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100 THE CONDUCTIVITY OF THIN METALLIC FILMS ACCORD- ING TO THE ELECTRON THEORY OF METALS BY K. FUCHS, H. H. Wills Physical Laboratory, University of Bristol [Communicated by MR N. F. MOTT] [Received 25 October, revised 25 November, read 6 December 1937] 1. INTRODUCTION The conductivity of thin films of the alkali metals has recently been measured in the H. W. Wills Physical Laboratory, Bristol*. It was found that as the thick- ness of the film is decreased to that of a few atomic layers the conductivity drops below that of the bulk metal. In the papers quoted the hypothesis was put for- ward that this effect is due to the shortening of the mean free paths of the con- duction electrons of the metal by collisions with the boundaries of the film. The experimental results were compared with a formula derived on the basis of this hypothesis. This formula was, however, obtained subject to a number of simplifying assumptions, and it is the first purpose of this paper to obtain a more accurate formula. I also compare this formula with experiment, and make certain deductions about the surfaces of thin films. We assume throughout that the electrons in the metal may be regarded as free, in the sense that their energy E is the same function of their wave number k as for free electrons, so that E = ^0 Evidence from the optical properties of alkali metals, as well as theoretical calculations of Wigner and others, suggest that this is a good approximation. We first discuss certain other formulae which have been derived for the resist- ance of thin films. 2. J. J. THOMSON'S FORMULA The first formula was given in 1901 by Thomsonf. To obtain this formula we make the following assumptions. (a) When an electron collides with the surface of the film, the probability * Lovell, Proc. Boy. Soc. 157 (1936), 311; Appleyard and Lovell, Proc. Roy. Soc. 158 (1937), 718. t Proc. Cambridge Phil. Soc. 11 (1901), 120.
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Conductivity of thin metallic films 101 that it is scattered into any solid angle cUo is do)j2n, and is thus independent of the initial and final directions of the motion of the electron. (/?) The free path of an electron in the bulk metal is a constant A o . We assume further that AQ is greater than the film thickness t. Consider an electron which starts from a point Pat a distance 2 from the surface Fig. 1. of the film, and moves in a direction making an angle 6 with the z-axis (cf. Fig. 1). Its free path A is given by ( {t-z)jcosd (O^d^dj), A o (6^6 ^d 0 ), (1) -Z/COSO (6 0 ^d^7T), where cos 6 X = (t — z)/Ao, cos^ 0 = -z/A 0 . The mean free path is obtained by taking the mean value of A over all angles 6 and all distances z. We obtain \ = ^-\ dz [" XsmOdd ttjo Jo 31 (2) Since a is proportional to the mean free path, we obtain for the ratio of the conductivity <r to the conductivity <r 0 of the bulk metal 1 1 (3) which is Thomson's formula. The assumptions used in deriving this formula are incorrect for the following reasons.
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This note was uploaded on 12/29/2011 for the course PHYSICS 731 taught by Professor Appelbaum during the Fall '11 term at Maryland.

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