electronic properties of stepped metal surfaces

The Inst. of Natural Sciences Nihon Univ Proc. of The Inst. of Natural Sciences Vol. 29 (1994) pp. 95~106

Electronic Properties of Stepped Metal Surfaces

Hiroshi ISHIDA

(Received October 31, 1993)

Results of systematic electronic structure calculations are presented for regularly stepped (vicinal)

jellium surfaces with varying step densities, step heights, and electron densities. First, we discuss

the ground-state electronic properties. The calculated results reproduce the experirnentally observed

linear dependence of the work function on the step density up to a very high step density. The

induced change in the electrostatic potential and in the density of states in the neighborhood of

the step site suggest higher chemical reactivity near steps than at low-index flat surfaces. Next,

we discuss linear and nonlinear response properties of the stepped surfaces to a static elecric field

oriented in the surface normal direction. The normal component of the polarization vector associated

with the nonlinear-induced charge is increased relative to that of a flat jellium surface in proportion

to the step density up to very high densities. However, the magnitude of this enhancement amounts

to only 15 o/o at maximum.

1. Introduction

There was remarkable progress in the study of clean solid surfaces for the last decade. For example,

the atomic structure of a number of reconstructed surfaces was determined by newly developed

experimental techniques such as the scanning tunneling microscopy (STM). As an important next step,

there has been growing interest in studylng surface defects such as stepsl'2). Steps exist more or less

in any surface in nature. Thus, it is important to clarify how the properties of clean surfaces are modified

by these surface defects. Also, the dynamics of steps plays an essential role in surface phase transitions

and in crystal growth3,4)

So far, most of the experimental studies of steps were conducted with use of vicinal surfaces, i. e.,

crystal surfaces whose surface normal is close to those of low-index planes. The vicinal surfaces take

a regularly stepped structure where neighboring steps are separated by a terrace surface of the flat low

-index planes with relatively low surface energies. The real-time observation of step motions has become

possible only recently by STM5). A number of experiments demonstrated that stepped surfaces show

markedly different electronic properties than flat low-index surfaces. Among them are the work function

reduction, the larger sticking probability for adsorbates, and the promotion of catalyiic reactions. In

addition, it has recently been found that the optical second-harmonic efficiency of metallic surfaces is

strongly influenced by the presence of monatomic steps6,7)

In contrast to the above-mentioned recent progress in experiments, theoretical efforts towards

understanding the electronic properties of steps were limited to rather simplified model analyses based

on the tight-binding Hamiltonian8,9) small cluster calculationsl0,11) and the jellium model combined with

7156 ;~~~:~~t~~El~~;~~'~;~'~71~3-25-40

Department of Physics, College of Humanities and Sciences,

Nihon University, 25-40, Sakurajousui 3-chome, Setagaya-

ku, Tokyo, 156, Japan

Hiroshi ISHIDA

approximate model electron densitiesl2,13). Recently, Nelson and Feibelmanl4) made a detailed study

of the structure relaxation of the A1 (331) stepped surface using the Car-Parrinello approachl5) in a

repeating-slab geometry. Nevertheless, the detailed knowledge of how the interaction between steps

changes the subtle charge rearrangements at the step site as a function of step density is still lacking.

Also, there has been no theoretical work on the response properties of stepped metal surfaces.

Instead of focussing on a particular surface, in the present article, we present results of systematic

self-consistent density functional calculations of the electronic properties of regularly stepped (vicinal)

Jellium surfaces with varying step densities, step heights, and positive background densrtres The vanous

electronic contributions to the interaction between steps are fully included in this approach. We discuss

not only the ground-state electronic structure but also the response properties of the stepped surfaces

to a static electric filed. Although the jellium model cannot take account of localized d states of transition

metals, it is a realistic model for simple metals such as A1 and Na. It may also serve to elucidating

various general properties of stepped metal surfaces in the same way as the work of Lang and Kohnl6)

for flat jellium surfaces played an important role in the study of low-index metal surfaces.

The outline of this article is as follows. In Sec. 2, we present the model for the stepped metal surfaces

and describe details of the calculational method. Sec. 3 and Sec. 4 are the main parts of the present

article. Sec. 3 contains results of the electronic structure calculations of the stepped surfaces in the ground

state. In Sec. 4, we discuss the linear and nonlinear response properties of the stepped jellium surfaces

to a static electric field oriented perpendicular to the surface. A sununary is given in Sec.5. Hartree

atomic units are used throughout this article.

