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PHYSICAL REVIEW B VOLUME 3, NUMBER 4 FEBRUARY 1971Theory of Metal Surfaces: Work Function~

D. LangIBM Watson I aboratory, Columbia University, Nese Fork, New cwork 10025and

W. KohnUniversity of California, San Diego, La Jolla, California 92037).and

Hebrew@ University, Je~salern, Israel(Received 16 October 1970)In a recent paper we presented a contribution to the theory of metal surfaces with emphasis

on the shape of the electron-density distribution and the surface energy. The present paperextends 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 pseudopotentialtheory, permitting the calculation of the variation of the work function from one crystal faceto another. For simple metals (Li, Na, K, Rb, Cs, Al, Pb, Zn, and Mg), agreement withavailable experimental data is good (5-10%); for the noble metals, the computed work func-tions are 150% too low.

I. INTRODUCTIONThe present paper represents a sequel to oneconcerned primarily with metal-surface chargedensities and surface energies. ' In the earlier

paper we presented, first of all, a theory of theelectronic structure of a model metal surface inwhich the lattice of positive ions was replaced by auniform background charge. Exchange and correla-tion effects were included using the self-consistentversion of the theory of the inhomogeneous electrongas. '~ Following this, the effect of the actual ionicstructure on the surface energy was taken into ac-count by calculating exactly a purely electrostaticcontribution (simila. r to a Madelung energy), andby evaluating the interaction of the electrons withthe ion cores using first-order pseudopotential the-ory.In the present paper, we shall follow a similarplan in developing a theory of the work function.This quantity, denoted by C', and defined preciselyin Sec. II in terms most useful for theoreticalanalysis, is equal to the minimum work that mustbe done to remove an electron from the metal at0 'K.We give first a rigorous demonstration thatwhere 4fI5 is the rise in mean electrostatic potentialacross the metal surface and p, is the bulk chemicalpotential of the electrons relative to the mean elec-trostatic potential in the metal interior. In spite ofits simple form, this expression includes all many-body effects, in particular, that of the image force.For the uniform-background model we evaluateAP from the electronic charge density n(r), com-

puted in LK-I, and we take p. from the availabletheory of exchange and correlation of a uniformelectron gas. This yields the work function Cofthis model as a function of the mean bulk densityn (or of the Wigner-Seitz radius r, ) 'These . resultsare compared with experiment and with the theo-retical calculations of Smith, who used a similarapproach but did not carry out a fully self-consis-tent calculation. 'Finally, we incorporate the effect of the actualion cores. We show first that when the differencebetween the pseudopotentials of the ion cores andthe electrostatic potential of the uniform chargebackground is treated as a small perturbation 5v(r),the change of the work function of a particular crys-tal face due to this perturbation is given to firstorder by the following rigorous expression:

M'= J 5v(r)n, (r) dr. (1.2)Here the integral is carried out over a slab whosesurface consists overwhelmingly of the face in ques-tion; and n, (r) is the change of the electron density,calculated in the uniform-background model, fol-lowing the removal of one electron from the system.This electron deficiency is of course localized nearthe metal surface. We have calculated the chargedensity n, by a method analogous to that of LK-I,except that we have now had to look for self-consis-tent solutions with a zero mean electric field deepinside the metal but a small finite field outside it.Since n, (r ) depends in fact only on the distancex from the nominal surface, it is possible to re-duce (1.2) to a one-dimensional quadrature. Inthis way we have calculated the total work function

1215

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1216 N. D. LANG AND W. KOHNfor the principal faces of nine simple metals Al,Pb, Zn, Mg, Li, Na, K, Rb, Cs and the noblemetals Cu, Au, and Ag. Experimental data aregenerally available only for polycrystalline samplesof unknown surface structure. This makes a de-tailed comparison between theory and experimentimpossible. Nonetheless, we can state the follow-ing conclusions.Simple metals: The measured work functionsrange over 2. 1-4.3 eV. With the possible exceptionof Li (where there is considerable uncertainty bothin the experimental data and in the pseudopotential),agreement between the full theory and experimentis typically within 5-10/0. The ionic lattice contri-butions 64, which are characteristically of the or-der of 10/o of the total work functions, contributeto establish this rather good agreement. Anisotro-pies among the different faces are typically also ofthe order of 10%%u() of the mean work function. In ac-cordance with the arguments of Smoluchowski, wefind the lowest work function to be associated withthe least densely packed face among those consid-ered [(110)for fcc, (111) for bcc].Noble metals: In view of the success with simplemetals, we have applied the same technique to thenoble metals to learn about the limits of validity ofour theory. Here the experimental work functionsrange over 4. 0-5. 2 eV, and the calculated valuesare 15-30%%uo too low. It may be assumed that thepresence of the filled d bands not far from the Fer-mi level makes our highly simplified theory, basedon the inhomogeneous-electron-gas model withsmall pseudopotential corrections, much less ap-propriate for these metals.In summary, the theory we have outlined appearsto describe well the work functions of simple met-als. Additional reliable experimental data for thisclass of metals would be highly desirable, partic-ularly data on the work functions of single-crystalfaces. In the case of the noble metals, on the otherhand, where, for metallurgical reasons, the ex-perimental data are much more consistent and re-liable, the present theory is less successful, andfurther theoretical work is needed. There is needalso for additional theoretical studies on the transi-tion metals, which are not discussed in this paper. '

