PHYSICAL REVIEW B VOLUME 3, NUMBER 4 15 FEBRUARY 1971 Theory of Metal Surfaces: Work Function* N. D. Lang IBM Watson Laboratory, Columbia University, New York, New York 10025 and W.Kohn, University of California, San Diego, La Jolla, California 92037 Hebrew University, Jerusalem, Israel (Received 16 October 1970) In a recent paper we presented a contributio to the theory of metal surfaces with emphasis on the shape of the electron-density distribution and the surface energy. The present paper extends this analysis to a consideration of the work function. Some general theoretical rela- tionships are established. Effects of the ions are included using a simple pseudopotential theory, permitting the calculation of the variation of the work function from one crystal face to another. For simple metals(li,na,k,rb,cs,l,pb,zn, and Mg), agreement with available experimental data is good (5-10%); for the noble metals, the computed work func- tions are 15-30% too low. I. INTRODUCTION puted in LK-I, and we take from the available theory of exchange and correlation of a uniform The present paper represents a sequel to one electron gas. This yields the work function f concerned primarily with metal-surface charge this model as a function of the mean bulk density densities and surface energies. In the earlier (or of the Wigner-Seitz radius).4 These results paper we presented, first of all, a theory of the are compared with experiment and with the theo- electronic structure of a model metal surface in retical calculations of Smith, who used a similar which the lattice of approach but did not carry out a fully self-consis- uniform background cha tent calculation. 5 tion effects were include Finally, we incorporate the effect of the actual version of the theory ion cores. We show first that when the difference gas. 23 Following this, the effect of etween the pseudopotentials of the ion cores and structure on the surface the electrostatic potential of the uniform charge count by calculating background is treated as small perturbation(), contribution(similar to the change of the work function of a particular crys- by evaluating the in tal face due to this perturbation is given to first the ion cores using f order by the following rigorous expression: ory. In the present p (1.2) plan in developing a Here the integral is carried out over a slab whose This quantity, denoted surface consists overwhelmingly of the face in ques- in Sec. II in terms mo tion; and() is the change of the electron density, analysis, is equal to the mini e uniform-background model, fol- be done to remove an electro ctron from the system. se localized near We give first a rigorous demo We have calculated the charge =△中一μ, s to that of LK- d to look for self-consis- where is the rise in mean electrostatic potential tent solutions with a zero mean electric field deep potential of the electrons relative to the mean elec- Since n () depends in fact only on the distance trostatic potential in the metal interio surface, it is possible to re- its simple form, this expression i duce(1. 2)to a one-dimensional quadrature.In body effects, in particular, that of the image force. this way we have calculated the total work function For the uniform-bac from the electronic charge density n( ckgroun 虫+ (1.3) 31215