PHYSICAL REVIEW 8 VOLUME 43, NUMBER 6 15 FEBRUARY 1991-II

Crystallinity effects on the surface optical response in metals: A preliminary calculation

J. T. Lee and W. L. SchaichDepartment ofPhysics and Institute for IrIaterials Research, Indiana University, Bloomington, Indiana 47405

(Received 28 September 1990)

The beginning of a practical evaluation scheme for dynamic screening at the surface of a crystal-line metal is developed. The derivation shows how band-structure effects on the d parameters maybe calculated. Simplified versions of the basic equations are evaluated for a model of the (110) faceof Li. Several features, absent from jellium models, are shown to make significant contributions.The theory is still too approximate for direct comparisons with experiment, but indicates a feasiblepath towards that goal ~

I. INTRODUCTION

For some time there has been a strong interest in and aconsiderable effort expended on the surface opticalresponse of metals. Even a list of review articles on thissubject is rather long. ' " However, at least for metals atthe microscopic level of interest here, theoretical workhas almost exclusively been devoted to jellium models.Such calculations can now be done with considerable so-phistication and relative ease, but they all omit from theoutset any microscopic allowance for crystallinity effects.This. kmitation has recently become a more pressing con-cern due to the surge in experimental work' ' that canbe directly compared with theoretical predictions. Cer-tainly a jellium model of Ag can only hope to be qualita-tive, and even for the alkali metals or Al it is not obviousthat subtle differences in the treatment of many-bodyeffects have more quantitative importance than the in-clusion of lattice scattering.

In this paper we begin the development of a theorythat will allow tractable estimates of the influence of bandstructure on surface optical response. Our specific em-phasis here will be on understanding the features that canappear far (on the scale of screening lengths) from thesurface. In Figs. 1 and 2 we illustrate some of thechanges that occur in switching from a jellium to a crys-talline substrate. The quantity plotted is the (speciallyscaled) component of the electric field normal to the sur-face as a function of depth into the sample. Such curvessummarize the mean-field linear response of the system toa long-wavelength external perturbation at a fixed fre-quency. In Fig. 1 the obvious new feature is the appear-ance of periodic oscillations tied to the lattice constant.These accompany any field that extends into the bulk andare the "local-field" terms that one suppresses in macro-scopic electrodynamics. '

In a different range of driving frequency additional os-cillations with periods larger than the lattice constant canappear in the asymptotic behavior. An example is shownin Fig. 2 where the frequency lies within the band ofzone-boundary collective states. The existence of thesemodes depends on the presence of gaps in the electronicenergy spectrum, hence on the deviations of the systemfrom a jellium model. Their influence on electron-

energy-loss spectra has been calculated before andconfirmed by experiment. Our theory shows how theyappear in the surface optical response.

In the rest of the paper we describe how these and fur-ther calculations can be done. Section II contains thebasic derivation and introduces the several approxima-tions necessary (at this stage) to obtain tractable formu-las. Then in Sec. III we describe the model calculationsthat we have done, illustrating various i&eoretical in-gredients and consequences. Although. our present nu-merical results only have a qualitative validity, this papershould be a useful theoretical guide to more sophisticatedevaluations.

II. FORMALISM

In an abstract sense, the problem of the surface opticalresponse of a crystal has already been solved, in thatseveral general (and essentially equivalent) schemes have

1.0

5 10 15x/a

20

FICr. 1. For a model of the (110) face of crystalline Li sub-jected to a perturbation at a frequency of

~the free-electron

Fermi energy cF, we plot the function q from Eq. (32) vs dis-tance into the substrate, with a the interplanar spacing alongthe surface normal. The solid curves give the real and imagi-nary parts of q, while the dotted curves are the correspondingjellium results.

43 1991 The American Physical Society

4630 J. T. LEE AND W. L. SCHAICH 43

1.5

1.0

0.5

0.0

0 5 I I I

10 15x/a

V ( t) I —K '(K x— t)e

where the three- i

(1)

e-dimensional vector x hainto a component x 1x a ong the normal to

r x as been separated

1

1 Th'

l l'nsiona projection X iria ies to the ri h

two-dimensionalna wave vector K (of mright of x =0 and th e

11 b tt . '0

frequency co, we stoe o zero. Since all quantities will be at

we stop writing factors of ee ependence on co implicit.

o e and leave

The densit r 'un

'y response is found

dependent Hartrun within the tim-

ar ree approximation orime-

(R )]by writingp-

induced at first order by an externalthe form

r er y an external potential energy of

FIG. 2. Same asb

' '0

arne as Fig. 1 exce t thenow .9cF. The two dorn

e externalP

the zonse o a and 5a/4 and hot

o ective state.e wit

5p(x) = d x'x Xo(x, x')[V,„,(x')+ V;„d x'1nd

where te t e induced potential energy is

V;„d(x)= f d x' 5 x'') .