DensityUniformSackground, n+ (x}Electrons, n (x)

(a)

Energy

$ (+(o )Fermi

y()() Level()()

FIG. 1. Schematic representation of (a) density dis-tributions at a metal surface and (b) various energiesrelevant to a study of the work function.

Z(n)= Jd(r)d(F)dF+ """," drdr'Ir-r'I

ative to the mean electrostatic potential there (seeFig. 1). It is important to know if this expressionincludes properly all many-body effects, in par-ticular, the work done against the image force inremoving an electron from the metal. This is infact the case. As we have not found any rigorousdemonstration of Eq. (2. 1) in the literature, wepresent one in this section.Since we are interested in removing one electronfrom the metal at O'K, we first develop a simpleextension of the theory of Hohenberg and Kohn (HK)to allow for a variable number of electrons, andthen use this theory in establishing the validity ofEq. (2.1).In HK, it was shown that for a fixed number ofelectrons N and arbitrary static external potentialv(r), there exists an energy expressione

II ~ RIGOROUS EXPRESSION FOR WORK FUNCTION+G[n], (2. 2)

Qualitative considerations, in the spirit of theSommerfeld electron theory of metals, stronglysuggest that the work function is given by the ex-pression(2. 1)

Here 4P is the change in electrostatic potentialacross the dipole layer created by the "spillingout" of electrons at the surface, and LU. is the chem-ical potential of the electrons in the bulk metal rel-

with the following properties: (a) G[n] is a univer-sal functional of n(r ), not explicitly dependent onv(r), given byG(d)-:td, (1' ~ 11)@)-f -, d jdr',(2. 3)

where the wave function )irefers to the (unique)electron ground state with density n(r ), and T andU are, respectively, the kinetic- and interaction-energy operators; (b) E[n] is equal to the correct

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THEORY OF METAL SURFACES: WORK FUNCTION 1217ground-state energy for a given v(r} when the cor-rect density n(r), corresponding to u, is used onthe right-hand side of the equation; (c) the firstvariations of E[n] about this density, consistentwith the restriction

5N = J-( r }d r = 0,vanish:

6E[n]= O.

(2. 4)

(2. 5)

5 Q[n]= 5 E[n]= 0. (2. 'I)Now let n(r) and n,, (r)be the correct elec-tron densities corresponding to a given v(r ) and,respectively, to p, and p, + 6p, . These two chemi-cal potentials describe two systems whose totalnumbers of particles differ by ~

&N= f [n,~ (r)n, (r)]dr J 2(F) dr.(2. 6)The corresponding first-order change of 0, ~ isgiven by

&aQ[n]=E[n,, ]E[n, ]p5N= 0, (2. 0)where the vanishing follows from the thermody-namic definition of the chemical potential at T= 0'K. Since an arbitrary small variation 5n is a(unique) sum of variations of the types 6n, and a,it follows that, in general,

5II[n]= O. (2. io)We now apply this theory to the work function.We consider a neutral slab of metal, all of whosedimensions are macroscopic but whose surfaceconsists overwhelmingly of two parallel faces thework function of which we wish to consider (thephysical properties of these two surfaces are takento be identical). Let n(r) be the correct electrondensity corresponding to the given nuclear poten-tial, the chemical potential p, , and a total numberof electrons N. By (2.6), (2. 2), and (2. 10) wehave, for a small variation in density 5n(r),

P(F)+ 5n(r) dr p6N=O,G[n]6n r (2. 11)where

Now consider an ensemble of macroscopic elec-tronic systems at the absolute zero of tempera-ture, specified by the external potentials v(F) andthe chemical potentials p.. The subsidiary condi-tion (2.4}will no longer be imposed on the densityvariations of interest. We define&[n]=E[n]p, J n( r ) d F. (2. 6)

Clearly, for a first variation of the density 5n, (F)which does satisfy the condition (2. 4),

(2. 12)is the total electrostatic potential. Now considerfirst a particle-conserving, but otherwise arbi-trary, variation 5n. Equation (2. 11) then givesP(F)+ [n]/(F) = p, ', (2. iS)where p, is some constant, independent of r.Next, consider a particle nonconserving variation;from (2. 11) and (2. 13)we then see that

p, = p, '= P(F)+ 6G[n]/5n(r) .The work function is, by definition,@= [4 (~) +Ex.i] E~ i

(2. i4)

(2. 15)where P(~) is the total electrostatic potential farfrom the slab considered above and E is theground-state energy of the slab with M electrons(but still with N units of positive charge). [BothP(~) and E~ depend on the choice of energy zero,but the combination (2. 15}does not. ] Using thedefinition of the chemical potential and Eq. (2. 14),this can also be written as4'=4(")- V=[4( )-4]9,where

e -=(~( )),u = u %=([n]/(F)) .