2. Model and Calculational Method

To simulate vicinal surfaces, we utilize the semi-infinite jellium whose positive background density

n+ (x, z) has a periodic modulation corresponding to the 90' step structure at the surface (the z axis

is chosen as the surface normal and the x axis is perpendicular to both the z axis and the ledges runnunig

parallel to the steps). One period consists of a terrace of width xw and a ledge of height xh, i. e.,

( L~: n+(x z) 77e [z (x) z]' zo(x) ::: O~x~ ) ' xhx xw xw ' l - , (2 ) zo(x) = xw(1 x) ~~'<x< l

xh I ~ ~ (1)

where 7T (>0) is the density of the background charge, I = f!r7 the length of a unit cell, and

the origin of the z axis is chosen as the lower edge of the steps. We show results for the jellium surfaces

with rs = 2 and 4, which approximately correspond to the free-electron densities of Al and Na, respectively.

xw and xh are chosen as m' ao and n'ao, where ao is the lattice spacing for the (OOl) plane, i. e. ao = 3.83

a.u. for Al and ao = 3.99 a. u. for Na. This configuration corresponds to the [m (OO1) X n (100)] structure

in the nomenclature of Lang et all?)

The ground-state calculations are performed within the local density approximation to the density

functional theoryl8). The one-electron wave functions of the semi-infinite system are calculated using

the embedding method of Inglesfieldl9). In the present case, the numerical problem is reduced to a

two-dimensional one because of the translational symmetry in the edge direction. We utilize the higher-

20) dimensional Anderson method reformulated by Blti gel for the iteration procedure toward self~consistency .

Convergence is assumed when the difference between the input and output dipole layers becomes less

than 0.0003 eV.

In Sec. 4, we study the change in the electronic structure of stepped surfaces due to a static electric

( 12 ) - 96 -


field. In order to apply a unifonn electric field in the z direction, we place a charge sheet in the vacuum

at a distance well separated from the surface21). The strength of the applied electric field is given by

Eapp = 27ca, where a is the surface charge per unit area. In the presence of a weak electric field, the electron charge density n(x) [x = (x, z)] can be expanded22)

as,

n(x, a) = no(x) + (;nl (x) + a2n2(x) + ""-- (2)

where no' nl' n2 are the ground-state, Iinear-induced, and second-order-induced charge densities,

respectively. For each step geometry, we perfonn three self-consistent calculations with a= O, ao' and - ao'

where ao is chosen such that the higher-order terms in eq. (2) can be ignored (Eapp was set equal to

1.5 X 10~3 in the actual calculations). nl and n2 are then obtained as

[n(x a ) n(x, -(Io)] 2ao

[n(x, ao) +n(x, -ao) ~ 2n(x, O) l. 2a2

As discussed later, the dipole moments of these induced densities,

1 1 f" l= fo p T dx dz z nl (x) , -"

and

* p2 T dx f dzz n2(x), = fo --1

are key parameters in discussing the linear and nonlinear surface response. The direct numerical

evaluation of these quantities is difficult because of the Friedel oscillations that nl and n2 exibit in the

interior of the bulk. As in the case of flat jellium surfaces23) and adsorbed alkali-metal layers24), the

dynamical force sum rule25) can be used to derive analytical fornrulae that relate pl and p2 to the partial

dipole momemts of nl and n2 only in the surface region. Following the derivation in Refs.2~ and 24,

we obtain in the static limit,

" p T dx f dz z ni(x) ,= fo 1

+ I I f a [n+(x)-7Te( z)] (5) * fo --dz q~l (x) dx 47c71 l az

with i = 1, 2 and {z)i is defined by

f I n (x ) (6) ~9 (x) =-2lcz6i,1 + J d r' r - r

where the first term is the applied linear potential and the second is the Hartree potential associated

with the induced charge ni.

3. Ground-state Electronic Structure

In Fig. 1 (a), we show the contour map of the electron density for Al (r* = 2) with (m,n) = (10, 1)

- 97 - ( 13 )

Hiroshi ISHIDA

10

Q

No o

1 2 0,003

3

CL02?

0.030 O (~) ~~~) (~~>

(Q)

o 20 40

10

:;

C,_

No <)

o~003

(~)(~

~027 O c) aC) oc 30 (~) c~) (:)

o a <~

(b)

o 20 40

10

Q~

No

2 ClOO037s

O003?5

3

c)

(c )

x(a.u.)

Fig. 1.

Contour maps of the electron charge density of regularly stepped jellium surfaces on vertically cut

planes perpendicular to the ledge direction. (a) Al

(r* = 2) single step, (b) Al (r:s = 2) double step,

and (c) Na ( r. = 4) single step. Contour spacings are 3.0XI0-3 a. u. and 3.75 x 10~4 a. u. for r* =

2 and 4. The dotted lines indicate the profile of

the background positive charge of jellium.