(2)

been derived H owever, there haese ormal theori

suits. Most of th e numerical work hasries into practical r-re-

conductors or insul tu a ors (see Refs. 29—r as focused on semi-

ol' dli h I

a y t e anal sis dn metals it is this case o

normal electric fieldprimary interest. So

e, that has been ofo in the lan ua e

eory, it is d rathe her t an d that weg ge of d-parameter

study. This limitedwe wish (initially) to

only needs to co 'dmi e scope has the aconsi er (scalar) densit

advantage that oone

t . Ith t 1

1 io h olOur

1

(11 f 11 d SC) icod fK dShc ale h.

A. Basic integral equation

We begin withg 'th a generalization of th pone seeks the electron dens't 5i y p x, t)

Using this roproperty of y, plus the tof 11o e Coulomb otentp ial,

—Q~x~ ig x2'A q Q

(5)

where 3 is the total sFo

o a surface area weurier transform of the X d

e obtain for thee dependence of 6p

In (2), is the independent-particle-p p

d t t 1 t'o 1'

e constants into theiscrete translatio 1'ona symmetr in x.

e material, also ay p o

is escribed b theth e projections of alon

an

g ogog, can e independentl . o a

mmetry in X implies that theproduces a response at all

e perturbationgiven by

(4)

5p(x, Q) = dx™x ~~XO(x, x', Q, K+G') I" e +G'IIx' —x"I

G'

—IK+ G'IIx' —x"

ie + i" " 5p(x", K+G') (6)

where

Xo(xx'Q Q')= " ' '

X x x' e'x ', ,'= fdX dX'e '~'

(

S( )=q = dx cosqxS(x),

I 2T(q~q )=— dx cos x '

qx cosqx dx'cosq' 'Tqx xx') . (9)

In a jelhum model thd f o of 5(

e, e integrals in E'), which in t

' in q. ) and replace Q and K+G'' with

Our method of solving Eq. (6 is based11

mso t exde en

'e

definitionsp n ence according to the

The transform of t e Coulomb po ential energyis

Vc q q' Q)=U(q, Q)5(q —q')—)'(Q)U(q, Q)U(q', Q),(10)

where

43 CRYSTALLINITY EFFECTS ON THE SURFACE OPTICAL. . . 4631

24meu qQ=

The physical interpretation of A, (Q) is that it representsa refiection amplitude in the sense that for x (0 (in vacu-um)

and

y(Q)=4~e

(12)

A(Q)l, e~ =f dx'V, (x,x';Q)5p(x', Q)

e~ f dx'e ~" 5p(x', Q) .0

(18)

V, (q, Q)=y, (Q)u(q Q» (13)

while the transform of the external potential energyV, (x)=l",e ~" is Changing to a q integration and using Eqs. (11)—(14) in

Eq. (18) gives Eq. (17). Indeed we can write the potentialenergy in vacuum as

where

r,y, (Q)=

4~e(14)

eiK x ' —Kx+yg(K+G)e K+G~x iG x '

G

(19)

Applying these results to Eq. (6) yields

5p(q, Q)= f dq'gyp(q, q', Q, K+G')0 G'

X u(q', K+G')5p(q', K+G')

+g 5pp(q Q K+G )[5G p A(K+G )]

where

5pp(q Q Q') = f, dq'Xp(q q';Q Q')u(q' Q')y, (Q')

(16)

so the normal component of the displacement (or electric)field is for x (0,

eD (x)=l, g e' +

G

X [5 e—&x g(K+ G)e IK+GIx]

Comparing with Eq. (15) we see that a weighted sum ofthe D„(x =O, K+G') acts as the driving term for 5p.This leads us to write

5p(q, Q)=g[~(q;Q, K+G ) —5

and XD (x =O, K+6')/47re, (21)

A, (Q)= f dq u(q, Q)5p(q, Q) .y, (Q)

(17) which when substituted in Eq. (15) gives an equation for3 in which 5p0 does not appear:

QA(q;Q, K+G')D (x =O, K+G')G'

=D„(x =O, Q)+ fdq'gyp(q, q';Q, K+G')u (q', K+G')g A (q', K+G', K+G")D (x =O, K+G") .G'

This may be further simplified by defining

v(q, Q)= g A (q;Q, K+G')D„(x =O, K+G')/D (x =O, K),

Gl I

(23)

so (22) becomes

v(q, Q)=D (x =O, Q)/D„(x =O, K)+g f dq'yp(q, q';Q, K+G')u(q', K+G') (q', K+G') .0

(24)

D (x =O, K+G)/D„(x =O, K)= —2y(K+G) f dq u(q, K+G)v(q, K+G),

0(26)

Working back through the algebra we have from (21)

4me5p(q, Q)=v(q, Q).D (x =O, K) D(x =O, Q) . —(25)

Combining Eqs. (17), (20), and (25) allows one to formallysolve for D„(x =O, Q) or A, (Q) in terms of v. These re-sults can be used to reexpress the "driving" term in (24)for G&0 as

which can be viewed as a modification of the effective g0..