(2. 16)

(2. 17)Here () denotes an average over the metal. p. isthe bulk chemical potential relative to the meaninterior potential; its independence of this poten-tial may be verified from the definition (2. 3) ofG[n]. Equation (2. 16) is equivalent to the postu-lated Eq. (2. 1).All many-body effects are contained in the ex-change and correlation contributions to p, and intheir effect on the barrier potential &P. In par-ticular, the image-force effect on 4 may be re-garded as contained in the disappearance of partof the correlation energy when one electron ismoved away from the metal surface.

III. UNIFORM-POSITIVE-BACKGROUND MODEL

Here kz = (3& n)'~ is the bulk Fermi wave number

We consider in this section a model of a metalsurface in which the positive ions are replaced bya uniform positive charge background filling thehalf-space x& 0. ' The electron density in thismodel is shown schematically in Fig. 1(a).We consider first the quantity p, in Eq. (2. 1).Since deep in the metal interior the electron den-sity has a constant value n, P takes on the simpleformp, = ,k~+ p,(n) .

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1218 N. D. LANG AND W. KOHN

22. 533.54. 555. 56

2k@2(eV)12.528. 015.574. 093.132. 472. 001.661.39

~(eV)9, 61-7.906.755. 925.284. 784.384. 043.76

JLt(ev)2. 91O. 111.181.832, 152.312. 382. 382 Q 37

(eV)6. 803.832. 321.430, 910. 560. 350. 160. 04

@u(eV)3.893.723.503.263, 062. 872. 732, 542, 41

TABLE I. The work function C'of the uniform-back-ground model, and its bulk and surface-barrier compo-nents. The Wigner-Seitz radius x, characterizes the in-terior density. @=~/JM, where p = 2k&+p. The bar-rier term &P is given with a self-consistency of 0. 03 eVor better (this is a somewhat greater self-consistencythan that of the preliminary report, Ref. 4).

Other, more recently suggested, forms of the cor-relation energy give substantially similar results. 'The quantity P and its two components ,k2r andLL(,are shown in Table I for r, in the metallicrange 2-6. By Eq. (2. 1), p, can be regarded asthe bulk contribution to 4'. It will be seen fromTable I that, for metals of low electron density(K, Rb, and Cs with r, =4. 96, 5. 23, 5. 63,respectively), p. is much larger than the otherterm nP. The rather good agreement with ex-periment (see Table II) constitutes therefore agood confirmation of the expression (3. 3) for e,.This is especially meaningful since the theory ofthe correlation energy is most difficult at low elec-tron densities.We turn now to the double-layer contribution

By Poisson's equation,

and p,(n) is the exchange and correlation part ofthe chemical potential of an infinite uniform elec-tron gas of density n. p, is given by the relationp.,(n) =n~,(n)], (3. 2)

with E, the exchange and correlation energy perparticle of the uniform gas. In our computations,we have used the expression"e,(n) =(0.458/r, )0.44/(r, + 7. 8) (3.3)

f ~r', =1/n (3.4)from Wigner's classic analysis of the electron gas;here the Wigner-Seitz radius x, is given by

@.= (&4' P)..gf., ~ (3.6)It will be noted that while b,P is negligible formetals of low electron density, it becomes domi-nant for high-electron-density metals. It is alsostriking that while p, and ~P separately change by5. 3 and 6. 8 eV, respectively, over the metallic

np=p( )p( )=4m J x[n(x) n, (x)]dx,(3. 5)with n(x) and n, (x), respectively, the electron-and positive-background densities in the uniformmodel. n(x), calculated in LK-I with a, self-con-sistency of better than I%%uq of n, was recalculatedfor the present work to an accuracy of 0. 2%%ug orbetter. The resulting values of &P are listed inTable I, as is the total work function in the uni-form model

e~, (eV)(polyc rys talline)@' (eV)(loo)ae (ev)(loo)'e (111) (11O)110)

