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)
4560 N. D. LANG AND W. KOHN introduced by the approximate treatment of corre- lations are not significant IIL ION LATTICE MODEL In the present section, we shall calculate the sur face energy on the basis of a model in which the ice, are represented by appropriate pseudopoten- tials. Such a model is known to be quantitatively successful for simple bulk metals in which the con- duction band has s-p character and is adequatel (bcc loL separated from d-like states. POTENTIAL For these metals, the difference between the Ifcc lll) total pseudopotential and the potential due to the -- UNI FORM POSITIV uniform charge background is small. Therefore, BACKGROUND MODEL taking advantage of the stationary propertyof ex- pression(2. 2)for E, n], we shall calculate all energies in the present model using the electronic density distributions n(x)of the uniform background model. In this way, we avoid the much more diffi- FIG. 4. Comparison of theoretical values of the sur- cult problem of solving truly 3-dimensional Schr face energy with zero-temperature extrapolations of ex- perimental results for liquid-metal surface tensions We adopt here the local ion pseudopotential pro (open circles). Dashed curve gives the surface energy for the positive background model. Vertical lines give osed by Ashcroft, which has the form eoretical values corrected for the presence of the lat (F) y≤c tice: The lower end point gives the value appropriate to point that appropriate to (3.1) bcc lattice. In both cases, the surface plane is taken to plane which is most densely pac where Z is the ionic charge and r is a cutoff ra- the alkali metals of lower density, the lines are con dius which has been determined for each metal to tracted almost to point give a good description of the bulk properties This is equivalent to representing each ion by an tive purposes. Finally, we note that, particularly effective charge distribution nion (r)which gives at higher densities, there are large cancellations rise to the potential(3.1 between positive and negative terms, making the Since in the present model, the electron densi- final results rather sensitive to small errors in ties n(x)of the uniform background case are em the individual terms. Over the range rs=3-5, ployed, the intrinsic electronic energies TIn]and covering the alkali metals Li, Na, K, for which Exc[n] are the same as before. The difference of Smith gives calculated surface energies, our re- the surface energies in the two models sults exceed his by about 50% In Fig. 4, the calculated surface energies are 6G=0-0g (3.2) compared with linear extrapolations to zero tem is therefore entirely due to the differences in elec perature of measured liquid-metal surface ten- trostatic interaction energies of all positive and sions. The agreement between theory and experi- is fair for the lo but for higher-density metals the measured surface tensions increase rapidly with density, while the calculated surface energies decrease towards larg negative values. This basic shortcoming of the uniform background model, which all previous cal culations have also encountered, will be corrected in the following section by going over to a model in which the positive ions are more realistically treated. The correlation contribution to the calcu lated o(Table Ii)is never more than about 15% of FIG. 5. Two steps for calculating the electrostatic imental value, indicating that contributions to the surface energy
THEORY OF METAL SURFACES: CHARGE DENSITY TABLE III. Cleavage energy constants a where o, is the surface energy in the uniform Lattice Cleavage planes background model and ooa and 8gpa are given by (111) Eqs.(3.3)and(3.4) 0.00325 0.01434 004407 B Computations and Comparison with Experiment 003100 We would like to thank M. Rao for his help with Since there are practically no data available for computations of a the surface energies of simple metals in the soli phase, we have computed values of o which would be most appropriate for comparison with mea negative charges [including the effective ionic surements of the surface tensions of liquid met arges nion G)] absence of a satisfac We recall the definition of surface energy as the the ionic configurations of liquid-metal surfaces, energy required, per unit area of new surface we have calculated o for such ordered lattice struc tures and cleavage planes as, in our view, re the two models, we calculate the electrostatic con- sembled most closely a liquid surtace. Exper g.5) similar to that in the solid state, provides some Step 1. We divide the crystal in two, holding the justification for this approach electron density uniform up to the nominal metal The coordination numbers, in the liquid state, boundary in each half. The contribution to 8o from of the metals considered lie between the