-atom step height E(m, n) = (10, 2) and r* = 2] shown in Fig.

lines start to display plateaus running parallel to the jellium profile not only along the terrace but also

along the ledge. Fig. I (c) shows the electron density map for Na (r* = 4) with (m, n) = (10, 1). The

charge distribution at the terrace is disturbed in a slightly larger range around the step as compared

with Fig. I (a) because of the larger screening length of Na.

Next we discuss the work function change which originates from the change in the electrostatic

potential confining electrons. First, we note that the stepped surface should have the same dipole potential

barrier as the flat surface if the terrace and ledge surfaces had the same dipole moment /lo (per unit

area) as the flat surface oriented in each surface normal direction. (In this case, the terrace and the ledge

each contribute 4lcllox~/ (x~+x~) and 47c,lox~ / (x~+x~), respectively, to the dipole potential barrier

of the macroscopic surface. Thus, the sum is 4lt;,lo' the flat surface value.) Hence, the work function

change in the presence of steps is a microscopic effect originating from the charge redistribution localized

near the step. We denote the dipole moment associated with this charge redistribution per unit edge

length by di ( d ) for the component perpendicular (parallel) to the terrace. The work function change

Ac is then written as

on the vertically cut plane perpendicular to the

ledge direction. The dotted lines show the profile

of the positive background charge. This figure

demonstrates how the electron charge which

cannot completely follow the profile of the

positive charge at the ledge, redistributes itself

in the vicinity of the step in order to lower the

overall kinetic energy. Essentialy, charge flows

from the top reglon of the step towards the lower

comer (Smoluchouski effect)26). As we will

discuss below, this redistribution is the origin of

the lowering of the work function observed for

stepped metal surfaces. Scattering of the one-

electron wave functions at the surface leads to

a superposition of Friedel oscillations in two

directions : orthogonal to the terrace and to the

ledge, as shown by the closed density contours

which form a two-dimensional pattem in the

interior of the jellium. Apart from these Friedel

oscillations, the electron density near the surface

is seen to distribute itself almost perfectly in one

-dimensional fashion along most of the terrace.

This means that effects on the electronic

structure of the terrace caused by the step are

confined to a small reglon of the order of tht

screening length. Actually, we found that, in

the vicinity of the step, the charge contour maps

of stepped surfaces with smaller terrace widths

(smaller m ) Iook very similar to those in Fig. 1

(a). This is also true for the surface with a two

1 (b). For this step height, the contour

( 14 ) - 98 -


Aep = 411: xwdi+xhd

x2w+x~ ' (7)

For low step densities (xw >>xh), we have Ac -

47rdil f-7, 1' e. Ac is proportional to the

step density.

In Fig. 2, we plot the calculated work

function change Ac as a function of 11 fT7 (-1/x~ for low step densities). Each

curve displays results for a series of stepped

surfaces with the fixed step height and the jellium

density as corresponding to the three cases

shown in Fig. I but with varying terrace widths

(2~m~10). It is seen that Ac fits a linear curve

up to a very high step density. For m~3 the

work function change becomes slower than linear

since now the redistributed charge densities on

neighboring steps start to overlap and may be

modified. The initial linearity of A ep implies (1)

that d is considerably smaller than di so that

the second term in eq. (7) can be outweighted

by the first one even at the high step density

(m/n - 3), and more importantly, (2) that_ di

remains essentially constant up to a very high

step density. The second aspect may be understood from Fig. I in which the disturbance

o

- 0.05

>G,

~:

<1 _O1

-0.1 5

*~~

*$~

*+~

**¥

',¥¥ Io

", ~~ lo

*¥ 10~ ¥

6 ¥

~te

** ¥ ** ¥

~'!~].

s

¥

4 ~]

¥4 ~~ ¥

¥

¥ 3 ¥~

~

~

2

1

11 ~~~(107 cn;jl)

Fig. 2.

Calculated work function change A c as a function

of 1/ J!･7, where xw is the terrace width,

and xh the step height. Data on solid, dashed, and dotted lines correspond to the Al single step,

Na single step, and the Al double step, respec-

tively. The small numbers indicate the parame-ter m for the terrace width. xw= m'ao where a0= 3.83 a. u. for Al and 3.99 a. u. for Na.

in the electronic charge distribution is highly localized near the step as stated previously. The dipole

moment di stems from the charge depletion near the upper edge of the step and the charge increase

near the lower corner of the step. The double step for Al ( r* = 2, Ie = 2) has a larger di than the

monatomic step because of the larger separation between these two regions. The larger di for Na

( r. = 4) as compared with that of Al may be attributed again to its larger screening length, i,e. the electron

gas with a lower density can follow the abrupt change of the positive charge profile at the ledge less

ef f iciently.