TABLE II. Theoretical and experimental work functions of nine simple metals. 4 is the work function for the uni-form-background model; (54 is the first-order pseudopotential correction: 4 = 4+64 (rounded to the nearest 0.05 eV).The pseudopotential core radii x~ are taken from the work of Ashcroft and Langreth (Befs. 1820). In the cases inwhich these authors give two possible values of x~ for a metal, the choice which yields agreement with experiment fora wider range of bulk properties is marked with an asterisk. Experimental values 4~t for polycrystalline sampleswere taken from Refs. 25-27 (see text for details of selection). [The most densely packed faces for the various struc-tures are: fcc (111), hcp (0001), bcc (110).]Metal Structure x~ @(eV)AlPbZnMgLlNaKRbCs

fccfcchcphcpbccbccbccbccbcc

2.072.302.302.653.283.994.965.23

3.873.803.803.663.373.062.742.632.49

1.121.121 47+1.271.391.06*2.00l.672, 142.6113+2.932.16+

0.21000.360.380.190.990.030.010.450.030.230.10

0.320.130.72for (0001)for (OOO1)0.050.950.290.340.530.260.610.21

0.19o.06"}.33faceface0.131.050.390.400.600.340.670.27

3.653.803.804. 154.053.552.403.102.752. 202. 652. 252. 60

4. 203.954.50for (OOOl)for (0001)3.302. 402.752.402.102.351.902.30

4.053.854.15faceface3.252.302. 652.352.052.30l. 802. 20

4, 194.014.333.662.32, 3 12.72. 392. 212. 14

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THEORY OF METAL SURFACES: WORK FUNCTION 1219

Ag 00 xxzn XMgPb -~

0Li 0Na5 0 xK 0Rb

Z0I-OxD2-LL - Uniform positivebackgr oun d mod el0 Experiment Cs

ion lattice model

density range, the total work function, given bytheir difference, varies only from 2.4 to 3.9eV."The values of 4given in Table I are similar tothose obtained by Smith, ' although Smith did notinclude in his calculations the Friedel density os-cillations near the surface. The reason for thisis the following: The contribution p, to 4 isidentical in both calculations. For high electrondensities, hQ is substantial, but since the Friedeloscillations at these densities are rather small, 'Smith's calculations give similar results to ours.For low electron densities, the Friedel oscilla-tions are important and our hP is substantiallysmaller than that of Smith (by about a factor of 2for Cs), but at these densities AQ is negligiblecompared with p, .Figure 2 compares the computed values of 4with recent experimental data on the work func-tions of polycrystalline simple metals. There israther good agreement between theory and experi-ment for these metals, to which the uniform-back-ground model would be expected to be best appli-cable. A more realistic treatment of the positiveions, presented in Sec. IV, improves this agree-

ment somewhat and also permits calculation, fora given metal, of the anisotropy of 4, i.e. , thevariation of 4 from one crystal face to another.IV. ION-LATTICE MODEL

A. Theory

54 = J 5v(r)n, (r) dr,which has already been explained in Sec. I [Eq.(1.2)], and which avoids the solution of three-dimensional wave equations. 'To derive (4. 1) we begin with the definition(2. 15) of the work function

(4. 1)

4 = [p(~) +Ex.t]EN EN Ex, (4. 2)where the system in question is a large metal slabwhose surface consists almost entirely of twoparallel faces of a given orientation relative to thecrystal axes. E is the ground-state energy ofthe neutral slab, containing N electrons; and gis the energy of an excited state of the N-electronsystem in which (N-1) electrons reside in the low-est possible state in the metal, while one electronis at rest at infinity. The first-order change of4 due to 6v is then, by standard perturbation the-ory,

When we pass from the idealized uniform-back-ground model to a more realistic model in whichthe effect of each metal ion on the conduction elec-trons is represented by a pseudopotential, astraightforward attempt to calculate C from Eq.(2. 1) would involve the prohibitively difficult taskof solving self-consistently a system of equationswhich no longer separate, but which are trulythree-dimensional. To avoid this problem we shalluse the fact that the replacement of the uniformbackground by the ion pseudopotentials representsa small perturbation 5v(r). To first order in 5v,the change of the work function 64 will be shownto be given by the expression

02 I4&s~FIG. 2. Comparison of theoretical values of the workfunction with the results of experiments on polycrystallinesamples (open circles). (The reason for the presence of

two experimental points for Li is discussed in the text. )@, the work function in the uniform-background model,is shown as a dashed curve. The C' values in the ion-lattice model were computed for the (110), (100), and(111) faces of the cubic metals and the (0001) face ofthe hcp metals (Zn and Mg). For qualitative purposes,the simple arithmetic average of these values for eachmetal is indicated by a cross (two crosses are shownfor the cases in which there were two possible pseudo-potential radii). The experimental and theoretical pointsfor Zn should be at x,=2.30; they have been shiftedslightly on the graph to avoid confusion with the data forPb.

