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Linear and non-linear response of stepped metal surfaces to a static electric field
::“.‘:‘:‘.““‘: .:.:.: .:::::y:.:::::::,> :::::: ~,,,:.,,, .’ “.“’ . ...‘. . .. . . . .,.... .,.,., : ,, ,, “.‘.““’ k.‘. “.‘.‘. ‘i.‘.‘. ..... . . . .. . .,., ,,,,., ,,,,,, .. ..”....i.......:.:.:.:.~,~.,~~~,~.‘.‘.’.. ,,,,, ,.,..~,,,,,, ,:,::i:i:i:i: “” ““““’ ::z+.:.:.:< :.:.... “““’“’ ‘.‘.‘. .. .....,.... .,..... ..... ...~.,.,.,
Surface Science North-Holland
$>,:.:.:
297 (1993) 106-l 11
.....
surface scienc;
Linear and non-linear response of stepped metal surfaces to a static electric field H. Ishida College of Humanities and Sciences, Nihon University, Sakura-josui, Setagaya-ku, Tokyo 156, Japan
and A. Liebsch Institut fiir FestkGrperforschung, Received
22 June
1993; accepted
Forschungszentrum for publication
Jiilich, D-5170, Jiilich, Germany 22 July 1993
The linear and non-linear response of vicinal metal surfaces to a static electric field oriented normal to the surface is studied by performing self-consistent density functional calculations. Regularly stepped jellium surfaces corresponding to the electron density of Al are used to model these surfaces. The normal component of the polarization vector associated with the non-linear-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% at maximum.
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 the non-linear optical polarizabilities that vanish in the bulk remain finite at the surface [l]. 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 non-vanishing elements of non-linear 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 incidence angle [2]. The sensitivity of the SH intensity to steps was recently utilized to determine the phase diagram of Cu(lll) vicinal surfaces [3]. 0039-6028/93/$06.00
0 1993 - Elsevier
Science
Publishers
So far, the most realistic calculations of the surface SHG have been carried out for the onedimensional jellium model by applying the timedependent density functional theory [4,5]. In this case, there is only one non-trivial surface polarizability, xzzr, which is determined by the rapid variation of the normal component of the electric field in the vicinity of the surface. Nearly quantitative agreement was achieved between these calculations and the experiments obtained for flat Al surfaces [6,7]. An important next step is to extend these calculations to more complicated systems with density corrugations in the surface plane, in order to investigate the influence of the corrugations on xrzz and to determine the remaining components of the surface polarizability. In the present work we address the first of these issues and study the linear and non-linear response of stepped metal surfaces to a static electric field. These calculations can be regarded
B.V. All rights
reserved
H. Ishida, A. Liebsch / Response of vicinal metal surfaces to a static electric field
as a preceding stage to the considerably more involved dynamical non-linear response calculations. We represent the vicinal metal surfaces by regularly stepped jellium surfaces whose groundstate electronic properties we have investigated previously [8]. These calculations reproduce the experimentally observed linear dependence of the work function on the step density. The (Y component of the surface-induced SH polarization, Pa, is expressed in terms of the non-linear polarizabilities Xap, as
pa= c XapyE$y
2
PTY
where E, denotes the (Ycomponent of the external electric field. The z-axis is chosen as the surface normal and the x-axis is perpendicular to both the z-axis and the ledges running parallel to the steps. For flat jellium surfaces, all but Xzzr and Xxxr vanish for p-polarized light incident in the xz-plane whereas all the six components ((.u, p, y =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 enormous enhancement of the SH intensity [9]. It is therefore of great interest to determine the modification of XZZZdue to the presence of steps. Second, the evaluation of Xzzr in the static limit can be reduced to a set of ground-state calculations [lO,ll] 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 Al surfaces since w/or, cz 0.1 for the typical experimental conditions Co and wP denote the frequency of the incident light and the plasma frequency of the metal, respectively). Hartree atomic units are used throughout this paper. 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 sur-
107
face [8]. One period consists of a terrace of width x, and a ledge of height x,,, i.e., 12+(x, z) =Ee[z,(x)
zdx) =
xw(E-x)
-z],
)
(+
(2)