The linear dependence of the work function on the step density has been reported so far for W,

Au, and Pt27-29). For W (110), Krahl-Urban27) obtained dl ~ O.065 D (per unit length) for the step running

parallel to [ool]. This value is much larger than 0.015 D, which we obtained for the Al double step

( r* = 2, n = 2). It might be that the relocation of the d charge as suggested by Desjonqueres and Cyrot

-Lackmann8) within the tight-binding calculation has a large contribution to the dipole moment in the

case of transition metals. An indirect support for this argument is the experimental result of Besocke

et al29) who showed that the dipole moment associated with the step for Au with its closed d shell

is more than two times smaller than that of Pt. We hope that the work function change due to steps

will be measured for simple metals to verify our predictions and to clarify the large difference as compared

to the changes observed for the transition metals.

The chemical behavior of surfaces is known to be strongly modified in the presence of steps. For

example, for simple metals, Testoni and Stair30) showed that the sticking probabilty of O on Al (111)

Hiroshi ISHIDA

:]

C,

N

lo

o

08

a7

~ =='=<L~~~.."~=r_.C=L~j)'1!

~"~

= _'_Q'~'_--~ f{ ~-~~~) -/~~~~- _= ~~~~,2'~~.=..= __/~~'-' ~ t (c

~:a t¥ [1) (~ ~

¥ cr x(a.L~)

Fig. 3.

Contour map of the electrostatic potential on the

vertically cut plane perpendicular to the ledge direction for the Na ( r* = 4) monoatomic step with

(m, n) = (10, 1). The solid, dashed, and dash-dotted

lines correspond to posive, negative, and zero values, respectively. The contour spacing is 0.1

eV.

,~, u) ~' 'c :)

Li

O 'v tf)

O O

(a)

+_: fi_

1

/// ~;2/*

･41' 3 ;_/

/ /

/// /~'~r

/

is increased by a factor of 4 by introducing steps

on the surface. Ibach31) argued that the activation

barxier for molecules in dissociating from a

precursor state into atoms may be very sensitive

to the electrostatic potential, which can be modified

by steps. In a similar way, Lang, Holloway, and

Ncrskov32) explained promotion (poisoning) of

catalytic reactions in the presence of electropositive

(negative) adatoms based on the induced dipole

field surrounding adatoms. In Fig. 3, we show a

contour map of the calculated electrostaiic potential

(Hartree potential associated with the electron

charge and the positive charge of jellium) for Na

(rs = 4) with (m, n) = (10, 1). Corresponding to

the charge map in Fig. I (c), the potential behaves

one-dimensionally along most of the terrace.

eV) in the interior of jellium near the ledge.

-12 -8 -4 o

(b)

1

/ / l~

/ / r) / l* L// l'

///~/~

/

ll /./'

/

-1

E - EF(eV)

Fig. 4.

Calculated density of states in a small sphere of radius 2 a.u. for three different sites on the

Al (a) and Na (b) stepped surfaces with ( m, n) =

(10, 1). The small number on each curve indicates the location of the sites marked in Fig.

1(a) and (c).

There appears a fairly deep potential minimum (--0.5

As pointed out by Kesmodel and Falikovl2) such a local

field created inside the metal by s- and p- electrons may affect the level and occupation of localized

d (or core) electrons of step atoms considerably. It is seen that the contour lines in outer regions protrude

markedly towards the vacuum above the ledge as a result of the reduced electron density in this reglon.

On the other hand, near the corner between the ledge and the lower terrace, these contour lines follow

the profile of the positive background charge much better ; towards the step they bend slightly inward

reflecting the charge increase near the corner. We found a similar pattern also for Al ( rs = 2).

Another quantity relevant to the chemical reactivity of surfaces is the density of states near the Femi

energy ( EF ) as suggested by Feibelman and Hamman33) in the study of catalyiic promotion or poisoning

induced by adatoms. They considered that a higher density of states at EF owing to the charge transfer

from adatoms may lead to higher chemical reactivity of the surface. In Fig. 4, we show the density of

states for Al and Na with (m, n) = (10, 1) calculated in a small sphere of radius 2 a. u. centered at three

sites along the terrace as marked in Fig. I (a) and (c). The distance between the sphere center and

( 16 ) - 100 -


the terrace surface is chosen as 2 a,u. for all of them. One sees a noticeable enhancement of the density

of states at the comer site between the ledge and the terrace. This may be understood from the increased

charge density near the corner shown in Fig. I . Thus the corner site may be chemically more reactive

than a site on the flat surface. On the other hand, the density of states on top of the ledge is slightly

lower than in the middle of the terrace, i. e., the chemical reactivity will be reduced due to the depletion

of charge (ballustrade effect). For Al, this effect is less pronounced than for Na because of the shorter

screening length. Our results agree qualitatively with those of Thompson and Huntingtonl3) who studied

the adsorption energy of atoms at a stepped Na surface using a simplified version of the local density

functional theory and an analyical variational jellium electron density. For stepped transition metal

surfaces, this effect might be more pronounced than for the simple metals, since the d-electron density

of states of step atoms may change quite a lot by losing more nearest neighbor atoms than the other

surface atoms.