54 = J 5v(i)n'(r) dr J 5v(r )n(i)dr, (4 3)where nand n~ are the electron densities in theuniform model, associated with E' and E, re-spectively. Now since 5v(~) = 0, the "escaped"electron does not contribute to the first integraland we may rewrite (4. 3) as

54= J 5v(i)n, (r) dr,where

(4. 4)

n, (r ) =n,(i)n(r ), (4. 5)

Jn. (r) dr= 1. (4. 6)with n, the density distribution of the (N 1)elec--trons in their ground state. Clearly the densitydeficiency n, satisfies the normalization

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1220 N. D. LANG AND W. KOHNNow in the (N -1)-electron ground state, associ-ated with Ethe electric field 8 inside the metalis zero, and outside each of the two parallel facesis given by 8 =4fT/2A (directed along the outwardnormal), with 2A the total surface area. Let ustake the x axis to have its origin in one of the faces

and point outwards [see Fig. 1(a)]. Then n, is afunction of x only. Further, let us denote the y-zaverage of 6v(r) by

I.O

0.5

EPOS IT IVEBAC KGROUNDIC

ba.0 I&v(x) =- J6Ef(x,y, z) d~f dz/Jdy dz.

Then, using (4. 6), Eq. (4. 4) reduces toM =AJ 6v(x)n, (x) dx

(4. 7) 0 I ~ ~ I~=8 -IO -5 ~ 0DiSTANCE (atomic units)Bulk density corresponds to r, =2(Surface charge densityiS I.oox IO s atOmiC unitS)

(a)=J bv(x)n, (x) dx/f n, (x) dx=J,

"&v(x)n, (x) dx/f n, (x) dx, (4. 8) I.O- ~POS I TIVEn(x) ~, BACKGROUNDwhere a is a length large compared with a screen-ing length but small compared with the thickness ofthe slab. In the last step both numerator and de-nominator have been reduced by a factor of 2. Since(4. 8) is homogeneous in nn, ( )x can now be takento be the surface-charge density in the semi-in-finite uniform-background model induced by an ar-bitrary weak electric field 8 perpendicular to thesurface.

0.5

-nQ-15 ~ID -5 0 .5DI sTANcE (atomic units)

Bulk density corresponds to r, = &(Surface charge densityis -s.oox IO-4 atomic units)

Iab.02

0, r&r,Z/0, r&r, ' (4. 9)These authors have determined the radius x, foreach metal to give a good description of bulk prop-erties.C. Results and Comparison with Experiment Simple MetalsOur calculated results for 4', 64, and 4'=++ ~4are given in Table II for nine simple metals. For

B. CalculationsTo evaluate (4. 8) we require n, ( )xand off(x). fs, (x)was determined as the difference between two self-consistent densities: one corresponding to a neu-tral metal with vanishing field outside, and theother corresponding to a metal with a small surfacecharge and finite field 8 outside. The method usedwas the same as that described in LK-I, and n, (x)was evaluated for x, between 2 and 6." The func-tions n, (x) for 0; = 2, 4, and 6 are shown in Figs.8(a), (b), and (c), respectively. They are ratherwide density distributions, which outside the metaldrop to half their maximum value at approximately2-3 a. u. (or 1.0-1.6 A) from the nominal surface. "The perturbation potential 6v(x), Eq. (4. 7), wasobtained, as explained in LK-I, Appendix D, '7 usingpseudopotentials of the form employed by Ashcroftand Langreth

IC

I.O

0.5

n(x) ,l

~POSITI VEBACKGROUNDIC

a0.05I I-IO -5 0 5,DISTANCE (Uf0mlC UIlffS)

Bulk density corresponds to r, =6(Surface charge densityis -Z.OO~ IO " atomic units)'=fS

(c)FIG. 3. The change n, (x) in the electron-density dis-tribution n(x) of the uniform-background model, induced

by a weak external electric field along the outward nor-mal to the surface (i.e. , along +x). Curves are shownfor (a) r,=2, (b) r, =4, and (c) r, =6. n is the meanelectron density deep in the metal.the cubic crystals the calculations were done forthe (111), (100), and (110) crystal faces, ' and forhcp crystals, for the (0001) face. In the cases forwhich Ashcroft and Langreth give two possible