xh
where ir (> 0) is the density of the background charge, I= /m 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 jellium surfaces with rs = 2 which approximately corresponds to the free-electron density of Al. x, and xh are chosen as ma, and na,,, where a, = 3.83 a.u. is the lattice spacing of Al for the (001) plane. The ground-state calculations are performed within the local density approximation to the density functional theory [12]. The one-electron wave functions of the semi-infinite system are calculated using the embedding method of Inglesfield [13]. In the present case, the numerical problem is reduced to a two-dimensional one because of the translational symmetry in the edge direction. In order to apply a uniform electric field in the z-direction, we place a charge sheet in the vacuum at a distance well separated from the surface [ll]. The strength of the applied electric field is given by Eapp = 27ru, where u 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 expanded [lo] as n( x, a) = n,(x)
+ an,( 1) + &z*(x)
+ ...) (3)
where n,, n,, 12* are the ground-state, linear-induced, and second-order-induced charge densities, respectively. For each step geometry, we perform three self-consistent calculations with u = 0, a,, and -a,, where a,, is chosen such that the higherAorder terms in eq. (3) can be ignored
108
H. Ishida, A. Liebsch
/ Response of vicinal metal surfaces to a static electric field
(J%Pwas set equal to 1.5 X lop3 in the actual calculations). n, and nz are then obtained as Qx)
= $
[n(x,
aa) - n(x,
0
h(X)
=
+[.(vO)+n(x, za2
-2n(x,
-%)I
9
(4)
--go)
O)].
(5)
As discussed later, the dipole moments of these
-6?& ,
, ,Q, 10
0
induced densities,
20
X(a.u.1
p1=:/ ‘dx / mdz z nl(x), lo
, , , , ,
-cc
and p2=I
1I m dx /0 / -cc dz z nz( x),
are key parameters in discussing the linear and non-linear surface response. The direct numerical evaluation of these quantities is difficult because of the Friedel oscillations that ni and nz exibit in the interior of the bulk. As in the case of flat jellium surfaces [S] and adsorbed alkali-metal layers [14], the dynamical force sum rule [15] can be used to derive analytical formulae that relate p1 and pz to the partial dipole moments of n, and n2 only in the surface region. Following the derivation in refs. [5,14], we obtain in the static limit,
: i;j ----0
20
10 X(a.u.1 I'
1 II
I
p,=~~d~~~dzzn,(x) 1 +- 4rnl
‘dx _-m - dZ4i(X) Io /
X&+(x,-q-z>],
(6)
with i = 1, 2 and $i defined by 4i(x)=-2”z6i,i+/
dr’]rlr,,ni(x’)Y
(7)
where the first term is the applied linear potential and the second is the Hartree potential associated with the induced charge ni. In fig. la we show the contour map of the ground state electron density no for Al (r, = 2) with (m, n) = (5, 1) in the xz-plane. Because of
0
10
X(a.u.)
20
Fig. 1. Contour maps of charge densities for the regularly stepped jellium surface with (m, n) = (5, 1) and rS = 2. The dotted lines indicate the profile of the background positive charge. (a) Ground-state charge “a. The contour spacing is 3 x 10P3 a.u.; (b) linear-induced charge n,. Solid, dashed, and dot-dashed lines correspond to positive, negative, and zero values of nt, respectively. The contour spacing is 0.05 a.u. (c) Nonlinear-induced charge nz. The contour spacing is 7 a.u.
the efficient metallic screening, the electron density distributes itself almost perfectly in one-dimensional fashion in most parts of the terrace
109
H. Ishida, A. Liebsch / Response of vi&al metaf surfaces to a static electric field
X(a.u.1 Fig. 2. Contour map of the linear change in the Coulomb potential, &, for the regularly stepped jellium surface with (mt n) = (5,lf and s-~= 2. The dashed and dot-dashed lines correspond to negative and zero v&es of &t. Tbe contour spacing is 10 ax.
surface. The charge redistribution in the immediate vicinity of steps on the other hand, leads to a work-function decrease that is proportional to l/r for m r 3. More details on the ground-state electronic structure are discussed in ref. [8]. Fig. lb shows the contour map of the linear-induced charge ~zt. According to eq. (4), n1 has a unit charge when integrated over unit area. The induced charge is located mostly outside the positive background charge n, and exhibits a Friedel oscillation toward the interior of the metal. A large peak appears in n, near the upper edge of the step. The height of this peak (N 0.5) is insensitive to the terrace width x, = ma, for m 2 3. The shape of the peak is strongly as~metric; IE~ decreases slowly along the upper terrace, while it drops steeply on the lower terrace side.
