PHYSICAL REVIEW B VOLUME 1, NUMBER 12 15june1970 Theory of Metal Surfaces: Charge Density and Surface Energy* N. D. Langt and.Kohn Department of Physics, University of California, San Diego, La Jolla, Califoria 92037 (Received 28 January 1970) The first part of this paper deals with the jellium model of a metal surface. The theory of the inhomogeneous electron relation energies, is used. Self- consistent electron density distrib ace energy is found to be neg- ative for high densities (rg2.5).In are calculated which arise ections to the surtace energy is replaced by a pseudopotential model of the ions. One c tralized lattice, the other an interaction ener Both of these correc- tions are essential at higl t rgy is in semiquan- titative agreement with surface-t eight simple metals (Li, Na, K, Rb, Cs, Mg, Zn, Al), typical errors being about 25%. For Pb there is a serious disagree- ment. I. INTRODUCTION ment over a wide range of densities. The calcu- The electron theory of metals has lated surface energy, however, while in fair agreement with experiment at low electron den- primarily concerned with properties of the m sities, fails completely-to the point of giving the interior. These bulk properties are, of course, of great fundamental interest, and, wrong sign -for higher-density metals such as Al The present paper is aimed particularly at the the theorist, the translati problem of the surface energy. In Sec. II, we ing inside the metal introdu of simplicity into describe fully self-consistent calculation for the years there has been exce round(or jellium) model of metal using the theory of Refs. 7 and 8. Nu- ments of both electr interactions, so that merical results for density distributions (including of giving quantita oscillations), potentials, and surface en- wide classes of m Theories of dy remarked, the uniform speaking, lagged primarily to the ground model is totally inadequate for de- bing the surface energy of high-density metals. duced by th In Sec. III, we supplement this model by first-or- near the su: using the zero- symmetry stributions of the uniform back- was animp and by the addition of the appropri- an appr Na. I energies. The r esulting gies are found to be in rather good very tey ith experiment over the entire range problem tion of elect In a subsequent paper we shall describe the ef- with systems nic lattice on the work function, par- was put forward ticularly on the anisotropies associated with dif- This formulation ferent crystal faces. roreatsns II. UNIFORM POSITIVE BACKGROUND MODEL formed approx using this the A. Mathematical Formulation surface in which torm seml-intinite po We address ourselves to the problem of deter- fully self-consister mining the surface electronic structure in the lines has been repo model of a metal in which the positive charges are rs. These studies give, the work replaced by a uniform charge background of den- function, good qualitative agreement with experi- sity 4555
N. D LANG AND KOHN where v[n;F]≡ =0,x>0. For orientation, we remark that a Thomas-Fermi Assuming that the form of Exon] and hence of calculation, f leads to an electron density distri vetrIn; F]is known, the solution of the following bution which decreases smoothly from its interior self-consistency problem gives the exact density value n to zero, over a distance of the order of the istribution of the system of n interacting ele Thomas-Fermi screening length[see Fig. 1(a) trons However, for quantitative purposes, such a calcu {-y2+le[n;F}如=∈;b lation is quite inadequate. It leads to a vanishing work function and negative surface energies, and n()=∑|(F)|2, does not exhibit the important Friedel oscillations of the electron density near the surface where the i are the N lowest-lying orthonormal oresented here uses the self-con- solutions of (2. 5a).The energy E,n] of the sys sistent equations of Kohn and Sham. These are tem is given by(2. 2),with based on the general theory of the inhomogenous electron gas, which includes exchange and corre- T[n]=∑∈;-∫ vet [ns;过]n()d (2.6) lation effects. We review these equations here It is convenient at this point to state a number of It is shown in Refs. 7 and 8 that the total elec facts Eqs. (2.7)-(2. 