4. Linear and Nonlinear Response of Stepped Surfaces

There has been growing interest in studying surfaces and interfaces by optical second harmonic

generation (SHG). Due to the lower symmetry at the surface, some of nonlinear optical polarizabilities

that vanish in the bulk remain finite at the surface34). Measurements of these components provide useful

information on the microscopic electronic structure in the surface region and on its optical response

properties. Surface defects such as steps and kinks can be additional sources of surface-induced SH

light because they lower the symmetry of flat low-index planes and increase the number of nonvanishing

elements of nonlinear polarizabilities. Janz et al. found that the SH intensity of Al (OO1) vicinal surfaces

changes by an order of magnitude depending on step density and incident angle6). The sensitivity of

the SH intensity to steps was recently utilized to deteunine the phase diagram of Cu(111) vicinal surfaces7)

So far, the most realistic calculations of the surface SHG have been carned out for the one dimenslonal

jellium model by applying the time-dependent density functional theory35,36). In this case, there is only

one nontrivial surface polarizability, X..., which is determined by the rapid variation of the normal

component of the electric field in the vicinity of the surface. Nearly quantitative agreement was achieved

between these calculations and the experiments obtained for flat Al surfaces37,38). An important next

step is to extend these calculations to more complicated systems with density corrugations in the surface

plane in order to investigate the influence of the corrugations on X..., and to determine the remaining

components of the surface polarizability. In this section, we address the first of these issues and study

the linear and nonlinear response of stepped metal surfaces to a static electric field. These calculations

can be regarded as a preceding stage to the considerably more involved dynamical nonlinear response

calculations.

The a component of the surface induced SH polarization, P~ , is expressed in terms of the nonlinear

polarizabilities X ~py as

p,y

where E~ denotes the (~ component of the external electric field. For flat jelhum surfaces, all but Xz"

and X.x' vanish for p-polarized light incident in the xz plane, whereas all the six components (a, p,

1/ = x or z) remain finite for stepped surfaces. In the present work we focus on the evaluation of Xzzz

for the following reasons. First, for flat simple metal surfaces, this polarizability component is by far

the most important one and is known to depend rather sensitively on the details of the electron density

distribution at the surface. For example, in case of adsorbed alkali metal layers, it is this component

that is responsible for the enonnous enhancement of the SH intensity39). It is therefore of great interest

Hiroshi ISHIDA

c5

~I

12

e

o

-e

c

cc)oC) Cll>

(a )

o 10

X(a u )

20

(:I

~i

12

e

o

-6

( b)

"~-"~=~-'-~~*=*-~~~~~~s~, ~~

~~~;~~;><~~~~::~¥'/ ~~; , ::;-1_~:- -~-:~ ¥¥ ::.'~' ~ ¥'~ ::=~~~ -~1:~~;;~;~~~~"'~'~'--~¥--" *=~r ¥_ ~~'~~ = r;~~:~L~_,¥::-_¥-)_ ¥

'..~

' ~ ~ ~:__7J l~ -'~~ J ~~ r j ~/~~ l~ /~ /r- ~~ r o 10

X(a,u )

20

(5

N

12

6

o

-6

(c)

_~ l_.!:~:~~:~rf=~'~'::･･l~";;'¥1,_~1_~~""

~~~~~¥1'~5'I~"~Ilt¥s~t~l¥

:'1;(~~"~~ ~;~]~~';' 3;~"~~I 1~ :1 <~_ /11 1 ¥'1~

~~" I / t - {1 ~~~I~~~'~':-rl~;::-~:~~~~~~~:;;:~':~~:'~:1~~;;~':"~'

--1~l Ct; 4~

X(a,u )

Fig. 5.

Contour maps of charge densities for the regularly stepped jellium surface with ( m, I~) =

(5, l) and rs = 2. The dotted lines indicate the profile of the background positive charge.

