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Local-field effect near the surface of dipolar lattices
Solid State Communications, Vol. 33, Pp. 645—648. Pergamon Press Ltd. 1980. Printed in Great Britain. LOCAL-FIELD EFFECT NEAR THE SURFACE OF DIPOLAR LATTICES N. Kar and A. Bagchi Department of Physics and Astronomy and Center for Theoretical Physics, University of Maryland, College Park, MD 20742, U.S.A. (Received 23 July 1979 byf. Tauc) The local-field correction near the surface of a dipolar crystal has been studied by explicitly computing the non-retarded electric field produced by a finite array of classical point dipoles induced by a time-varying electric field. Crystals of simple cubic, f.c.c. and b.c.c. structures are considered, and results presented for the self-consistent dipole moment and the local electric field at a lattice point. The utility of the method and its possible applications are discussed. THE SELF-CONSISTENT electric field at a lattice point of an infinite lattice of polarizable molecules (without permanent dipole moments), under the influence of an external electric field, has been well-studied in the past [1,2], and leads to the familiar Lorentz—Lorenz relation [3] for the refractive index. The object of this paper is to calculate the self-consistent electric field near the surface of a lattice of dipoles under the influence of an external electric field. The motivation of the work comes from the possibility of using our method of calculation to study a wide variety of phenomena, e.g. light propagation near a semiconductor surface, importance of local.fields in photoemission from insulators and adsorbate-covered metals, etc. We shall consider a slab geometry of a finite number of lattice planes which are, however, infinite in extent parallel to the surface. We imagine molecules to be located at discrete lattice sites corresponding to simple cubic (s.c.), face-centered cubic (f.c.c.) or body-centered cubic (b.c.c.) structure. The molecules constituting the slab are polarizable by an applied electric field which may be decomposed into components parallel and normal to the lattice planes. For simplicity, the polarizabiity tensor will be assumed to be diagonal, with coordinate axes coinciding with the usual crystal axes. To calculate the self-consistent field, one has to, as a first step, compute the field due to all the dipoles induced in the slab by the applied electric field. We regard these as classical point dipoles, thus ignoring quantum-mechanical effects [4, 5]. Since the polarizability tensor is assumed to be diagonal, all the dipoles lying on a lattice plane would be oriented in the same direction and have the same magnitude. The dipole moment may vary, however, from one plane to another. The basic problem is thus of computing the field at a given point from an infmite two-dimensional lattice of dipoles of equal moment, all oriented in the same direction. 645
In what follows we shall restrict ourselves, for simplicity, to the case where the surface of the slab coincides with the [001] face of the appropriate crystal structure. With suitable modification, our procedure can be generalized to other faces as well. We are interested, in general, in the field at a lattice point. Considering the origin to be at the lattice point at which the field is being evaluated, the contribution of all the other dipoles to the electric field, which we call EthP, is given by =
~‘
k,1,m
3~k1mT~m)~1m ~lmPk1m
(I)
r~,m
where the prime indicates that the term having k = I = m = 0 must be excluded. Here PkIm denotes the dipole moment of the molecule located at the lattice point rkzm [= a(k, I, m)], a being the lattice constant and k, 1, m the indices describing the lattice point. Note that while the indices will be integers for the s.c. structure, they may be half-integral for the other two structures. In the bulk, there have been two ways of performing sums of this type. One method is to evaluate part of the sum directly by summation inside a finite region and approximate the rest by integration with a uniform distribution of dipoles of appropriate density. Such a technique was used by several authors [6—8]in the study of cohesion and van der Waals interaction energy at a crystal surface. The alternate approach is to convert the discrete lattice sum into a highly convergent sum analytically, and then evaluate the latter up to a finite number of terms. This technique was used by Nijboer and dé Wette [9], who transform the sum into the reciprocal space, and van der Hoff and Benson [10]. It was also used by Mahan [11] in his calculation of the van der Waals energy in solids using both instantaneous Coulomb and retarded interactions. Our procedure for evaluating the sum (1) is to carry out the summation plane by plane parallel to the
646
LOCAL-FIELD EFFECT NEAR THE SURFACE OF DIPOLAR LATTICES
Table 1. Values of~~ Plane index
that to find the dipolar field at a lattice point in the slab problem, we must consider only a finite number of lattice planes. Guided by Table 1, we conclude that
s.c.
f.c.c.
b.c.c.