coordina- this step is a classical cleavage energy which will tion numbers of the bcc and fcc lattices, which are be denoted by 6o 8 and 12, respectively. Therefore, calculations Step 2. Next we change, in both models, the were carried out for both of these two lattice electron density from its step-function form to its types. The faces selected were those most dense- actual form n(x). The contribution from this step ly packed,(111)for fcc and(110)for bcc.Such to 5g will be called 8o a choice has been considered reasonably repre Step 1 requires no energy in the uniform back sentative of a liquid surface by various authors, o ground model. In the ion lattice model we may, in and we have verified by sample calculations that calculating the energy required for this step, re place the pseudopotentials by point-charge poten orgies and hence would be expected to appear or tials.a dimensional argument shows that, for the surface a given lattice type and a given cleavage plane We have calculated surface energies for the 8 metals listed in Table IV. These include all of 80 =azn, (3. 3) those considered in the bulk pseudopotential calcu- where a is a dimensionless constant. The compi tation of this constant is described in Appendix c The results for the body-centered and face-cen tered cubic lattices with cleavage planes perpen 8v(x)with rc=0 dicular to the [100], [111], and [110] directions n(x}-百 0.2 are given in Table Ill Step 2, as inspection of Fig. 5 shows, contrib utes the following term to 60 COps- 6v()n(x)-n.()]dx Here v(x)is the average, over the y-z plane, of he sum le ionic pseudopotentials of the half- 06 lattice, minus the potential due to the semi-infinite round. The algebraic expres sion for 6v(x)is derived in Appendix D. The two STANCE (Bohr radii factors entering the integrand in (3. 4)are plotted FIG. 6. Factors in the integrand giving dopg [Eq. in Fig. 6 (3.4)1. The case of potassium is shown here. In the The total surface energy in the present model is absence of pseudopotential cancellation, &v(x)is the then given by unction represented by the deeply cusped dashed line (the lattice planes are at the cusps). The presence of substantial oscillations in n(x)-n,)has in general an 0=y+601+b0p, (3. 5) important effect on the value of &
4562 N. D. LANG AND W. KOHN 1 TABLE IV. Surface energies in the ion lattice model for eight simple metals. The table gives total surface energy o nt parts (a=Ou+dops +80e1) for the lattice structures and surface planes indicated, Units are ergs/em Also included are rs values, and values for the pseudopotential radius r (Ref. 3) Metal fcc(111) bcc(110) fcc(111) bcc(110) 2.07 1.12 408 2.65 30 0000 35 19 2.14 140 2.61 110 120 293 70 100 100 lations of Ashcroft and Langreth, with the excep typical error for these metals being about 25% tion of pb to which we shall return later. The We omitted the case of pb from the above consider calculations were carried out for mean densities tions. For this metal the measured surface ten n appropriate to the solid, using the values for the sion extrapolates to 620 ergs/ cm at zero temper pseudopotential radius re, given in Ref 3 ature, while our calculations gave a mean value of <a Since for given n, the lattice-plane spacings of of 1400 ergs/cm2. We have no real explanation of this large discrepancy at this time. It may be another, a difference which has been neglected noted, however, that ashcroft and langreth, in Oops, shown in column 5 of Table Iv, is the same their pseudopotential calculations of bulk energies, for the two types of faces. On the other hand, the also found rather less satisfactory agreement for classical cleavage energies shown in Pb than for the other metals. We also remark columns 6-9, differ rather considerably. This that, unlike the other metals considered, Pb is difference is reflected in the total surface ene tetravalent, and that furthermore, it has by far gies g listed in columns 8 and 9 the highest atomic number. We note that both 6pg and &]a are positive and In all the above described calculations the ion half-lattice was undistorted. In Appendix E, we Together they more than compensate for the neg how that allowing the surface plane of ions to ative values of ou. relax to a position of lower energy has a negligi Our final results are plotted in Fig. 4, ble effect on the calculated surface energies they are denoted"pseudopotential theory. "The results for fcc(111)and bcc (110)faces are IV CONCLUDING REMARKS joined by vertical lines. These lines may be re This paper reports the results of calculations garded as a rough measure of the uncertainties of the electron density distributions and surface introduced into our estimates for liquid-metal energies of simple metals. Many-body effects surface tensions by our present very incomplete are taken approximately into account by the use knowledge of the ionic configurations near a liquid of local effective exchange and correlation ener- surface. The same figure shows also the results gies [Eq.