3.4 ’ 0.0
0.2
0.4
0.6
0.8
I 1.0
In fig. 2 we show the corresponding linear change in the Coulomb potential, st)i, defined by eq. (7). 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 417 and 0 in the vacuum and in the interior of the metal, respectively. In contrast to the associated induced charge in fig. lb, the contour lines of #i very smoothly follow the surface profile. For flat jellium surfaces, it was shown that & = 0 at the edge of the positive background charge [ 161. By rewriting the sum rule eq. (61, we obtain the corresponding formula
which is applicable not only to the present stepped surfaces, but also to positive background charges with any surface profile. One may expect that +t takes a constant value along the jellium edge of the terrace surface for a larger terrace width. If this is the case, eq. 18) indicates that this value must be zero. Actually, as seen from fig. 2, the contour line corresponding to +i = 0 (dot-dashed line) coincides with the profile of the positive background charge in most parts of the terrace surface. This result also demonstrates that our numerical calculation is highly accurate (note that the sum rule eq. (6) is not enforced to hold in the iteration procedure toward self-consistency). In fig. 3a the calculated centroid of the linearinduced charge, pl, is shown for monatomic steps (n = 1) for various terrace lengths (m ranges from 1 to 10). p1 defines the position of the classical
28’ 0.0
J 0.2
0.4
0.6
0.8
1.0
l/m Fig. 3. (a) Calculated image-plane position p, and (b) non-linear response parameter a = 47ip, for regularly stepped jellium surfaces (rs = 2). The step height x,, is a0 = 3.83 au. and the terrace width n, is ma, with 1 I m I 10.
110
H. Ishida, A. Lie&h
/ Response of vicinal metal surfaces to a static electric field
image plane in the static limit 1171. 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, p1 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, p1 reaches the as~ptotic value p,(m) = OSa, + z0 = 3.48 au. where no = 3.83 a.u. is the step height as discussed above and z,, = 1.57 a.u. is the static image-plane position for flat jellium with rS = 2 [l?]. Next, we discuss the non-linear response properties. Fig. lc shows the contour map of the second-order-induced charge n2 for Al (f; = 2) with (m, n) = (5, 1) in the xz-plane. As seen from eq. (5), n2 has a 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 n, in fig. lb. The planar average of the z-component of the non-linear polarization vector is given by p,(z)
= fk’
dxlm dz’nz(x,
z’).
From eq. (1) P, = Irn dzpz(z) --m
=xzrzE&.
(10)
From eqs. (9) and (10) we have X
**=
=-
P2
(2+
*
Instead of xZZZ, one often uses a dimensionless parameter a originally introduced by Rudnick and Stern as a measure of the normal component of the SH surface current 1181. At low w, a is given by a = 4Ep, [93. In fig. 3b, we show the
calculated a parameter for Al (rS = 2) vicinal surfaces with n = 1 and 1 5 m < 10. Unlike the linear moment pl, pz and, therefore, also a, reaches the flat surface value in the limit of large m (see the solid square on the vertical axis). As m decreases, a increases monotonically and approaches its maximum value for m = 1, i.e. for the highest step density. In this limit, the positive background forms 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 ra for Cm, n) = (1, 1) is enhanced by only about 15% 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 r, [lo], such a density decrease would give an enhancement of pz by about 20% (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 layers [9]. In the latter case, the Iarge values of a are caused by the much lower average density in the overlayer (factors of 8 to 16 less than that of Al). In summary, using the density functional theory we have studied the linear and non-linear response of vicinal jellium surfaces to a static electric field oriented normal to the surface. There is a moderate enhancement of up to 15% in the non-linear polarizability xZZZ at high step densities. At lower step densities, corresponding to terrace lengths greater than m = 10, the enhancement is less than 4%. In the recent measurements by Janz et al. [2], SHG from vicinal Al(OO1) surfaces was investigated for various step densities. For m 2 10 they found that the intensity of the SH light changes greatly when the incident electric field is reversed from (E,, E;) to ( -E,, E, 1. Among the six non-vanishing components, xXxX, xxzZ, and xrxz contribute to the asymmetry of the SH intensity with respect to the incidence angle. While we cannot yet address the magnitude of these anisotropic polarizability components, we can, however, conclude from the
H. Ishida, A. Liebsch / Response of vicinal metal surfaces to a static electric field
present calculations that the normal component of the non-linear surface polarization is enhanced relative to the flat surface value. Thus the near cancellation of isotropic and anisotropic components observed in ref. [2] must be caused by very large anisotropic surface polarizabilities. The evaluation of these elements for stepped jellium surfaces will be discussed in a forthcoming publication. One of us (H.I.) would like to thank the Alexander von Humboldt foundation for its support during his stay in Jiilich.