12) which are strictly cor tronic round-state energy of a many -elec rect for the present model [Eq.(2.1)], including system in an external potential v(r)can be written all many-body effects. Some of these statements in the following form are illustrated in Fig. 1 E!1在+/m件FmP The electrostatic potential energy difference of an electron between x=+o and x=-oo, the so +Tsn]+ExIn called electrostatic dipole barrier, which we de note by△q, is given by Here the functional Ts[n] is the kinetic energy of noninteracting electron system of density dist △q≡cp(+∞)-q(-∞) bution n(), and the functional E[n] represents 4 mLo dx fdx[n(x”)-n(1 the exchange and correlation energy(Hartree theo y corresponds to setting Exc=0). One then defines =4丌Cx{n(x)-n,(x)d (2.7) The chemical potential u of this system, de- ntm;=()+Fdy+m;1,(2.3) fined, as usual, as the ground-state energy differ ence of the N+ l and N electron systems(with the background charge fixed at Nlel )is given by (-∞)+以 Background, n+(x) where u is the intrinsic chemical potential of the Electrons, n(xI infinite system (relative to the electrostatic po tential in this system). 16 From its definition, u is given by where kF is the Fermi momentum of a degenerate electron gas of density n and ux(n)is the ex change and correlation part of the chemical po tential of an infinite uniform electron gas of den sity n. If the exchange and correlation energy per particle of such a gas is denoted by Ex (n),then from the definition of Exe[n] FIG. 1. Schematic representation of (a)density dis tributions and (b) various energies relevant to the metal The work function, defined as the minimum en ergy necessary to eject an electron, is
THEORY OF METAL SURFACES: CHARGE DENSITY 4557 更=φ(+∞)-=△q-μ (2 relation energy give, within a few percent, the same results In the interior of the metal, Vett approaches a constant value [see(2. 3),(2. 4), and(2. 8c)] We can now rewrite the self-consistency problem (2. 5)in a form specific to the present problem: Ue→q(-∞)+μi) Hence the eigenfunctions of (2. 5a)can be labeled 2 dx tuer In;x]vo2(x)=2 by the quantum numbers k, k ku, with the fol where has the asymptotic form(2. 11b).vett is aP*, R,, k, =P, (x)exp[i(R, y +ksz)] (2.11a given by where,forx→-∞ e;x=4团n]-4Cax":dx W(x)=sink -y(k)] [n(x”")-n,、(x)+μn(x),(2.16b) Here y(k)is the phase shift which is uniquely de termined by the conditions that y(o)=0 and that th型n]=△n]-a r(k)be continuous. The eigenvalues of(2 a are The density is in turn given by then [from(2. 10) 2广a (2.16d) ∈,A,An=p(-∞)+μx如)+k2+k2+k2),(2.12a) If for convenience we choose the zero of energy The numerical solution of these equations requires careful treatment of quantum oscillations which are present in the density and potential(see Appen 0 (2.13) dix A1). Details concerning the method of solution then by(2.8),q(-∞)+μ)=-k2,and(2.12a) are given in Appendix B B 2+k2+k2-12) (2.12b) The self-consistent system of Eqs.(2. 16)was In order now to make practical use of the theory solved for the bulk metallic densit embodied in Eqs. (2. 2)-(2. 6), some approximate 2-6 at intervals of 0.5. The degree of self- form of the exchange and correlation energy func- consistency achieved in n(x)varied from 0.08% tional is required. For a system with very slowly (for rs=2)to 0. 7%(for rs=6)of the asymptotic density En[n]=∫∈min(式)n(F)d式 (2.14) Table I gives n(x)for rs=2, 2 displays nx)for rs=2 and 5. It will be observed with errors proportional to the squares of the that for the low mean density corresponding to density gradients. Following Refs. 8 and 18, we rs=5, there are sizeable Friedel oscillations shall use this form for the present problem, even ng an ot of n by 120 though in the surface region of a typical metal the hand, at the high mean density corresponding to density varies quite rapidly. a"control"calcula s=2, the density distribution begins to resemble tion, to be described below, and the fact that the the monotonically decreasing form of the Thomas final results are in rather good agreement with Fermi theory (ef. discussion in Appendix A2) experiment suggest that the errors introduced by Figure 3 shows the electrostatic potential ener approximation(2. 