(2. 14)]. The interactions of electro of the uniform model and experimentally mea and ions are represented by pseudopotentials sured surface tensions. 31 It should be mentioned taken from theories of bulk metals. All electro- that particularly for Zn, there are still consider static energies, including an important classica able discrepancies among the data obtained by cleavage energy, are included different workers The results are compared with experimental We note that passing from the uniform to th data on surface tensions of eight liquid metals, ion lattice model leads to relatively small changes K, Rb CS, Mg, Zn, and Al, whose sur r the low-density alkali metals Cs, Rb and K, face electron densities vary by a factor of 20 and with an indication of slightly improved agreement whose surface energies range from 80 to 1000 with experiment. On the other hand for the der ergs/cm". For this entire set of metals, we find ser metals Na, Li, Mg, Zn, and Al, the differ- ood semiquantitative agreement, typical ences between the two models become progres- being about 25%. In the of the lower-density sively greater, and while the uniform model alkalis the agreement is especially close. For a lattice model ninth metal, Pb, the theoretical surface energy is follows the experimental trends quite well, a too high by a factor of about 2
THEORY OF METAL SURFACES: CHARGE DENSITY 4563 In spite of the unexplained failure for pb, we Taking proper account of these o(x"2)correc believe that we have come a substantial step tions was found to be important in the actual cal- closer to a quantitative theory of the electronic culation. The computations implied by Eqs (2.16)are carried into the metal to a point xmls, Effects which we have presumed to be small but at which p and the wave-function renormalization which need to be further examined include second constant are presumably chosen so that vett e order pseudopotential terms and contributions to xmin]=-2kp and a ()matches a pure sine wave the surface energy from changes of the zero point of unit amplitude. In order that these choices lattice vibrations 35 represent an adequate approximation to the More fundamental remaining questions concern the use of local exchange and correlation ener irst be separated out from the computed values gies; the use of pseudopotentials, designed for of vott [n; x] and th (x). Making I xmin I so large bulk metals, in the surface region; and the simu- that these terms become unimportant leads to lation of a liquid-metal surface by an appropriate face of a solid metal. It is hoped that additional good experimental data for both liquid and solid metals will become The condition that the friedel oscillations in available to allow a wider test of the theory and n(x)be self-consistent in the asymptotic region perhaps to suggest necessary modifications leads to the requirement that the s of Eq.(Al) Te would like to thank J. Rudnick for helpful 多=[1+krpk()/(2n2)]1 discussions. The assistance of the staff of the This result is obtained by substituting the form UCSD Computation Center is also very much given for y,x)in Eq. (A3)into Eq.(2. 16d).The appreciated parameter s is found to increase from 1.004 at APPENDIX A: FRIEDEL OSCILLATIONS rs=2 to 1. 07 at rs=6 (using Wigner's formula for ex to obtain u ) The fact that the oscilla General Forms: Importance in Numerical Calculations tions in the rs=2 curve of Fig. 2 are so much maller than those at r.=5 is clear Substitution of the sine-wave form of a, (x)into counted for by this small variation in 5.The Eq.(2. 16d)yields the well-known result reason for the difference is rather as follows From Eq. (3. 16)of Ref. 36, it is seen that qu (Al) tum density oscillations of the type analyzed here are reduced in general by inverse powers of the where t=l and yp =y r).Use of this form for integral n(x)in Eq.( 2. 16b)then gives k2 (which tends toward Ixl as x -oo). Here xo is the i +o (A2) tr[n; xo]=H=0). If vet is take exhibit simple exponential decay toward-tkF to with u()the derivative with respect to density the left of xo, with decay length A, then x=lx-xpI2 of the correlation part of uxe n). The oscillatory for large Ixl, with xp=21n2+xo erms in the electrostatic and exchange potentials In the present calculation, xo (equal to 1.4 at r it is found, cancel each other exactly in the 6 and 2.4 at rs=2)and the characteristic length asymptotic region. The solution ,x)of Eq a over which vett varies, increase slowly as rs de 16a)for the potential of Eq. (A2), in turn, also creases. This means that in terms of the wave exhibits a correction of o(x-2) length of the Friedel oscillations(/kp),xF,the effective origin of x, rapidly moves further and further to the right as r decreases. This implies, vR)- sin [kx-y(e) in turn, that the first oscillations to appear to the left of x=0 become smaller and smaller relative to i as n increases )sin[(+2kr)x-y()-2r] APPENDIX B: SELF-CONSISTENCY PROCEDURE sin[(-2kp)x-y()+2yE (A3) The set of equations(2. 16)in the order (c),(b) (a),(d)[with n replaced by n, in(a),(b),(c)and by
