrostatic field cannot be maintained in stable equilibrium under the influence of the electrostatic forces alone is not applicable. Both diffusion and electrostatic forces are acting, and the equilibrium is dynamic, being maintained by diffusion into and through the conduction levels of the insulator. The electrodes at either end of the insulator furnish the high concentration regions which maintain the external contribution to the dynamic equilibrium.
20.In the case of the circular functions this is evident by applying the requirement that for each boundary the value of the expressions in the first half of (17) should be the same at that boundary whether the origin is taken initially at that boundary or, initially, at the other boundary and moved to the first boundary with the direction of the x‐axis reversed. The same value must then be used in the argument for the hyperbolic functions to assure analytic continuation. This may also be shown directly since the circular and hyperbolic functions involved here are actually the first terms of an asymptotic expansion. The requirement in the first sentence of this footnote, taking account of Stokes’ phenomenon for the asymptotic expansions of the Bessel functions involved, yields the value of a or g. This requires, of course, that be given its value obtained from the last expression in the second half of Eq. (17) by letting
21.The charge per of surface contained in the plastic from one metallic contact is Values of and are given in Table I, reference 2.
22.With even somewhat extreme values of the constants the inner potential will not be above that at the interface by more than 0.7–1.0 ev.
23.This is discussed in a forthcoming article by the authors of reference 2.
24.See, for example, Bobalek, LeBras, Powell, and von Fischer, Ind. Eng. Chem. 46, 572 (1954), in particular, the last paragraph of the article.
25.However, it is known that electron bombardment of a metal electrode de‐gasses hydrogen ions from the interior.
26.W. J. Russell, Proc. Roy. Soc. (London) A61, 424 (1897), and seq.
26.A summary is given by G. L. Keenan, Chem. Rev. 3, 95 (1920).
27.L. Grunberg, Proc. Phys. Soc. (London) B66, 153 (1953). Zn, Al, Mg, Ni; in the case of Zn, the number of molecules of formed was 1.5 times the number of Zn atoms exposed.
28.The effect of semiconducting layers between metal and dielectric will be treated in a separate paper.
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