Xp Ip 25G G (1 5G,p)y(K+G)

One next needs to find a reasonable yp and to solve (24)for v(q, Q). To ease the computational effort we willmake severe approximations at both of these steps. Sinceour emphasis is on d~, we drop all crystallinity effects inX; i.e. the only deviation from the jellium model in thebulk is a periodic variation in x. Momentum parallel tothe surface is then conserved through the screening pro-

4632 J. T. LEE AND W. L. SCHAICH

cess, so (24) simplifies to

v(q)=1+ f dq'yo(q, q')U(q')v(q'),0

(27)

In the present notation this appears as

(1—1/ei )di = f dx [E,(x) —E,(x)]/D, (0),where, as done earlier for the frequency ~, we havesuppressed all reference to the common parallel wavevector K—+0. Equation (27) is identical in form to thejellium result, cf. Eq. (24) in Ref. 32. The differentanswers for the two models depend on differences in yo.

We remark that a much earlier paper on surface opti-cal response also approximated the surface of a three-dimensional crystal as a system with only one direction ofdiscrete translational symmetry. However, they wereonly concerned with the surface-plasmon dispersion andtheir direction of periodicity was parallel, rather thanperpendicular, to the surface plane.

B. Calculable quantities

To further emphasize the formal similarity with equa-tions of the jellium model, we show how the function vdetermines all the quantities of interest. Equation (25)now appears as

4rre op( q) = [v(q) —1]D„(0), (28)

where D (0)=D (x =O, K—+0). Then inverting thetransform,

QO

4vreop(x) =D (0)—f dq cosqx [v(q) —1],7T 0

(29)

which describes the distribution of screening charge den-sity inside the metal. The left-hand side of (29) also ap-pears in Poisson's equation

4irefip(x)=V E . (30)

Since the perturbation is long wavelength (and since wehave neglected transverse umklapp processes) we can re-place V.E=BE„/B„and use (29) and (30) to find

E, (x)=E„(0)+4vre f dx '5p(x ')

=D, (0)+D (0)—f dq [v(q) —1]VT 0 g

=D (0)—f dq v(q) .o q

(31)

To within normalization terms this result describes allthe curves shown in Figs. I and 2. Indeed the g plottedthere is given by

2 ~ sinqx v(q) —1rI(x) =— dq

o q v(0) —1(32)

which vanishes at x =0 and whose value, averaged over aunit cell, tends to one deep in the bulk. It is an artifact ofthe semiclassical infinite barrier model (SCIB) approxi-mation introduced below that g has a nonzero derviativeat x =0+ (see Figs. 1, 2, and 8), which implies an unphys-ical discontinuity in 5p there. The spatial distribution ofthe near surface behavior of the response is not well de-scribed by the SCIB, but this will be repaired by bettertheories of v(q).

Finally we reduce the formal prescription of SC for d~.

where

E„(x)=6(x)E„(x)+6( x)—E, (x)

with

(33)

(34)

0, x & —a/2e(x ) = . x /a + —,', —a /2 (x (a /2

1, a/2&x(35)

and close (on the scale of transverse wavelengths) to thesurfaceE„(x)=D (0), (36)

E~(x)= eo '(0,0)+2 g cos(ngox)eo '(n, O) D„(0) .n)0

(37)The system in bulk has a lattice constant along x equal toa and we represent the corresponding reciprocal-lattice"vectors" by g =ngo, where n is an integer andgo=2~/a. We assume that the Hamiltonian in bulk hasreAection symmetry about at least one point in the unitcell and choose the origin for x at an integer multiple of aaway from such a symmetry point and far enough out invacuum so that no significant equilibrium or inducedcharge density exists there. This latter choice makesE„=E„ for x &0. The symmetric placement of the ori-gin also simplifies the expansion of the bulk "referencefield" E . The e's that appear there are from the micro-scopic dielectric function to be analyzed below. Theironly property that we need here is

eo '(0, 0)=1/ei ',where ez ' is the macroscopic dielectric function for fieldsalong x. The subscript l is necessary in (38) because ourneglect of crystallinity effects in X has made the systemoptically anisotropic.

