Anisotropy of the Electronic Work Function of Metals

Abstract

Work function is experimentally known to be different for different faces of a crystal by amounts ranging from one-tenth to half a volt. For tungsten the faces can be arranged according to decreasing work function as follows: 110, 211, 100 and finally 111. The explanations so far suggested for the differences of the work function are discussed and shown to give either an incorrect sequence or a wrong order of magnitude of the observed differences. The author uses the picture of Wigner and Bardeen according to which the work function is a sum of a volume contribution and a contribution due to a double layer on the surface of the metal. The origin of the latter can be described in the following manner. With every atom one can associate a polyhedron ("$s$-polyhedron") with the atom at its center, such that it contains all points nearer to the atom under consideration than to any other atom. If the distribution of the electron density within these polyhedra of the surface atoms was the same as for the inside atoms then there would be no double layer on the surface. However, this is not the case since the total energy is lowered by a redistribution of the electron cloud on the surface. There are two effects: the first is a partial spread of the charge out of the $s$-polyhedra and the second is a tendency to smooth out the surface of the polyhedra. In consequence of the second effect the surfaces of equal charge density are more nearly plane than in the original picture. The two effects have opposite influences and since they are comparable in magnitude, it is not possible to predict the sign of the total double layer without numerical computations. Some general formulae for the double layers are derived and discussed more fully in the case of a simple cubic and a body-centered cubic lattice. The minimum problem of the surface energy is solved for four faces of a body-centered crystal and the results are applied to the case of tungsten. One obtains the differences between the work functions for different directions. The results agree satisfactorily with the experimental data: assuming a reasonable density of the free electrons, one obtains the correct sequence of faces and the correct differences of the work function. The surface energies are calculated an d found in agreement with the observed stability of certain crystal faces.