(a) Ground-state charge no' The contour spacing is 3 x 10-3 a. u. (b) Linear-induced

charge nl' Solid, dashed, and dot-dashed lines correspond to positive, negative, and zero values of nl' respectively. The contour

spacing is 0.05 a. u. (c) Nonlinear-induced charge n2' The contour spacing is 7 a. u.

( 18 )

to detenuine the modification of Xz" due to the

presence of steps. Second, the evaluation of Xzzz in

the static limit can be reduced to a set of ground state

calculations21'22) while a perturbation approach is

necessary for the evaluation of the other components

even in the adiabatic limit. This static approximation

is adequate for the low frequency response of A1

surfaces since (o /(vp/¥'/¥O. I for the typical experimental

conditions (co and (op denote the frequency of the

incident light and the plasma frequency of the metal,

respectively) .

In Fig. 5 (a) we show the contour map of the ground

state electron density no for Al (rs = 2) with (m, n) =

(5, 1) in the xz plane. As discussed in the preceeding

section, the electron density distributes itself almost

penfectly in one-dimensional fashion in most part of the

terrace surface.

Fig. 5 (b) shows the contour map of the linear-

induced charge nl' According to eq. (3), nl has a unit

charge when mtegrated over unrt area. The induced

charge is located mostly outside of the positive

background charge n+ and exhibits a Friedel oscillation

toward the interior of the metal. A Iarge peak appears

in nl near the upper edge of the step. The height of

this peak (-0.5) is insensitive to the terrace width x = w

m'ao for m ~ 3. The shape of the peak is strongly

asymmeinc ; nl decreases slowly along the upper

terrace, while it drops steeply on the lower terrace side.

In Fig. 6 we show the corresponding linear change

in the Coulomb potential, q'l defined by eq. (6). These

contours illustrate how the applied electric field is

screened by the metal. In the static limit, the magnitude

of the total electric field approaches 471~ and O in the

vacuum and in the interior of the metal, respectively.

In contrast to the associated induced charge in Fig. 1

(b), the contour lines of ~l behave very smoothly

following the surface profile. For flat jellium surfaces,

it was shown that q~l= O at the edge of the positive

back ground charge23). By rewriting the sum rule eq.

(5), we obtain the corresponding formula,

fo 1 1 ' o (9) dx ~ (x z (x))=0,

which is applicable not only to the present stepped

surfaces but also to positive background charges with

any surface profile. One may expect that ~1 takes a

- 102 -


12

6

C~

N o

-e

-If'~~; ¥ ~ - ."'~,,･'~= ' ¥ ~L ¥--1' ~~~~~~ /~~ ~ '='~ ____ _/~~---~ *+.~'

=1 *~~

C~ ~~ ~'*~* _l J /~~1 ~-,> =1 ~ r _~ ~ r--~~, ¥.~ ~

X(a.u.)

Fig. 6.

Contour map of the linear change in the Coulomb potential> ~1' for the regularly stepped jellium surface with (m, n) = (5, 1)

= 2. The dashed and dot-dashed lines and r*

correspond to negative and zero values of ~1'

The contour spacing is 10 a. u.

constant value along the jellium edge of the terrace

suriace for a larger terrace width. If this is the case,

eq. (9) indicates that this value must be zero. Actually,

as seen from Fig. 2, the contour line corresponding to

q~1 = O (dot-dashed line) coincides with the profile of

the positive background charge in most part of the

terrace surface. This result also demonstrates that our

numerical calculation is highly accurate (note that the

sum rule eq. (5) is not enforced to hold in the iteration

procedure toward self-consistency).

In Fig. 7 (a) the calculated centroid of the linear-

induced charge, pl' is shown for monatomic steps (n =

1) for various terrace lengths (m ranges from I to 10).

pl defines the position of the classical image plane in

the static limit40). For m = 1, the image plane lies closest

to the surface since the electron distribution can be

efficiently polarized along both sides of the symmetrical

cusp forming the step. With increasing m the image

plane moves initially outwards because screening at the upper portions of the terraces becomes more

important. For m larger than 3 to 4, however, pl begins to diminish again since the density distribution

near the steps does not vary very much any longer while the step density decreases. In the limit of

large m, pl reaches the asymptotic value pl (oo) = 0.5ao + zo = 3.48 a. u. where ao = 3.83 a. u. is the step

height as discussed above and zo = 1.57 a. u. is the static image plane position for flat jellium with rs = 223)

Next, we discuss the nonlinear response properties. Fig. 5 (c) shows the contour map of the second

-order-induced charge n2 for Al (rs = 2) with (m, n) = (5, 1) in the xz plane. As seen from eq. (4), n2

has dipolar character with vanishing integrated charge. The calculated n2 has a positive peak on the

vacuum side of the upper edge of steps, a second negative peak with the largest amplitude, and subsequent

Friedel oscillations in the bulk. The magnitude of these peaks becomes insensitive to the terrace width

x~ for larger m. The outermost peak of n2 is located 2-3 a, u. farther on the vacuum side than the

main peak of nl in Fig. 5(b).