_____________________________________________________
0
9.0336 0.3275 0.0005 0.0000 0.0000 0.0000
2 3 4
— —
6.3877 1.0107 0.0160 0.0000 0.0000 0.0000
—4.5 168 2.0882 0.1638 0.0062 0.0003 0.0000 —
—
—
in equations (4) one needs to retain only terms for which v~~N, where N = l(s.c.), 2(f.c.c.) and 3(b.c.c.). This is indeed an approximation, but it enormously simplifies the task of finding the self-consistent or local electric field. The familiar identity ~ = —8ir/3, which follows directly [9] from Lorentz’s formula for internal field in the bulk of a cubic lattice, is satisfied by our approximation to better than 0.02%. Let us now consider a slab of L lattice planes (labeled by 1, 2, L) under the influence of a spatially uniform field in the z-direction, E0 = E0~.The
...
surface of the slab. We first note that the dipole moment depends only on the index m specifying a lattice plane, i.e. Pklm = Pm~Next we consider the applied field to be in the direction of the surface normal (call it z), so that p~= (0, 0,Pm). It follows quite simply that only the z-component of Ed~Pis then nonzero. Noting that rk,m = a(k, 1, m), we have 2 (k2 + 12 + m2) ‘~dip,Z = ~ Pm a~~c 3m(k2 + 12 + rn2)512 (2) —
Let us introduce two notations for the sake of compactness. Label the lattice planes by an integer v such that v = m(s.c.) and v = 2m (f.c.c. and b.c.c.). Also define as P~,the local dipole moment density on a lattice plane, i.e. 3, ~
=
Vol. 33, No.6
l(s.c.), 4(f.c.c.), 2(b.c.c.).
(3)
= ~PmIa Equation (2) may then be rewritten as
,
local electric field at a lattice point on the pth plane is then given by ~ = E0 +~~ (5) where the sum over v is restricted by 1 ~ v ~ L and p vi ~N. Let a be the polarizability of each molecule in the z-direction and F the volume polarizability, i.e. F = ~a/a3. If the applied field varies in time with a frequency w, both a and F will depend on o.~and change sign near a resonance. The self-consistency condition for the local dipole moment density, viz. PIA = FE~JC,Z, now yields the matrix equation —
~.MILVPV
FE
=
0,
p
=
1,2,... ,L,
(6a)
“
where = (1 I’~o)&~v—F~l,L_vI(1 ~ (6b) Although the matrixMis an (L x L) matrix, only a small fraction of its terms is finite because of the nature of the ~‘s (cf. Table 1). Inverting the matrix M where it is permissible gives us the dipole moment density on each plane of the slab. For comparison, the uniform dipole moment density in the bulk for the infinite-lattice problem is given, within our approxi—
Ethpz
=
~ ~VPV,
(4a)
v
where ~ comes from summation over a single lattice plane. When the applied field is parallel to the surface, lying along the x-axis, for example, one similarly finds =
~ ~
~
=
—
~
(4b)
mation, by
p
vClearly = 0 it ~, is depends apart from only sign on the andmagnitude geometricof factors v, and for —
—
—
—
pb
=
FE 0
essentially the sum first evaluated by Topping [12]. For other, non-vanishing v’s, the ~‘s can be evaluated for the different crystal structures by either of the two methods mentioned above. In practice, the technique outlined by van der Hoff and Benson [10] gives a highly convergent series whose accuracy is easy to control. The ~‘s computed by that method are given in Table 1 for various plane indices v. We note again that for f.c.c. and b.c.c. structures, odd values of v correspond to lattice planes at half-integral lattice distances. An outstanding feature of Table 1 is that the ~‘s go extremely rapidly to zero as v increases. This means
/
Ii
—
~
].
(7)
We have carried out the explicit inversion of the matrix to obtain P,2 for a 21-layer slab problem for s.c., f.c.c. and b.c.c. lattices. We regard F as a free parameter that can be varied in the problem [13]. The inversion procedure breaks down for a range of values of F for which det MI = 0. For these values of F, the system can support spontaneous polarization modes, which are the counterparts, within our approximation, of the excitonic normal modes of an infinite lattice as discussed by Hopfield [14] and Mahan [11]. Outside of
Vol. 33, No. 6
LOCAL-FIELD EFFECT NEAR THE SURFACE OF DIPOLAR LATTICES .6
647
.6
(a)
(b)
4
4
12
12 F~O.5
r~o5
Layer Number (~)
Layer Number (M)
Fig. 1. (a) Plots of the dipole-moment density on a lattice plane p vs the plane index, for selected values of the volume polarizability I’, for the simple cubic structure. The dipole-moment density is normalized to the bulk moment density ~b for the uniformly polarized infinite lattice [cf. equation (7)]. (b) The dipolar contribution to the local electric field on a lattice plane p, in the simple cubic case, normalized to the dipolar field in the uniformly polarized bulk system, plotted against the plane index for selected values of the volume polarizability F. (.6
(.6
(a)
~ ~F0.25
)b)
r~-O.25
~
O2~
Layer Number (~)
Layer Number (p.)