References 111J.E. Sipe, D.J. Moss and H.M. van Driel, Phys. Rev. B 35 (1987) 1129.
111
[2] S. Janz, D.J. Bottomley, H.M. van Driel and R.S. Timsit, Phys. Rev. Lett. 66 (1991) 1201. 131 S. Janz, G. Liipke and H.M. van Driel, Phvs. Rev. B 47 (1993) ;494. _ [4] A. Liebsch, Phys. Rev. Lett. 61 (1988) 1233. [5] A. Liebsch and W.L. Schaich Phys. Rev. B 40 (1989) 5401. [61 R. Murphy, M. Yeganeh, K.J. Song and E.W. Plummer, Phys. Rev. Lett. 63 (1989) 318. [7] S. Janz, K. Pedersen and H.M. van Driel, Phys. Rev. B 44 (1991) 3421. [8] H. Ishida and A. Liebsch, Phys. Rev. B 46 (1992) 7153. [9] A. Liebsch, Phys. Rev. B 40 (1989) 3421. [lo] M. Weber and A. Liebsch, Phys. Rev. B 35 (1986) 4711; 36 (1987) 6411. [ll] H. Ishida and A. Liebsch, Phys. Rev. B 42 (1990) 550. [12] W. Kohn and L.J. Sham, Phys. Rev. 140 (1965) A 1133. [13] J.E. Inglesfield, J. Phys. C 14 (1981) 3795; G.C. Aers and J.E. Inglesfield, Surf. Sci. 217 (1989) 367. [14] R. Ishida and A. Liebsch, Phys. Rev. B 45 (1992) 6171. [15] R.S. Sorbello, Solid State Commun. 56 (1985) 821. [16] A. Liebsch, Phys. Rev. B 36 (1987) 7378. [17] N.D. Lang and W. Kohn, Phys. Rev. B 7 (1973) 3541. [181 J. Rudnick and E.A. Stern, Phys. Rev. B 4 (1971) 4274.
Surface Science North-Holland
$>,:.:.:
297 (1993) 106-l 11
.....
surface scienc;
Linear and non-linear response of stepped metal surfaces to a static electric field H. Ishida College of Humanities and Sciences, Nihon University, Sakura-josui, Setagaya-ku, Tokyo 156, Japan
and A. Liebsch Institut fiir FestkGrperforschung, Received
22 June
1993; accepted
Forschungszentrum for publication
Jiilich, D-5170, Jiilich, Germany 22 July 1993
The linear and non-linear response of vicinal metal surfaces to a static electric field oriented normal to the surface is studied by performing self-consistent density functional calculations. Regularly stepped jellium surfaces corresponding to the electron density of Al are used to model these surfaces. The normal component of the polarization vector associated with the non-linear-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% at maximum.
