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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 -
Electronic Properties of Stepped Metal Surfaces
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 -
Electronic Properties of Stepped Metal Surfaces
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 -
Electronic Properties of Stepped Metal Surfaces
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 -
Electronic Properties of Stepped Metal Surfaces
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.
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(22)一106㎝