4564 D. LANG AND W. KOHN 2 in( d)] may be taken to define a functional F that properties of n(x)are already close to those of the transforms one electron density into another true self-consistent solution. The phase difference n2(x)=FIn; x] between the oscillations in n(x)and n(x)was in fact found never to be more than 5(this reflects an The self-consistent solution to Eqs. (2. 16)is then accurate choice of trial density), and the oscilla n(x)=FIn; x]. The trial density no(x)employed infind- tion amplitude for n(x)[corresponding to 5=1 in ing this solution consisted of an exponential decay- Appendix A, since these oscillations are identical metal, matched to a linear combination of two ex- sons to be quite close to the correct value i.e, ing toward n in a Thomas- Fermi length inside the to those in n,(x)]is constrained for theoretical potentials with adjustable decay lengths outside the self-consistent s is close to unity (see Appen- A Gaussian (with its parameters adjustable)was dix A) added to simulate roughly the first large peak of the expected density oscillations, and the neutrality APPENDIX C: EVALUATION OF THE TERM So IN THE SURFACE ENERGY condition S.no(x)-n (x)]dx=0 was imposed Straight iteration was found not to be a conver It is useful for the following discussion to intro gent solution procedure, and so an analog of the duce the abbreviation E(S, s)to refer to the inter Newton-Raphson method, based on the use of the action energy of S and S per unit area normal to inear response function &F/6n, was employed the x-direction, where the symbols S and S may An no(x)was chosen and n,(x)=Fino x] evaluated be replaced by"*"(denoting a rigid, uniform A neutralizing charge distribution was added to n1 positive or negative background)or"lat"(denoting in the surface region, in order that dvet[n1i x// the lattice). E(S, S)is a self-energy. The charge xIxs. vanish, since the fact that no obeys distributions S and S' will be taken to occupy the eutrality condition does not guarantee that n, will half-space x<0, unless the subscript "inf"is af- y it also. The function n2(x)=FIn1: x] was then fixed to e, to indicate that they fill all of space [(i. e, that they refer to the uncleaved crystal The trial density no was readjusted until n, and cf. Fig. 5]. The classical cleavage energy dal n2 were close to one another implying that eac was near the true solution. ni was then corrected by the addition of a linear combination of functions 60el=E(lat, lat)+E(, lat)+E(, Ehs1auui(r) with the ar determined by the self-cor MEint (lat, lat)+Eint(-, lat)+Eint( The electron distribution of step 1 in Fig. 5 is n4x)+2a11(x)=n2(x)+∑a considered to interact with the lattice via a pur Coulomb potential, rather than via a pseudo tential. This distinction affects the value of bocl (),(1)22222x occur in the present calculations projected onto a set of M orthogonal functions The ur were taken to be the derivatives(so as to e Simple considerations of electrostatics lead to preserve charge neutrality) of the first M har 60 =E(lat, lat)+2 E(, lat) monic oscillator functions, with width and center chosen so as to localize them in the surface re -EInt(lat, lat)+2Eint(-, lat).(C1) gion, and Eq .(B1)was projected onto the oscilla tor functions themselves. The integral So(oF/ In evaluating this expression it is convenient to 8n,)ur(x)dx'was found by computation of the ex let APua(v) represent the Coulomb potential at pression x(F[n1+Aui; x]-n2(x))with Au,(x)<<n lattice site v due to all the ions of the semi-infi- Evaluating n(x)=Fli; x](which is the function ac nite lattice, minus that due to the ions of the infi tually given in Table I)provided a direct check nite (uncleaved)lattice. Here the semi-infinite on the self-consistency of n. The procedure de lattice is taken to be in the left half-space(x<O scribed here could be used repeatedly, but it as is site v, which implies that Aiat(v)is simply proved unnecessary to do so(with M= 8) the negative of the Coulomb potential t v due to a It will, of be recognized that the as semi-infinite lattice in the right half-space. It ptotic phase and amplitude of the Friedel oscil- is convenient in addition to introduce the symbol lations in n(x)are not affected by the addition of P ( v) to represent the corresponding difference unctions localized in the surface region the u(x)]. of potentials due to semi-infinite(x<0)and infi This is not important, however because these nite tive backgrounds. This difference is then