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 the non-linear optical polarizabilities that vanish in the bulk remain finite at the surface [l]. 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 non-vanishing elements of non-linear 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 incidence angle [2]. The sensitivity of the SH intensity to steps was recently utilized to determine the phase diagram of Cu(lll) vicinal surfaces [3]. 0039-6028/93/$06.00
0 1993 - Elsevier
Science
Publishers
So far, the most realistic calculations of the surface SHG have been carried out for the onedimensional jellium model by applying the timedependent density functional theory [4,5]. In this case, there is only one non-trivial surface polarizability, xzzr, which is determined by the rapid variation of the normal component of the electric field in the vicinity of the surface. Nearly quantitative agreement was achieved between these calculations and the experiments obtained for flat Al surfaces [6,7]. An important next step is to extend these calculations to more complicated systems with density corrugations in the surface plane, in order to investigate the influence of the corrugations on xrzz and to determine the remaining components of the surface polarizability. In the present work we address the first of these issues and study the linear and non-linear response of stepped metal surfaces to a static electric field. These calculations can be regarded
B.V. All rights
reserved
H. Ishida, A. Liebsch / Response of vicinal metal surfaces to a static electric field
as a preceding stage to the considerably more involved dynamical non-linear response calculations. We represent the vicinal metal surfaces by regularly stepped jellium surfaces whose groundstate electronic properties we have investigated previously [8]. These calculations reproduce the experimentally observed linear dependence of the work function on the step density. The (Y component of the surface-induced SH polarization, Pa, is expressed in terms of the non-linear polarizabilities Xap, as
pa= c XapyE$y
2
PTY
where E, denotes the (Ycomponent of the external electric field. The z-axis is chosen as the surface normal and the x-axis is perpendicular to both the z-axis and the ledges running parallel to the steps. For flat jellium surfaces, all but Xzzr and Xxxr vanish for p-polarized light incident in the xz-plane whereas all the six components ((.u, p, y =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 enormous enhancement of the SH intensity [9]. It is therefore of great interest to determine the modification of XZZZdue to the presence of steps. Second, the evaluation of Xzzr in the static limit can be reduced to a set of ground-state calculations [lO,ll] 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 Al surfaces since w/or, cz 0.1 for the typical experimental conditions Co and wP denote the frequency of the incident light and the plasma frequency of the metal, respectively). Hartree atomic units are used throughout this paper. 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 sur-
107
face [8]. One period consists of a terrace of width x, and a ledge of height x,,, i.e., 12+(x, z) =Ee[z,(x)
zdx) =
xw(E-x)
-z],
)
(+
(2)
xh
where ir (> 0) is the density of the background charge, I= /m 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 jellium surfaces with rs = 2 which approximately corresponds to the free-electron density of Al. x, and xh are chosen as ma, and na,,, where a, = 3.83 a.u. is the lattice spacing of Al for the (001) plane. The ground-state calculations are performed within the local density approximation to the density functional theory [12]. The one-electron wave functions of the semi-infinite system are calculated using the embedding method of Inglesfield [13]. In the present case, the numerical problem is reduced to a two-dimensional one because of the translational symmetry in the edge direction. In order to apply a uniform electric field in the z-direction, we place a charge sheet in the vacuum at a distance well separated from the surface [ll]. The strength of the applied electric field is given by Eapp = 27ru, where u 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 expanded [lo] as n( x, a) = n,(x)
+ an,( 1) + &z*(x)
+ ...) (3)
where n,, n,, 12* are the ground-state, linear-induced, and second-order-induced charge densities, respectively. For each step geometry, we perform three self-consistent calculations with u = 0, a,, and -a,, where a,, is chosen such that the higherAorder terms in eq. (3) can be ignored
108
H. Ishida, A. Liebsch
/ Response of vicinal metal surfaces to a static electric field
(J%Pwas set equal to 1.5 X lop3 in the actual calculations). n, and nz are then obtained as Qx)
= $
[n(x,
aa) - n(x,
0
h(X)
=
+[.(vO)+n(x, za2
-2n(x,
-%)I
9
(4)
--go)
O)].
(5)
As discussed later, the dipole moments of these
-6?& ,
, ,Q, 10
0
induced densities,
20
X(a.u.1
p1=:/ ‘dx / mdz z nl(x), lo
, , , , ,
-cc
and p2=I
1I m dx /0 / -cc dz z nz( x),
are key parameters in discussing the linear and non-linear surface response. The direct numerical evaluation of these quantities is difficult because of the Friedel oscillations that ni and nz exibit in the interior of the bulk. As in the case of flat jellium surfaces [S] and adsorbed alkali-metal layers [14], the dynamical force sum rule [15] can be used to derive analytical formulae that relate p1 and pz to the partial dipole moments of n, and n2 only in the surface region. Following the derivation in refs. [5,14], we obtain in the static limit,
: i;j ----0
20
10 X(a.u.1 I'
1 II
I
p,=~~d~~~dzzn,(x) 1 +- 4rnl
‘dx _-m - dZ4i(X) Io /
X&+(x,-q-z>],
(6)
with i = 1, 2 and $i defined by 4i(x)=-2”z6i,i+/
dr’]rlr,,ni(x’)Y
(7)
where the first term is the applied linear potential and the second is the Hartree potential associated with the induced charge ni. In fig. la we show the contour map of the ground state electron density no for Al (r, = 2) with (m, n) = (5, 1) in the xz-plane. Because of
0
10
X(a.u.)