We begin the simplification of (33) by expressing it as aprimary term plus a remainder:

(1 —1/e' ')d = f dx[E„(x) E(x)]/D —(0)+R(39)

Substituting from (31) and (37) the explicit integral in (39)becomes

f dx [E (x) E(x)]/D, (0)—0

[v(q) —1/ei ]dq, (40)0

where we have used the fact that the cosine terms in (37)integrate to zero and have replaced J o dx with

2 f ~ sinqx 2 f ~ dq0 7T 0 P 77 0 q

The result (40) is meaningful if (q v~0)~/ Ie, iwhichwe will show later is true. The remainder term may bewritten as

CRYSTALLINITY EFFECTS ON THE SURFACE OPTICAL. . . 4633

RD„(0)=f dx [6( x—)E (x)—6( x—)E (x)+ [6(x)—6(x)]E„(x)I—a/2

dx 0 —x —6 —x D 0+ Ox —ex E x—a/2= f dx [[6(x)—6(x)][E (x) —D (0)]],—a/2

(41)

where 6(x)=Lt, pB(x). The field difference in the lastline has both constant and fluctuating parts, but whenmultiplied by the difference of 8 functions all parts in-tegrate to zero. Hence the remainder term vanishes so

(1 —1/e', ')d, =—f, [v(q) —1/e,' '],0 q

(42)

identical in form to the jellium result. Our derivation of(42) used a specific choice of origin, but in fact the func-tional form of (42) is independent of that choice. Thefunction v(q) changes in a complicated way when the ori-gin is translated by b, but as SC showed d~ merely shiftsto d~ —b.

As a slight digression we note here for completenessthe form of d~~ for our present model which only has crys-tallinity effects along the surface normal. The response toelectric fields parallel to the surface is given by (local)free-electron formulas so d~j is easily found. From SC theanalog of (33) is

(1 —EI~ ')d~~ = f dx [D~~(x) —D~~ (x)]/E~~(0), (43)

which reduces to the frequency-independent real-valuedresult

where

q =k +g, q'=k'+g', (47)

with 0(k, k'(g0 and g =ng0, g'=n'g0 with n and n'non-negative integers. The yp ii(q, q') are Fourier trans-forms of the bulk susceptibility:

yp ii(k +ng p, k +n'gp)

yk(n, n')—= —f dx f dx'e

i ( k + n'go)x'Xgp ~(x x )e

garded as allowing only qualitative insights. ' Ouropinion is that any complete theory must evaluate bothtypes of contribution to g0, but one must start somewheretractable.

The appearance of the singular terms is more involvedin the presence of crystallinity effects than for jellium,since wave vectors are only conserved to within multiplesof g0 =2~/a. We claim that y0 separates into

Xp(q, q') =Xp,~(q, q'»(k —k')

+yp ii(q, —q')6lk +k' —gp)+gp(q, q'), (46)

d~~~~

= f dx[6(x)np g(x) np(x)]/np ii (44) (48)

where np(x) is the equilibrium density, n p ii(x) describesits variation in an infinite "crystal, " and n0 & is the aver-age of np ii(x) over a unit cell. The macroscopic dielec-tric function for fields parallel to the surface is

4~n0 ze(45)

and similar equations relate the D~~'s in (43) to E~~, whichis nearly constant. An evaluation of (44) requires onlythe ground-state density profile, but represents merely aslight improvement over a full jellium model for whichn0 z is constant.

C. Integral equation solution

Having shown via Eqs. (29), (31), (32), and (42) the util-ity of the function v(q) we consider next its evaluationfrom (27). Again a drastic (but improvable) approxima-tion will simplify the initial estimate. The susceptibilityyp(q, q') has both smooth and singular contributions,with the latter coming solely from the bulk response ofthe system. Our approximation is to keep only the singu-lar terms, which has an uncontrolled effect on the surfacecontributions. In a jellium calculation such an approxi-mation yields the so-called semiclassical infinite barriermodel, which has often been used but is now generally re-

where

equi(q)= 1 —

yp ii(q, q')v (q) (50)

is (in the RPA) the Lindhard bulk dielectric function.We used (49) and (50) to determine the jellium curves in

Figs. 1 and 2.The SCIB solution for v when one keeps (one-

dimensional) crystallinity effects is more involved. It ishelpful to define a symmetrized bulk dielectric functionmatrix as

5(k —k')equi(q, q')

—= ek (n, n ')6(k —k')

=5(k —k')[5„„.—u'~ (q)yp i, (q, q')U'~z(q')],

where

i yz (4vre )'~

k +ng0

(51)