Fig. 2. (a) Plots of the dipole-moment density on a lattice plane p vs the plane index, for selected values of the volume polarizability F, for the face-centered cubic structure. The dipole-moment density is normalized to the bulk moment density ~b for the uniformly polarized infinite lattice [cf. equation (7)]. (b) the dipolar contribution to the local electric field on a lattice plane p, in the f.c.c. case, normalized to the dipolar field in the uniformly polarized bulk system, plotted against the plane index for selected values of the volume polarizability F. this range [which, for the s.c. system, is (~ + 2~i)_1 ~ F ~ (~— 2~~)’} our method gives well-behaved results. Figure 1(a) shows a plot of ~ vs the lattice-plane index p for selected values of F in the s.c. case. We emphasize that only points corresponding to integral values of p are meaningful, and that we have drawn lines through them as an aid to the eye only. For most values off, the local dipole moment density remains close to the bulk value, differing from it by only a few per cent on the surface layer. However, as F approaches the critical value for the excitation of the uniform polarization mode, Fl/Pb differs dramatically from unity,
becoming as low as 20%. When 0 > F>
2~ 1, 1j 1, exhibits an oscillatory behavior. As Fig. 1(a) shows, ~1/~b can be substantially larger than I when F is close to the upper edge for the band of polarization modes. To obtain a better perspective of our results, we have computed the dipolar contribution to the local electric field on a lattice plane, i.e. ~ = ~ ~ and compared it to the bulk result for a uniform distribution of dipole moments, viz. ~ = (~o+ Figure 1(b) shows the ratio of the two for the s.c. structure as a function of the plane index, again with selected Fl/Pb is greater than unity and P,2/P
(~
—
648
LOCAL-FIELD EFFECT NEAR THE SURFACE OF DIPOLAR LATTICES
values of F. The ratio is seen to be close to unity, except when I” is close to the edges of the polarizationmode band, on either side of which the dipolar field on the surface layer greatly diverges from the bulk value. It is interesting to note, however, that the curves of Figs. 1(a) and (b) corresponding to the same value of F are quite similar, leading us to the approximate identification ~ (81r/3)P,2. This would tend to justify the model used by some investigators [15] studying the propagation of light near a semiconductor surface that the local field near the surface may be written as E + 4irP(z)/3, in terms of a z-dependent dipole.moment density. Unfortunately, though, the situation is found to be quite different for f.c.c. and b.c.c. structures. As representative examples, we have plotted ~M/~b and E~jP.Z/E~iP,Zas functions of p, in the f.c.c. case with certain choices of F, in Figs. 2(a) and (b) respectively. The general features are the same as for the s.c. lattice, viz, the ratios are close to unity except when F approaches its critical range for the excitation of spontaneous polarization modes. On one side of the range, the ratios oscillate around unity, but on the other side they change monotonically with p, the changes being most noticeable near the edges of the polarization band. For this system, however, the identification ~ —(87T/3)P,2 cannot be made without entailing significant errors (‘~20%). This calls into question the prescription for using a local-dipole-moment-density model for treating the local field effect near a surface. We note that our results near the surface are not affected qualitatively on going to a 31-layer slab. In summary, then, we have presented the first results of the local electric field near the surface of a dipolar lattice, in the classical, point-dipole approximation, under the influence of an external electric field. Although the external field was taken to lie along the —
—
—
surface extension toona field an arbitrary directionnormal, is straightforward usinginequation (4b). Our calculation predicts large differences of the local field at the surface from the bulk result for certain values of the volume polarizability F. It will be of interest to see if the effect can be observed experimentally, either in photoemission or in reflectance studies, where [‘(w) can be varied continuously and often rapidly by changing the frequency of light. Our method may be used to
Vol. 33, No.6
define a local dielectric constant on a plane-by-plane basis near the surface, and this may, in turn, be used in a numerical study of light propagation. The method may also be extended to investigate the effect of local fields on photoemission from an ordered overlayer of absorbates on a metal surface, within some simple model of the dynamical response of the metal to change fluctuations. Suggestions about the importance of the local electric field in the chemisorption problem have indeed been offered [16, 17] in connection with surface enhanced Raman spectroscopy. Acknowledgement This work was partially supported by NSF under Grant No. DMR 76-82128. One of us (A.B.) wishes to acknowledge the support of a summer grant the Graduate of the University offrom Maryland. Thanks Research are due toBoard the Computer Center of the University of Maryland for allowing a generous use of their facilities. —
REFERENCES 1. 2. 3. 4. S. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.