14)are not too serious. This o()and the effective potential vern; x]for question is discussed again later on in the present r.=5. It will be noticed that the electrostati section, and in the concluding remarks arrier△φ=q(∞)-q(-∞) is very small, but For ex(n), the exchange and correlation energy that in the vicinity of the surface, o(x)exhibits per particle of a uniform electron gas, we use the substantial oscillation. The corresponding oscil approximation due to wigner. In atomic units, it lation in vet is considerably smaller. This can be explained by the fact that, for large negative 0.458 x, the oscillatory terms of and of the exchange ∈x(n)=- part ofμ cancel exactly( Appendix A).Bothφ and vett are given in Table I for integral rs values where r (n)is defined by from 2 to 6 (4丌/3)[yam)3=1/mn Approximation (2. 14)for Exen] is based on the assumption of a nearly uniform gas. It leads to Other more recently suggested forms of the co an effective exchange and correlation potential
4558 n. d. LANG AND W. KOHN 计计计 学计计9宁9宁心后出 点:3 ·::: 2:: R?77777?9?7 身当 N:88A下 到3号8 68888444440988824244-44808888889888888888888 三当写 3号到 a........44.4444 引9;ss 运 器图到5 6乱 注÷沪宁宁宁ss vxc which vanishes exponentially as x-o, where- ployed for vett. However for x>xo, vott was as one would expect that the correct vxc would be taken to have the image form -1/4x, with p have like the classical image potential, i.e computed from(2. 16c)and(2.7). The problem was then solved, for rs=2. 5 and 5, requiring self- consistency of n(x). Fortunately, the densities in To assess the quantitative importance of this fail- these calculations were found to differ from those ure of our approximation, we have carried out the previously obtained by no more than 1. 2% of n for following control calculations. Up to the point xo rs=5, and no more than 0. 3% for rs=2. th where Dett =0, we used the form previously em appears that our use of the form (2. 16b)for vett
THEORY OF METAL SURFACES: CHARGE DENSITY 4559 Here the first two terms represent, as before, kinetic, exchange, and correlation contributions to the electronic energy [see Eq.(2. 2).The last term e is the total classical electrostatic en ergy of all positive and negative charge densities (n(F)-n,(F) n,(F) BACKGROUND =「q[;F](m()-n,()症,(219) of an electron, is given bye tic potential energy where the total electrosta pIn; r (2.20) Corresponding to(2. 18), the surface energy of FIG.2. Self-consistent charge density near metal ne uniform background model may be written urface for rs=2 and rs=5 (uniform positive background (2.21) also in the region outside of the metal surface For as we can take over the analysis presented by where it is not correct, does not introduce seri- Hunting hich gives ous errors into the density distributions. In ad dition, we shall see in Sec. Ic that the correla- tion contribution to the surface energy is a rela- 0(4-7/a)-k2)kd tively small fraction of the experimental value and thus, in discussing surface energies, errors j fvott[n; x]-vott[;-oo]] n(x)dx ue to an inadequate treatment of correlation ef (2.22) fects should not be important The other two terms are, in the present model, C Surface Energies [∈xa(x)-∈l)n(x) The surface energy o of a crystal is the energy required, per unit area of new surface formed and g=h∫。φv;x](n(x)-n,(x)dx.(2.24) to split the crystal in two along a plane. The total energy of the crystal, split or unsplit,can Table ii lists the magnitudes of ou and its three be written as a sum of three terms components for different values of rs.First,we E=Tsn]+Ere n]+eesIn observe that the kinetic-energy contribution os is negative, reflecting the fact that in the split crystal, the electron density is more spread out 0. Second, we note that over the entire density range Uxe >>0es, showing that Thomas-Fermi or Hartree calculations are completely useless for quantita TABLE I. The surface energy O, and its components in the uniform background model. 0xe 0x+Oc; O =0s+oxc 1330 1350 430 DISTANCE(FERMI WAVELENGTHS 05050 380 FIG.3. Effective one-electron potential veff, with electrostatic part near metal surface (positive back- 6.0 10 vs=5)