(52)

and the y0 function contains the nonsingular contribu-tions that we ignore here. For a jellium model the g'swould not appear, which would remove the second singu-lar term and make the first diagonal in the total wavevector, rather than diagonal only in the reduced wavevector. In this limit the SCIB solution for v is trivial:

v'""," (q)=1/e (q),

4634 J. T. LEE AND W. L. SCHAICH 43

Then the combination of Eqs. (27) and (46) can be writtenas

with n and n' integers. The SCIB solution then appearsas

f dk'[u '~ (q)Ek(n, n')u'~ (q')5(k —k')n')0 vsciB( lq I )=&Bk '(n, n')

n'(62)

+u ' (q)Ek(n, n—' —1)

Xu' (q')6(k+k' —go)]v(q')

=1+f dqgo(q, q)u(q)v(q) . (53)

We can make the second term on the left-hand side of(53) look like the first by some notational manipulation.After the integration over k', replace in the second termn' with —m —1:

g u' '(q)Eii(q, —(go —k+n'go))

n') 0

X u'

( —(go —k+ n 'go ) )v(go —k + n 'go )

u '~(q)E~(q, k+mgo)m(0

Xu'~'(k+mgo)v( —(k+mgo)) . (54)

Then (53) becomes

gu ' (q)Ek(n, n')u' (q')v( q'l)n'

=1+f dq fo(q, q)u(q)v(q) (55)

where now q'=k +n'go and the suII1 on n' runs over allintegers. To make the left-hand side of (55) appear as amultiplication by a square matrix, consider the changesin (55) under the replacement k~go —k. At the sametime let n~ —m —1 with m (0 and n'~ —m' —1 withm ' unrestricted. Then

q ~(go —k) —(m + 1)go = —(k + mgo) = lk + mgo l

and the same matrix inverse would be needed for the gen-eral solution of (60).

D. Subtraction terms

As we show in Sec. III, the matrix in (61) is readily cal-culated. However, its inverse can fail to exist at particu-lar values of q. Since our formulas for, say, g and d~, re-quire integrals over q, these singular points cause numeri-cal difhculties. We shall treat them by the same methodsused above the bulk plasmon threshold in jellium wherethe E~(q) of (50) has a single zero. ' This involves sub-tracting from the exact B k

' functions which reproduceits pole structure, but which are analytically tractable.Our symmetrized definition of ek is especially useful inthis task since it leads to

BI, '(n, n')=u '~ (q)Ek '(n, n')u'~2(q') (63)

Bk '(n, n ')=, Eii '(q, q')q'

in which the two sorts of poles that occur do so inseparate factors, either u'~ (q') or El, '(n, n'). We treatthese in turn.

The singularities due to u' (q') are easy to locate andare always present. Since q =k+ngo and q'=k+n'gowe have when q'~0 that q~(n —n')go. Hence nearthese singularities

and

(56)(n —n')go

Ez '((n —n')go, O) .q

—n n' go— (64)

(g, —k) —(m'+ l)g, = —(k +m'g. ) . (57) Summing over n ', as required by (62), we define

Our assumption of an inversion symmetry in the bulkprovides the relation

u'~ (q)Ei, (q, q')u'~ (q')

(n —n')gov"'(q) =g, E~ '((n n')go, O—)

q—n n' go—

=u '~( —q)Eii( —q, —q')u' ( —q') .

Combining all these changes, (55) becomes

g u' (q)Ek(m, m')u'~ (q')v(lq' )

m'

(58) E~ '(g, O)~0) q g

Eo '(n, O),2

g)0 q g(65)

=1+ dqyo q, q U q vq (59)

where the square matrix Bi, (n, n') is defined by

Bk(n, n')=u '~ (q)Ek(n, n')u' (q'), (61)

where now q =k +mgo, q' =k +m 'go, and m is a nega-tive integer. Together (55) and (59) imply that

g Bk(n, n')v(lq'l) =1+f dq go(lql, q )u(q )v(q ), (60)n'

where the last step follows from the symmetryEii (g, O)=Eii ( —g, O). We refer to the singular struc-tures in (64) and (65) as zone-boundary singularities.They are responsible for the local-field oscillations evi-dent in both Figs. 1 and 2. Their periods are independentof frequency and of the form a/n with n an integer, buttheir amplitudes depend on co since the Eo (n 0) do.When we need to integrate over q, the singular structuresin (65) are treated as requiring principal value integrals.To illustrate we obtain for

43 CRYSTALLINITY EFFECTS ON THE SURFACE OPTICAL. . . 4635

g'"(x)= —$ dq""q v"'(q)(m) q

v' '(q) =R Q g N/, (m, m ')m, m'

l

q—(ko+ mgo )

and for

= I dx' —y dq cosqx'v'"(q)o

=2 g eo '(n, O)(cosngox —1)n)o

(66)

where

l

q + (ko+ mgo)

2(ko+mgo)=Rog Mk(m),q' —(ko+mgo)(73)

M/, (m)=yN/, (m, m') .m'

(74)

k —kodet(ek)= +O(k —ko)

Ro(68)

then the values of ko and Ro must be found numerically.The functional relation between ko and co defines thedispersion relation of these so-called zone-boundary col-lective states (ZBCS's). ' Since they are excitationsof the bulk, (68) also implies that

= g eo '(n, O)—f- =0 . (67)00

q2

g2

The standing-wave form of the fluctuating structure in(66) is what we expected to obtain, but (67) shows that itspresence has no direct effect on d~.