H.A. Lorentz, Wiedem. Ann. 9, 641 (1880). L. Lorenz, Wiedem. Ann. 11,70(1881). M. Born & E. Wolf, Principles of Optics, Section 2.3. Pergamon, New York (1970). S.L. Adler, Phys. Rev. 126,413 (1962). J. Heinrichs,Phys. Rev. 188, 1419 (1969). J.E. Lennard-Jones Soc. 24, 92 (1928). & B.M. Dent, Trans. Faraday W.J.C. Orr, Trans. Faraday Soc. 35, 1247 (1939). R. Shuttleworth, Proc. Phys. Soc. (London) A62, 167 (1949). B.R.A. Nijboer & F.W. de Wette,Physica 23,309 (1957);Physica 24,422 (1958). B.M.E. der Hoff& G.C. Benson, Can. J. Phys. 31 1087Van (1953). G.D. Mahan, J. Chem. Phys. 43, 1569 (1965). J. Topping, Proc. Roy. Soc. (London) A114, 67 (1927). For the single-oscillator model forgoes dispersion, w2) and F(~) rapidly a(w) e2/m(w~ over a=wide range of values near the absorption frequency. See, e.g. [3, Section 2.3.4]. J.J. Hopfield,Phys. Rev. 112, 1555 (1958). M.J. Frankel & J.L. Birman,Phys. Rev. B13, 2587 (1976). F.W. King, R.P. Van Duyne & G.C. Schatz,J. Chem. Phys. 69, 4472 (1978). G.L. Eesley & J.R. Smith, General Motors Research Publication No. GMR-2894 (preprint). —
In what follows we shall restrict ourselves, for simplicity, to the case where the surface of the slab coincides with the [001] face of the appropriate crystal structure. With suitable modification, our procedure can be generalized to other faces as well. We are interested, in general, in the field at a lattice point. Considering the origin to be at the lattice point at which the field is being evaluated, the contribution of all the other dipoles to the electric field, which we call EthP, is given by =
~‘
k,1,m
3~k1mT~m)~1m ~lmPk1m
(I)
r~,m
where the prime indicates that the term having k = I = m = 0 must be excluded. Here PkIm denotes the dipole moment of the molecule located at the lattice point rkzm [= a(k, I, m)], a being the lattice constant and k, 1, m the indices describing the lattice point. Note that while the indices will be integers for the s.c. structure, they may be half-integral for the other two structures. In the bulk, there have been two ways of performing sums of this type. One method is to evaluate part of the sum directly by summation inside a finite region and approximate the rest by integration with a uniform distribution of dipoles of appropriate density. Such a technique was used by several authors [6—8]in the study of cohesion and van der Waals interaction energy at a crystal surface. The alternate approach is to convert the discrete lattice sum into a highly convergent sum analytically, and then evaluate the latter up to a finite number of terms. This technique was used by Nijboer and dé Wette [9], who transform the sum into the reciprocal space, and van der Hoff and Benson [10]. It was also used by Mahan [11] in his calculation of the van der Waals energy in solids using both instantaneous Coulomb and retarded interactions. Our procedure for evaluating the sum (1) is to carry out the summation plane by plane parallel to the
646
LOCAL-FIELD EFFECT NEAR THE SURFACE OF DIPOLAR LATTICES
Table 1. Values of~~ Plane index
that to find the dipolar field at a lattice point in the slab problem, we must consider only a finite number of lattice planes. Guided by Table 1, we conclude that
s.c.
f.c.c.
b.c.c.