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THEORY OF METAL SURFACES: WORK FUNCTION 1221choices for the pseudopotential radius xwe havecarried out the computation for both. In these in-stances, the r, value marked with an asterisk inthe table is generally found to yield agreement withexperiment for a wider range of bulk propertiesthan the other. '9'~0 There exist at present practical-ly no experimental data on the work functions ofsingle-crystal faces of the metals listed. Of thosefew recent data, which we could find, ~ almost nonewere determined by more than one group of workersand some appear to disagree with accepted resultsfor polycrystals. We therefore felt that comparisonof our results with single-crystal measurementswas premature. We remark however that the gen-eral trend of a decrease of the calculated 4 withdecreasing density of ions in the lattice plane2' isin keeping with the arguments of Smoluchowski andwith the extensive body of single-crystal work onmetals such as W, Mo, and Ni. ~4 For the polycrys-talline data we have, so far as possible, used recentresults which agreed substantially with earlier mea-surements. With the exception of Li, we have usedonly experimental data obtained by the photoemissionor Kelvin contact-potential-difference methods. 'These methods are in wide use, they yield the workfunction in a direct way, and they avoid high tem-peratures. For Li, however, the most frequentlyquoted result was obtained over 20 years ago, by atype of contact-potential-difference method, 26 anddisagrees seriously with that of a very recent field-emission experiment by Ovchinnikov and Tsarev. ~7In this case we have included both experimental re-sults. The reader is referred to the article byRiviere for a general review of the experimentalsituation.The measured work function of a polycrystallinesample represents a certain average over variouscrystal faces, which depends on the technique used,the conditions of the measurement (e. g. , tempera-ture), and the relative proportion of each face onthe surface. ~~ For qualitative orientation the simplearithmetic average of the 4 values computed forthe several faces of each metal is shown in Fig. 2as a cross (there are two crosses for the cases inwhich there were two r, values).However, it should be remarked that for higher-density metals, theory predicts that the moreclosely packed faces will have substantially lower

surface energies than the others, ' and thereforeshould be preferentially present. These surfaceshave the highest work functions. Thus, in the casesof Al and Pb, for which the work-function measure-ments were made on evaporated films, the appro-priate theoretical estimates are probably somewhathigher than the crosses. For Zn and Mg (whoseexperimental work functions were also obtainedusing evaporated films) the crosses already cor-respond to the most closely packed faces, whichwere the only ones calculated.It should also be mentioned that photoemissionmeasurements weight the low-@ faces most strong-ly. This effect is probably of greatest importancefor the lower-density alkali metals whose quotedwork functions were obtained by this method andfor which the surface-energy anisotropy is small.Hence, for these metals, the appropriate theoreti-cal estimates are probably somewhat lower thanthe crosses.In view of the uncertainties concerning the cor-rect crystal-face average for polycrystalline ma-terials, it is not possible to state just how much,if at all, the ion-lattice corrections 64 to the uni-form model improve the agreement with experi-ment. However, we note (see Fig. 2) that for thealkalis, for which 4&4, the average 6& isnegative, as it should be, and of the right order ofmagnitude, while for Al, Pb, and Zn, 4&+,~and the average 64 is properly positive and againof the right order. (For Mg, Cagrees exactly with4, and the addition of 54 produces a 10% error. )Thus it appears that, all in all, the inclusion of theion potential in the theory improves the agreementwith experiment to within 5-10/0. '0D. Results and Comparison with Experiment - Noble MetalsOur calculated results for the noble metals Cu,Ag, and Au are given in Table III. Again we haveused local pseudopotentials of the simple form(4. 9), with values of r, taken from Ref. 19." Thecalculated work functions are seen to be 15-30%too low compared with experiment, the absolutediscrepancies being in the vicinity of 1 eV. Forthese metals the experimental data are consistent

and appear to be reliable, so that this error is al-most certainly due to the theory. We do not knowat present how this error is apportioned between

TABLE III. Theoretical and experimental work functions of the three noble metals. See Table II for explanation ofsymbols. Experimental values for polycrystalline samples were taken from Ref. 32,

CuAuAg

fccfccfcc2.673.013.02

3.653.493.490.81 0.080.81 01.04 0, 15

0.140.180.070.260.300.19

Metal Structure x~ 4'(eV) ee (ev)(100) (111) 4 (ev)(0) (100)3.55 3.803.50 3.653.35 3.55

3.903.803.70

e, (ev)(polycrystalline)4.655.224.0

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1222 N. D. LANG AND W. KOHNp. and the barrier potential dP. (Significant out-ward movement of the last layer of atoms wouldreduce 4Q and thus further increase the error. ")It is not clear whether more sophisticated pseudo-potential theory would remove most of the discrep-ancy or whether these metals, with hybridized s-dbands, require a different approach.

ACKNOWLEDGMENTS

One of us (W. K. ) wishes to thank the HebrewUniversity in Jerusalem for their hospitality duringa sabbatical leave. We would also like to expressour appreciation to S. Berger for assistance witha part of the numerical work.