The other singularities in (63) arise from the diver-gences of Vk '(n, n'), more specifically from the zeros ofdet(e k ). Hence their location only depends on k, not onn or n'. If we write

To make the subtraction function (73) appear similar tothat in the jellium model we further define

2RQMk (m)

(1/ei ' —1)(ko+mgo)(75)

so

y (ko+mgo)v"'( )=(1/eI ' —1)gq

2 —(ko+ mgo )(76)

il' '(x)= —J dq v' '(q)o q

(77)

The singularities here are treated in integrals by replacingko —+ko+i0+ to represent the creation of a ZBCS propa-gating away from the surface. This yields for

k —(go —ko)det(ek ) = +O(k —(go —ko))

( —RQ)

the simple result(69)

i.e., the poles always appear in "conjugate" pairs. To ap-

proximate B k near them we recall that the inverse of amatrix can be expressed as

gI '(x)=gy (e ' ' —1),

and for

(78)

B„'(n,n') =N(q, q')/det(B„), (70)

where Nk is the classical adjoint matrix. Then for knear ko or go

—ko

d/2) — f q Lv/2/(q) v/2)(0) j

N„(l, n')

+lo oN k (l, n')

q—(go —ko+ lgo )

(71)

the finite contribution

(2)'Y

~ i (ko+mgo)

(79)

(80)

Since N/, has the same symmetries as Bk, see (58),

„(l,n') =N(gQ —kQ+lgQ, gQ—ko+n'go)

=N(k, +mg„k, +m g, )=N„(m, m ),(72)

where m = —(l +1) and m'= —(n'+1). Then if we sum(71) over n

' to define v' '(q), it can be reduced to

To complete the evaluation of g and d~ for the SCIB ap-proximation is now a well-defined numerical task. Onewrites

vsciB(q) =v(q)+ v" '(q)+ v' '(q),where v is determined by substituting from (62), (65), and(76). By construction, v(q) has no singular structures.This makes all required integrals well behaved if we notethe following limiting values. As q —+~, v"' and v' '

vanish while vsc&z and v tend to l. As q ~0,

4636 J. T. LEE AND W. L. SCHAICH 43

vscIB(q) =X, ~kq

—n n—' go

—+g, ek (0 n')kk +n'go

=eo '(0, 0)+O(q )

(m) (82)

v"'(q ~0)~(1—1/eI™)g y +O(q') . (84)

From (82) —(84) the limiting behavior of v immediatelyfollows. To make contact with jellium results define

S(q)=v(q) —1 —2 g eo '(n, O) .n&0

Then

(85)

7m&k, + g,

2y

dq 1+~ S(q)1 y&(m)

(86)

where

S(q~0) = — 1— 1 1++y +O(q ), (87)

which compares directly with Eq. (21) in Ref. 34.To end this section we remark that we have formulated

the subtractions from B k' so the results would still apply

if go were retained. The same general scheme could alsobe used when three-dimensional lattice structure is al-lowed.

The identification in the last line of (82) is the appropriateversion for our anisotropic system of the well-known con-nection between microscopic and macroscopic e's. "'

For us it ensures that the integral (42) for di is conver-gent. The limiting values of the v" are

v'"(q~O)~ —2 g eo '(n, O)+ O(q ),n &0

band model used by Sturm and Oliveira. Although bothmodels allow just one Fourier component in V(x), ourband structure is explicitly periodic in an extended zonescheme. This is necessary since we eventually need in-tegrals over q of the response function. For the samereason we also keep all local-field effects. Finally, we notethat although we are using a pseudopotential for the lat-tice scattering, we do not try to correct for the differencebetween pseudocharge densities and true charge densities.