4.Q

5 38 ,5

~ 1-a

3.6

3.4

o.o o 2 o 4 o 6 o 8 1 1 Im

Fig. 7.

.O

34

32

~ :5

'~I

~ ~~ 30

28 O. o 0.2 0.4 o 6 o 8

1 Im

1.0

Calculated image plane position pl (a) and nonlinear response parameter a = 4~mp2 (b) for regularly

stepped jellium surfaces ( rs = 2). The step height xh is ao = 3.83 a. u. and the terrace width

xw is m'ao with 1~m~10.

- 103 - ( 19 )

Hiroshi ISHIDA

The planar average of the z component of the nonlinear polarization vector is given by

pz(z) T fo ; 1 f~ = dx J dz le2(x, z'). (10) From eq. (8),

" Pz f ::: dzpz (z) = Xz~zEapp (11)

From eqs. (10) and (11), we have

X - p2 zzz~ (27c)2

Instead of Xzzz, one often uses a dimensionless parameter a originally introduced by Rudnick and Stem

as a measure of the normal component of th SH surface current41) At low (o a is given by a ~ 4np

In Fig. 7 (b), we show the calculated a parameter for Al (r. = 2) vicinal surfaces with n = I and 1~m

< 10. Unlike the linear moment pl' p2 and therefore also a, reaches the flat surface value in the limit

of large m (see the open circle on the vertical axis). As m decreases a increases monotonically and

approaches its maxirnum value for m = 1, i. e. for the highest step density. In this limit, the positive

background forrns symmetrical cusps and the electronic charge can be polarized along both sides of the

step. In spite of the rather pronounced lateral corrugation of the ground state density profile, it is

remarkable that the absolute value of of a of for ( m. ?~) = (1, 1) is enhanced by only about 15 o/o relative

to the flat surface value. Qualitatively, the presence of steps implies an average electron density in the

selvedge region of one half of the bulk value. Considering the trend of p2 with rs22) such a density

decrease would give an enhancement of p2 by about 20 o/o (from 239 for rs=2 to about 290 for rs = 2.5).

The results of our microscopic calculations are consistent with this estimate. This enhancement is very

much smaller than that obtained for adsorbed alkali metal layers39). In the latter case, the large values

of a are caused by the much lower average density in the overlayer (factors of 8 to 16 Iess than that

of Al).

5. Summary

In summary, we used the density-functional approach to study the electronic properties of the regularly

stepped (vicinal) jellium surfaces as a function of step height, terrace width and bulk electron density.

In the first part of this article, we discussed the ground-state electronic properties. The results obtained

reproduce the observed linear dependence of the work function on the step density. The disturbance

of the electronic structure due to the presence of steps was found to be highly localized in the inunediate

vicinity of the steps. The analysis of the electrostatic potential and of the local density of states suggests

that the chemical behavior of the surface may be strongly modified near the step. In the second part

of this article, we we have studied the linear and nonlinear response of vicinal jellium surfaces to a static

electric field oriented normal to the surface. There is moderate enhancement of up to 15 o/o in the nonlinear

polarizability X... at high step densities. At lower step densities corresponding to terrace lengths greater

than m = 10, the enhancement is less than 4 o/o. In the recent measurements by Janz et al6).. SHG

from vicinal Al (OO1) suriaces was investigated for various step densities. For m ~ 10 they found that

the intensity of the SH Iight changes greatly when the incident electric field is reversed from ( E., E. )

to (-E.. E.). Among the six nonvanishing components, X.*., X..., and X,x' contribute to the asynunetry

of the SH intensity with respect to the incident angle. While we cannot yet address the magnitude

of these anisotropic polarizability components, we can, however, conclude from the present calculations

Electronic　Prope而es　of　Stepped　Metal　Surfaces

that　the　no㎜al　component　of　the　nonhnear　surface　polarizadon　is6πhαπ6θ4re1＆tive　to　the　flat　surface

value．Thus　the　near　cancellation　of　isotropic　and　anisotropic　components　observed　in　Ref。6must　be

cαused　by　very　large　anisotropic　surface　polarizabilides．

Ac㎞owledgement：

　　Most　part　of　this　ardcle　is　based　on　two　o亘gin＆1papers42・43）by　the　author　and　A。Liebsch，