*Supported in part by the National Science Foundationand the Office of Naval Research.~Present address: IBM Thomas J. Watson ResearchCenter, Yorktown Heights, New York 10598.~Permanent address.N. D. Lang and W. Kohn, Phys. Rev. B ~1 4555(1970); this paper will be denoted as LK-I.P. Hohenberg and W. Kohn, Phys. Rev. 136, B864(1964).3W. Kohn and L. J. Sham, Phys. Rev. 140, A1133(1965).4A preliminary account of this part of the calculationhas been given in N. D. Lang, Solid State Commun. ~71047 (1969).J, R. Smith, Phys. Rev. 181, 522 (1969).R. Smoluchowski, Phys. Rev. ~60 661 (1941).An example of a semiempirical study of the work func-tion using an approach very different from that taken inthis paper is given by D. Steiner and E. P. Gyftopoulos,in Proceedings of the Twenty-Seventh Annual Conference

on Physical Electronics, Massachusetts Institute ofTechnology, Cambridge, Mass. , 1967 (unpublished).The extension of HK to the grand canonical ensembleat arbitrary temperatures has already been given byN. D. Mermin, Phys. Rev. 137, A1441 (1965). [On theright-hand sides of Eq. (9) and the second unnumberedequation of this paper, the term p f n(r)dr must beadded. ] We present de novo the specially simple case ofzero temperature.

We use atomic units, with l e I =m =k=1. In thissystem the unit of energy is 27. 2 eV.The first wave-mechanical computation of the workfunction using this model was given by J. Bardeen, Phys.Rev. 49, 653 (1936),.~ For this corrected form, see Eq. (8.58) of D. Pines,

Elementary Excitations ~n Solids (Benjamin, New York,1963).~ See, e.g. , K. S. Singwi, A. Sjolander, M. P. Tosi, 'and R. H. Land, Phys. Rev. B ~1 1044 (1970).30ne can easily show that in the high-density limitthe leading terms of p, and ~Q are both 2k&, they divergeindividually but cancel each other in (3.6). We conjec-ture that in the uniform model the work function tends to

a finite limit (in the vicinity of 4 eV) as the interiorelectron density becomes infinite.' In connection with the material of this section wenote that the effect of an array of planar 6-function po-tentials on the double-layer moment has been consideredby A. J. Bennett and C. B. Duke, Phys. Rev. 188,1060 (1969).'5The values of g were chosen large enough that n~(x)was well above the "noise level" associated with lack ofself-consistency, and small enough to be in the linear-response region. These conditions were checked by re-peating the calculations with the values of g doubled.The entire theory of this section was also checked by

using an artificial small tv(x) localized near the sur-face, and doing a direct self-consistent calculation of ~Pand hence of @=4/ p for this case. The result for i54agreed within a few percent with the perturbation result(4. 8).The surface-charge distributions n, (x) also play animportant role in theories of the image force and ofsmall-gap condensers, which will be discussed in a forth-coming paper.The case x~ &2d (d is the spacing of the lattice planes),which did not occur in LK-I, does occur in some of thepresent calculations.N. W. Ashcroft and D. C. Langreth, Phys. Rev. 155,682 (1967).'~N. W. Ashcroft and D. C. Langreth, Phys. Rev. 159,500 (1967).

N. W. Ashcroft, J. Phys. Cl 232 (1968).~Of these three faces, the (111) face is the mostclosely packed and the (110) face the least closely packedfor the fcc structure; for the bcc structure the roles arereversed.Al [(111), (100)]: B. Sieroczynska-Wojas, Fiz. Tverd.Tela ~10 693 (1968) [Soviet Phys. Solid State 10, 544(1968)]; Zn (0001): L. P. Mosteller, T. Huen, and F.Wooten, Phys. Rev. 184, 364 (1969);K[(110, (100)]:J. C. Richard and P. Saget, J. Phys. Suppl. 31 Cl155 (1970). Professor G. -A. Boutry (private communi-

cation) has described current work on the (110) faces ofK, Rb, and Cs. Single-crystal data on the noble metalsCu, Au, Ag [(111), (100)] is given by P. Vernier, E.Coquet, and E. Boursey, Czech. J. Phys. B19, 918(1969); see also, for Cu, P. Kohler, Z. Angew. Phys.191 (1966).Note, however, the exceptions to this trend for the(100) faces of Al and Pb.For example, W, Mo: O. D. Protopopov, E. V, Mik-heeva, B. N. Sheinberg, and G. N. Shuppe, Fiz. Tverd.Tela 8, 1140 (1966) [Soviet Phys. Solid State 8, 909(1966)];Ni: A.. Kashetov and N. A. Gorbatyi, Fiz. Tverd.Tela ~10 2135 (1968) [Soviet Phys. Solid State 10, 1673(1969)].Al: J. C. Riviere, Proc. Phys. Soc. (London) B70,676 (1957) (as corrected in Ref. 28); Pb: F. Baumann,