A picture of the (one-dimensional) band structure isshown in Fig. 3. The first Brillouin zone has been chosenso the reduced wave vector remains positive, as is ap-propriate for the cosine Fourier transforms of our formal-ism. The Fermi energy is determined by requiring thatthe average bulk density be given by 3/(4irr, ao); itsvalue is cF =4.04 eV, slightly smaller than the jellium re-sult v+=4. 33 eV. The Fermi wave vector p~=1.20 Ais larger than pF, but a bit smaller than go/2= 1.27 A

The bulk susceptibility defined in (48) is calculatedfrom

~ =2 fpl, P fp' I ', P

P e, +fico e I+—IO

X QADI(x)P, I.(x)Q*,I,(x ')P I (x '),(89)

where p and p' label wave vectors in the first zone, I andl' are band indices, P is the common wave vector parallelto the surface, and the overall factor of 2 is for spin. Thef's are Fermi occupation factors and will be evaluated atzero temperature. Their argument is the total energy ofan electron state, whereas only the difference of "normal"energies survives in the denominator. The wave func-tions are also one dimensional (Bloch waves). The planewave variation in X has been Fourier transformed away.

In Fig. 4 we compare the continua of single-particlebulk excitations for our band-structure model and its jel-

III. MODEL CALCULATION

To illustrate the formalism we now describe a pseudo-potential evaluation of the basic formulas. The parame-ters are chosen to model the (110) face of Li since thissystem has relatively strong lattice scattering and showssignificant deviations from jellium predictions. The lat-tice constant along the surface normal is a =2.48 A sogo =2m /a =2. 54 A '. With one conduction electronper atom the jellium r, =3.26ao, where ao is the Bohr ra-dius. The lattice potential energy is described by a singleFourier component pseudopotential

V(x) =2 Vocos(gox ) (88)

whose strength is Vo=1.44 eV. This choice, plus the as-sumption of an isotropic effective mass I */m =1.09, al-lows a reasonable representation of the band dispersion atlow energy in the (110) direction. ' ' ' Our treatmentis similar in spirit, but different in detail, from the two-

0—I I I I I I I I I I I I I

0.0 0.2 0.4 0.6p/g.

O. B 1.0

FIG. 3. Band structure along the (110) direction in Li. Thesolid curves give the electron energy E~ vs (reduced) wave vectorp. The dashed horizontal line is the Fermi level.

43 Ts ON THE SURFACE OPTICAL{RYSTALLINITY EFFEC 4637

I I I I I I I I I

0 I I I I

0.0 0.5I I I I I I

1.0 1.5q/g.

2.0

I I I I I I I II I I I I I I I

(b)

oW

V ~0 and m *~m. The bounded re-lium limit, when Vo 0 an m . nded rehere the imaginary part o

Th ' " "'tin int')] is nonzero. e gadirect consequenceexcitations is a ir

o' ' . We also show in ig.g.

d bulk collective mo es.~

=0. Fo h llib re uiring det e&are determined by qthe bulk plasmon,ode that appears is emodel the only mo

d to a damping of this— tructure effects lea to awhile the band-struns and the appearance ofmode ue od t interband transitions an

tive mode in the continuum g Pa the ZBCS.~ 41

a new collective of he dielectric matrixWe next show se

ts through Fig- 4.vera& views o t e

long certain cuplotting its va ""g2) is iven in Fig

ylimit, defined by, is gThe macroscopic im,

h d from the jellium(m) is s]ightly c angee real part of &y 't,. makes the imaginarybut the interban

ove a threshold.part nonzero abovd frequency plotIn Fig. 6 we comp

'b 1 w the interband

are severa xe ro 0 5 one is eow„(00). At &~/ F ' '

d lhum picture»threshold so theo h are considerable

band structure an jeAt A'co/E+ =0.9 t ere ay

pe in articu ar eto the o an go-

local-field efi'ects —specifica y

They occur close tothe ZBCS since loca — e

n e (0,0) and det ek —a~

—are not large atdifference between ek=2.0 one is in thethis frequency. Finally, at Ace/c~ =

no complete zeroplasmon range for jellium,m butt ereisn

In Fig. 7 we compm are the d~'s. or eto +4go were kept.1-lattice vectors out tocase, reciproca — a ithat begins sharplyadditional structure

dThere is clearly a

d ntinues through anat the interband threshold an con inuband. We have separaarately plotted thebeyond the ZBCS an .

ion of the ZBCS to d . It begins and endsdirect contribution on th, 'ust as in eelectron-loss spec-wi'th vanishing streng, j

a lnaly part, oNote that the imag y ptra.lasmon. In the presence o an sp

ar art of d~ is no ature the imaginary pd analogous calcula-We have also done anathe absorption. e

ose seudopotentia s'

1 strength is an orderfLi Th d

' t'o b-g lier than that o i.~ ~

Na results and a jellium imitween the a red 1 roughly with Vo.ects on d~ sca e rou

7 it is interesting o nReturning to Fig. ih to significantlyrin is strong enoug

~ ~ ~

interband scatteringhat a pears in the jelli-lasmon divergence t a ap

h 1'fti bo dum dz. We emp hasize that t is i e iming calcu a e1 t d microscopically here, no

31 i I !