20
Fig. 1. Contour maps of charge densities for the regularly stepped jellium surface with (m, n) = (5, 1) and rS = 2. The dotted lines indicate the profile of the background positive charge. (a) Ground-state charge “a. The contour spacing is 3 x 10P3 a.u.; (b) linear-induced charge n,. Solid, dashed, and dot-dashed lines correspond to positive, negative, and zero values of nt, respectively. The contour spacing is 0.05 a.u. (c) Nonlinear-induced charge nz. The contour spacing is 7 a.u.
the efficient metallic screening, the electron density distributes itself almost perfectly in one-dimensional fashion in most parts of the terrace
109
H. Ishida, A. Liebsch / Response of vi&al metaf surfaces to a static electric field
X(a.u.1 Fig. 2. Contour map of the linear change in the Coulomb potential, &, for the regularly stepped jellium surface with (mt n) = (5,lf and s-~= 2. The dashed and dot-dashed lines correspond to negative and zero v&es of &t. Tbe contour spacing is 10 ax.
surface. The charge redistribution in the immediate vicinity of steps on the other hand, leads to a work-function decrease that is proportional to l/r for m r 3. More details on the ground-state electronic structure are discussed in ref. [8]. Fig. lb shows the contour map of the linear-induced charge ~zt. According to eq. (4), n1 has a unit charge when integrated over unit area. The induced charge is located mostly outside the positive background charge n, and exhibits a Friedel oscillation toward the interior of the metal. A large peak appears in n, near the upper edge of the step. The height of this peak (N 0.5) is insensitive to the terrace width x, = ma, for m 2 3. The shape of the peak is strongly as~metric; IE~ decreases slowly along the upper terrace, while it drops steeply on the lower terrace side.
3.4 ’ 0.0
0.2
0.4
0.6
0.8
I 1.0
In fig. 2 we show the corresponding linear change in the Coulomb potential, st)i, defined by eq. (7). 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 417 and 0 in the vacuum and in the interior of the metal, respectively. In contrast to the associated induced charge in fig. lb, the contour lines of #i very smoothly follow the surface profile. For flat jellium surfaces, it was shown that & = 0 at the edge of the positive background charge [ 161. By rewriting the sum rule eq. (61, we obtain the corresponding formula
which is applicable not only to the present stepped surfaces, but also to positive background charges with any surface profile. One may expect that +t takes a constant value along the jellium edge of the terrace surface for a larger terrace width. If this is the case, eq. 18) indicates that this value must be zero. Actually, as seen from fig. 2, the contour line corresponding to +i = 0 (dot-dashed line) coincides with the profile of the positive background charge in most parts of the terrace surface. This result also demonstrates that our numerical calculation is highly accurate (note that the sum rule eq. (6) is not enforced to hold in the iteration procedure toward self-consistency). In fig. 3a the calculated centroid of the linearinduced charge, pl, is shown for monatomic steps (n = 1) for various terrace lengths (m ranges from 1 to 10). p1 defines the position of the classical
28’ 0.0
J 0.2
0.4
0.6
0.8
1.0
l/m Fig. 3. (a) Calculated image-plane position p, and (b) non-linear response parameter a = 47ip, for regularly stepped jellium surfaces (rs = 2). The step height x,, is a0 = 3.83 au. and the terrace width n, is ma, with 1 I m I 10.
110
H. Ishida, A. Lie&h
/ Response of vicinal metal surfaces to a static electric field
image plane in the static limit 1171. 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, p1 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, p1 reaches the as~ptotic value p,(m) = OSa, + z0 = 3.48 au. where no = 3.83 a.u. is the step height as discussed above and z,, = 1.57 a.u. is the static image-plane position for flat jellium with rS = 2 [l?]. Next, we discuss the non-linear response properties. Fig. lc shows the contour map of the second-order-induced charge n2 for Al (f; = 2) with (m, n) = (5, 1) in the xz-plane. As seen from eq. (5), n2 has a 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 n, in fig. lb. The planar average of the z-component of the non-linear polarization vector is given by p,(z)
= fk’
dxlm dz’nz(x,
z’).