Z. Physik 158, 607 (1960) (as quoted in Ref. 28); Zn:R. Suhrmann and G. Wedler, Z. Angew. Phys. ~14 70(1962); Mg: R. Garron, C. R. Acad. Sci 258, 1458(1964); Na: R. L. Gerlach and T. N. Rhodin, SurfaceSci. ~19 403 (1970), and private communication (valueused is average of high-coverage-limit work functionsfor Na deposited on three different faces of No; K, Rb,G. -A. Bountry and H. Dormount, Philips Tech. Rev.an, 225 (1969).26P. A. Anderson, Phys. Rev. 75 1205 (1949) (ascorrected in Ref. 28).A. P. Ovchinnikov and B. M. Tsarev, Fiz. Tverd.Tela 9, 3512 (1967) [Soviet Phys. Solid State ~9 2766

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THEORY OF METAL SURFACES' WORK FUNCTION(1968)]. Value used is average of high-coverage-limitwork functions for Li deposited on various faces of %and Be.8J. C. Biviere, in Solid State SN/ace Science, editedby M. Green (Dekker, New York, 1969), Vol. 1.

~C. Herring and M. H. Nichols, Bev. Mod. Phys.21, 185 0.949).It might be mentioned that T. Schneider tPhys. StatusSolidi ~32 323 (1969)] has shown second-order effects in6v to reduce P, the bulk contribution to the work func-tion, by 0.1-0.2 eV for the alkalis andby 0.3 and 0. 8 eVfor Al and Mg, respectively. The second-order effectof 6u on the surface term &~tj must be determined, how-ever, before any conclusions can be drawn concerning

the net importance of (6v) ~ terms in 4'.BlBeference 19 gives also a second, much larger, valueof x~ for each of these metals, which leads to somewhatless satisfactory results for bulk properties. The workfunctions calculated with these r~'s are about 1 eV lowerthan those corresponding to the preferred set, giving aneven greater discrepancy with experiment.

3 Cu and Ag: D. E. Eastman, Phys. Bev. B ~2 1(1970); Au: E. E. Huber, Appl. Phys. Letters 8, 169(1966}.There is some theoretical evidence suggesting theexistence of occupied surface states in the noMe metals;this could of course also affect 4P, See F. Forstmannand J. B. Pendry, Z. Physik 235, 75 (1970).

PHYSICAL BEVIE% B VOLUME 3, NUMB EB 4 15 FEBBUABY 1971Some Formal Aspects of a Dynamical Theory of Diffusion*

Michael D. FeitDepartment of Physics and Materials Reseurek Laban atty,University of Illinois, Uxbana, BBnois 61801(Heceived 2 October 1970)A classical dynamical theory of diffusion is presented in which the reaction coordinate and itscritical value are expressed in terms of a 3N-dimensional vector B. The Slater approximationto the Kac equation is shown to be exact for classical statistics, and the jump rate is calcu-lated accordingly. The jump rate ean be expressed in terms of the vector 8, and the dynamicmatrix; this leads to a frequency factor different from that obtained from the reaction-ratetheory, and an indication that the system does not jump through the relaxed saddle-point con-figuration. The self-diffusion isotope effect is also considered. It is shown explicitly that themigration energy is mass independent, and that the effect may be expressed simply in termsof the reaction coordinate. In terms of the phonons, the isotope effect depends on a weighted.average of the fraction of the energy carried by the jumping atom in each mode. Quantumcorrections are discussed and a connection is made between the isotope effect and thermomi-gration.

I. INTRODUCTIONThere has been increasing interest in recentyears in formulating the theory of diffusion jumprates in a way that avoids some of the conceptualdifficulties of the absolute-rate theory. ' The ratetheory was applied to diffusion by Wert and Zener,and elegantly formulated by Vineyard to show ex-plicitly how the motions of many atoms are involvedln the ]ump process. This can be seen ln Vine-yard's form for the preexponential or frequencyfactor

in which v, and v, are the normal-mode frequenciesof the equilibrium and saddle-point configurations,respectively, with the unstable "mode" of the sad-dle point left out of the product in the denominator.The many-body nature is also apparent in the iden-tification of the migration energy with the potential-energy difference of the relaxed saddle-point and

equilibrium configurations.The troubling aspect of rate theory is that itfocuses so strongly on properties of the relaxedsaddle-point configuration. In a quantum-mechani-cal theory it would not only be impossible to treatpositions and velocities independently, but, moresignificantly, it would be completely inappropriateto speak of the properties of the intermediate state,as has been pointed out by Flynn and Stoneham.Attempts to circumvent these difficulties havebeen made by several authors. ' These "dynami-cal" theories view the jump process as resultingfrom a special kind of fluctuation from equilibrium,and attempt to calculate the frequency of such fluc-tuations. In the earlier work, fluctuations wereconsidered that carried the system to a definiteconfiguration, e.g. , the relaxed saddle-point con-figuration. Glyde~ has shown that, for this case,the dynamical and rate theories have the sameformal content. More recently, Flynn' has con-sidered fluctuations in a reaction coordinate, madeup of a linear combination of particle displacements,