0 I

0.0 1.0q/g.

1.5 2, 0

FIG. 4. Comparisonn of electron- o e-h 1 pair continuum of exci-(b) the Li(110) band-'ellium model andtation for (a) the je

the boundaries inhe solid curves areor s ace between regions

t cture model. T e sof zero and finitef quency-wave-vector space

c' ' '. In (a) the excitations occcur between the

h b lo h 1

band) or above the upper so'

Thethe w tor,b) is erio ic inpane the dispersion o un amdashed curves describe t e is

'am

from the zone boundaries a(b) the ZBCS begins and ends away from t e zon

q =0 and go.

uenc forFIG. 5. Macroscop ic die ec ric1 t 'function vs freq y

1. Both the real and imaginaryTh d hd

e surface normal. Both t e reaive(m) n with solid lines.

t e je '; e that its imaginary pthe jellium result; note azero.

4638 J. T. LEE AND W. L. SCHAICH 43

hoc parameter. The appearance of the field when thedamped plasmon is present is shown in Fig. 8, which likeFigs. 1 and 2 is determined from (32) with v replaced byv+v'"+v' '. The plasmon oscillation is evident, but itdecays as one moves into the bulk. In Fourier spacethere is a (rounded) polelike structure in the e &

' versus kcurves which we can handle numerically without subtrac-

tions. However, the neglect of zone planes other than(110) is a serious omission in this higher-frequencyrange. '

To sum up, we would describe our results as a promis-ing beginning on a difFicult problem. The derivation inSec. II has shown how to reduce a formal prescriptionto a practical evaluation scheme. Several severe approxi-

40 I I I I

(o)I I I I I I I I I I I I I I I I I I } I

20— 10—

0

40

0

20

20— 10—

0—C)

0

—20 —10—

I

4O0.0 0.2 0.4 0.6 0.8 1.0

q/g.0.0 0.2 0.4 0.6 0.8 1.0

q/g.

(c)

0—

0

—20.0

I I I I I I I I I I

0.2 0.4 0.6q/g.

1.0

FICx. 6. Comparison of microscopic dielectric functions. In each panel both the real and imaginary (bottom and top) parts ofez(0, 0) are plotted for both the band-structure and jellium (solid and dashed) models. The fixed frequency is (a) 0.5c+, (b) 0.9m+, and(c) 2.0cF. The singularities that go off scale are inverse square-root divergences.

43 CRYSTALLINITY EFFECTS ON THE SURFACE OPTICAL. . . 4639

I I I I I I I i I

0

00

00000

CL~~ (xBD+4\ 0

0 0000~ ~ ~ ~

0 5 10x/a

20

8~ ~ ~ 0

t I

0o 0

I I I I I I I

FIG. 8. Same as Fig. 1 except the frequency of the externalperturbation is here 2.0c„. The local-field oscillations, which incontrast to Fig. 1 are now more evident in the imaginary part ofg, still have period a.

1

f1&/ep

FIG. 7. Comparison of d~ for the band-structure (opensquares) and jellium (solid line) models. The real and imaginaryparts of d, are plotted separately. The solid squares give thedirect contribution (80) of the ZBCS over the frequency rangewhere it exists as an undamped mode.

mations were made in this first effort, but there appearsto be no fundamental impediment to their removal. Wecan envision the retention of yo, starting first with aninfinite barrier model and then progressing to finite andself-consistently determined barriers. Also, an analysisthat accounts for crystallinity effects in three dimensionsshould be a feasible, but considerable challenge. One cango beyond a RPA description of many-body effects too,incorporating, say, a time-dependent local-density-functional scheme. Our confidence in the possibility ofthis succession of improvements rests on the facts that asimilar path was followed for the jellium models' " andthat the formal developments here show that crystallinity

effects do not change the basic structure of the theory.The awareness of this possibility plus the existence ofrelevant experimental data' ' will hopefully lead tosteady improvements. Although the present numericalresults cannot be directly compared with experiment,they do give first measures of the existence and impor-tance of effects totally absent in jellium models. Theirevaluation has also provided a calculational basis ontowhich one can build systematic refinements. We lookforward to this continuing progress.

ACKNOWLEDGMENTS

We wish to thank Dr. Wei Chen for help at the begin-ning of this work. We are also grateful to Dr. K. Sturmand Dr. L. E. Oliveira for useful discussions of theirwork. This effort was supported in part by the NationalScience Foundation through Grant No. DMR-89-03851.Some of the calculations were done on the CrayResearch, Inc. X-MP/48 system at the National Centerfor Supercomputing Applications at the University of Il-linois at Urbana-Champaign (Champaign, IL).

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