From eq. (1) P, = Irn dzpz(z) --m
=xzrzE&.
(10)
From eqs. (9) and (10) we have X
**=
=-
P2
(2+
*
Instead of xZZZ, one often uses a dimensionless parameter a originally introduced by Rudnick and Stern as a measure of the normal component of the SH surface current 1181. At low w, a is given by a = 4Ep, [93. In fig. 3b, we show the
calculated a parameter for Al (rS = 2) vicinal surfaces with n = 1 and 1 5 m < 10. Unlike the linear moment pl, pz and, therefore, also a, reaches the flat surface value in the limit of large m (see the solid square on the vertical axis). As m decreases, a increases monotonically and approaches its maximum value for m = 1, i.e. for the highest step density. In this limit, the positive background forms 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 ra for Cm, n) = (1, 1) is enhanced by only about 15% 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 r, [lo], such a density decrease would give an enhancement of pz by about 20% (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 layers [9]. In the latter case, the Iarge values of a are caused by the much lower average density in the overlayer (factors of 8 to 16 less than that of Al). In summary, using the density functional theory we have studied the linear and non-linear response of vicinal jellium surfaces to a static electric field oriented normal to the surface. There is a moderate enhancement of up to 15% in the non-linear polarizability xZZZ at high step densities. At lower step densities, corresponding to terrace lengths greater than m = 10, the enhancement is less than 4%. In the recent measurements by Janz et al. [2], SHG from vicinal Al(OO1) surfaces was investigated for various step densities. For m 2 10 they found that the intensity of the SH light changes greatly when the incident electric field is reversed from (E,, E;) to ( -E,, E, 1. Among the six non-vanishing components, xXxX, xxzZ, and xrxz contribute to the asymmetry of the SH intensity with respect to the incidence angle. While we cannot yet address the magnitude of these anisotropic polarizability components, we can, however, conclude from the
H. Ishida, A. Liebsch / Response of vicinal metal surfaces to a static electric field
present calculations that the normal component of the non-linear surface polarization is enhanced relative to the flat surface value. Thus the near cancellation of isotropic and anisotropic components observed in ref. [2] must be caused by very large anisotropic surface polarizabilities. The evaluation of these elements for stepped jellium surfaces will be discussed in a forthcoming publication. One of us (H.I.) would like to thank the Alexander von Humboldt foundation for its support during his stay in Jiilich.
References 111J.E. Sipe, D.J. Moss and H.M. van Driel, Phys. Rev. B 35 (1987) 1129.
111
[2] S. Janz, D.J. Bottomley, H.M. van Driel and R.S. Timsit, Phys. Rev. Lett. 66 (1991) 1201. 131 S. Janz, G. Liipke and H.M. van Driel, Phvs. Rev. B 47 (1993) ;494. _ [4] A. Liebsch, Phys. Rev. Lett. 61 (1988) 1233. [5] A. Liebsch and W.L. Schaich Phys. Rev. B 40 (1989) 5401. [61 R. Murphy, M. Yeganeh, K.J. Song and E.W. Plummer, Phys. Rev. Lett. 63 (1989) 318. [7] S. Janz, K. Pedersen and H.M. van Driel, Phys. Rev. B 44 (1991) 3421. [8] H. Ishida and A. Liebsch, Phys. Rev. B 46 (1992) 7153. [9] A. Liebsch, Phys. Rev. B 40 (1989) 3421. [lo] M. Weber and A. Liebsch, Phys. Rev. B 35 (1986) 4711; 36 (1987) 6411. [ll] H. Ishida and A. Liebsch, Phys. Rev. B 42 (1990) 550. [12] W. Kohn and L.J. Sham, Phys. Rev. 140 (1965) A 1133. [13] J.E. Inglesfield, J. Phys. C 14 (1981) 3795; G.C. Aers and J.E. Inglesfield, Surf. Sci. 217 (1989) 367. [14] R. Ishida and A. Liebsch, Phys. Rev. B 45 (1992) 6171. [15] R.S. Sorbello, Solid State Commun. 56 (1985) 821. [16] A. Liebsch, Phys. Rev. B 36 (1987) 7378. [17] N.D. Lang and W. Kohn, Phys. Rev. B 7 (1973) 3541. [181 J. Rudnick and E.A. Stern, Phys. Rev. B 4 (1971) 4274.