Forschuロgszentrum　J廿Hch．

Re£erences

1）K．Wandelt，Su㎡，Scl．251／252，387（1991）．

2）H．Wagner，in　Sρ励gθ7Tzα伽oヅMlo4θ7πPh卿65，Vol・85（Sp血ger，Berhn・1979）・

3）E．Pehlke　and　J．Tersoff，Phys．Rev。Lett．67，465（1991）；必’4。68，816（1992），

4）C．Roland　and　G．H．Gi㎞er，Phys．Rev．B46，13437（1992），

5）」．Frohn，M．Giesen，M．P㏄nsgen，J．F。Wolf，and　H，Ibach，Phys。Rev，Lett．67，3543（1991），

6）S．Janz，D．」．Bottomley，H．M．van　Drie1，and　R．S．T㎞sit，Phys・Rev・Lett・66，1201（1991）・

7）S．Janz，G．LUpke，and　H．M．van　Dhe1，Phys。Rev，B47，7494（1993）。

8）M．C．Desjonqueres　and　F．Cyrot－Lackmam，Solid　State　Commun，18，1127（1976）。

9）V．T．Rajan　and　L．M．Falikov，」．Phys．C9，2533（1976）．

10）G．S．Painter，P．J。Jennings，and　R　O．Jones，」。Phys。C8，L199（1975）．

11）T．Yamaguchi　and　N．Fujima，Surf。Sci。242，233（1991）・

12）L．L．Kesmodel　and　L．M．Falikov，Solid　State　Commun．16，1201（1975）．

13）M．D．Thompson　and　H。B・Huntington，Surf・Sci・116，522（1982）・

14）　J．S．Nelson　and　P。」．Feibe㎞an，Phys．Rev．Lett．68，2188（1992）。

15）R，Car　and　M．Parrinello，Phys．Rev．Lett．55，2471（1985）．

16）N。D．Lang＆nd　W．Kohn，Phys．Rev．B1，4555（1970），

17）B．Lang，R．W．Joyner，and　G．A．Somo均ai，Su㎡．Sci．30，440（1970）。

18）W．Kohn　and　L．J．Sham，Phys．Rev，140，A1133（1965），

19）」．E．Inglesfield，J．Phys．C14，3795（1981）．

20）　S。Bl廿gel，unpublished．

21）H．Ishid旦and　A．Liebsch，Phys．Rev．B42，5505（1990）

22）M，Weber　and　A．Liebsch，Phys．Rev．B35，4711（1986）1弼4．B36，6411（1987）．

23）A．Liebsch，Phys，Rev・B36，7378（1987）．

24）H．Ishida　and　A．Liebsch，Phys．Rev．B46，7153（1992）・

25）R．S．Sorbello，Solid　State　Commun．56，821（1985）。

26）R．Smoluchowski，Phys。Rev．60，661（1941）．

27）B．Krah1－Urban，E．A．Niekisch，and　H．Wagner，Su㎡．Sci。64，52（1977）．

28）K．Bes㏄ke　and　H．Wagner，Phys．Rev．B8，4597（1973）．

29）H．Bes㏄ke，B．Krahl－Urban，and　H．Wagner，Surf．ScL68，39（1977）。

30）A．L．Testoni　and　P．C．Stair，Surf．Sci。171，L491（1986）・

31）H．Ibach，Surf．Sci．53，444（1975）．

32）N．D．Lang，S．Holoway，and　J．K．Nφrskov，Surf．Sci。150，24（1985）。

33）P。」．Feibelman　and　D．R。Hamman，Phys．Rev．Lett。52，61（1984）．

34）」．E．Sipe・D．J．Moss，and　H．M．van　Dhel，Phys・Rev・B35，1129（1987）・

35〉A．Liebsch，Phys．Rev．Lett．61，1233（1988）．

36）A．Liebsch　and　W．L．Schaich，Phys．Rev．B40，5401（1989）・

37）R。Murphy，M．Yeganeh，K．J．Song，and　E．W．Plummer，Phys，Rev。Lett。63，318（1989）。

38）S．J＆nz，K．Pedersen　and　H。M，van　Dhel，Phys．Rev．B44，3421（1991）。

39）A。Liebsch，Phys・Rev．B40，3421（1989）。

一105一（21）

Hiroshi　ISHIDA

40）　P．J．Feibehnan，Prog．in　Su㎡．Sci．12，287（1982）．

41）J．Rudnick　and　E．A．Stem，Phys、Rev．B4，4274（1971）。

42）H，Ishida　and　A．Liebsch，Phys．Rev．B46，7153（1992）．

43）H，Ishida　and　A．Liebsch，Surf．Sci．297，106（1993）．

（22）一106㎝

electronic properties of stepped metal surfaces

Documents