_____________________________________________________
0
9.0336 0.3275 0.0005 0.0000 0.0000 0.0000
2 3 4
— —
6.3877 1.0107 0.0160 0.0000 0.0000 0.0000
—4.5 168 2.0882 0.1638 0.0062 0.0003 0.0000 —
—
—
in equations (4) one needs to retain only terms for which v~~N, where N = l(s.c.), 2(f.c.c.) and 3(b.c.c.). This is indeed an approximation, but it enormously simplifies the task of finding the self-consistent or local electric field. The familiar identity ~ = —8ir/3, which follows directly [9] from Lorentz’s formula for internal field in the bulk of a cubic lattice, is satisfied by our approximation to better than 0.02%. Let us now consider a slab of L lattice planes (labeled by 1, 2, L) under the influence of a spatially uniform field in the z-direction, E0 = E0~.The
...
surface of the slab. We first note that the dipole moment depends only on the index m specifying a lattice plane, i.e. Pklm = Pm~Next we consider the applied field to be in the direction of the surface normal (call it z), so that p~= (0, 0,Pm). It follows quite simply that only the z-component of Ed~Pis then nonzero. Noting that rk,m = a(k, 1, m), we have 2 (k2 + 12 + m2) ‘~dip,Z = ~ Pm a~~c 3m(k2 + 12 + rn2)512 (2) —
Let us introduce two notations for the sake of compactness. Label the lattice planes by an integer v such that v = m(s.c.) and v = 2m (f.c.c. and b.c.c.). Also define as P~,the local dipole moment density on a lattice plane, i.e. 3, ~
=
Vol. 33, No.6
l(s.c.), 4(f.c.c.), 2(b.c.c.).
(3)
= ~PmIa Equation (2) may then be rewritten as
,
local electric field at a lattice point on the pth plane is then given by ~ = E0 +~~ (5) where the sum over v is restricted by 1 ~ v ~ L and p vi ~N. Let a be the polarizability of each molecule in the z-direction and F the volume polarizability, i.e. F = ~a/a3. If the applied field varies in time with a frequency w, both a and F will depend on o.~and change sign near a resonance. The self-consistency condition for the local dipole moment density, viz. PIA = FE~JC,Z, now yields the matrix equation —
~.MILVPV
FE
=
0,
p
=
1,2,... ,L,
(6a)
“
where = (1 I’~o)&~v—F~l,L_vI(1 ~ (6b) Although the matrixMis an (L x L) matrix, only a small fraction of its terms is finite because of the nature of the ~‘s (cf. Table 1). Inverting the matrix M where it is permissible gives us the dipole moment density on each plane of the slab. For comparison, the uniform dipole moment density in the bulk for the infinite-lattice problem is given, within our approxi—
Ethpz
=
~ ~VPV,
(4a)
v
where ~ comes from summation over a single lattice plane. When the applied field is parallel to the surface, lying along the x-axis, for example, one similarly finds =
~ ~
~
=
—
~
(4b)
mation, by
p
vClearly = 0 it ~, is depends apart from only sign on the andmagnitude geometricof factors v, and for —
—
—
—
pb
=
FE 0
essentially the sum first evaluated by Topping [12]. For other, non-vanishing v’s, the ~‘s can be evaluated for the different crystal structures by either of the two methods mentioned above. In practice, the technique outlined by van der Hoff and Benson [10] gives a highly convergent series whose accuracy is easy to control. The ~‘s computed by that method are given in Table 1 for various plane indices v. We note again that for f.c.c. and b.c.c. structures, odd values of v correspond to lattice planes at half-integral lattice distances. An outstanding feature of Table 1 is that the ~‘s go extremely rapidly to zero as v increases. This means
/
Ii
—
~
].
(7)
We have carried out the explicit inversion of the matrix to obtain P,2 for a 21-layer slab problem for s.c., f.c.c. and b.c.c. lattices. We regard F as a free parameter that can be varied in the problem [13]. The inversion procedure breaks down for a range of values of F for which det MI = 0. For these values of F, the system can support spontaneous polarization modes, which are the counterparts, within our approximation, of the excitonic normal modes of an infinite lattice as discussed by Hopfield [14] and Mahan [11]. Outside of
Vol. 33, No. 6
LOCAL-FIELD EFFECT NEAR THE SURFACE OF DIPOLAR LATTICES .6
647
.6
(a)
(b)
4
4
12
12 F~O.5
r~o5
Layer Number (~)
Layer Number (M)
Fig. 1. (a) Plots of the dipole-moment density on a lattice plane p vs the plane index, for selected values of the volume polarizability I’, for the simple cubic structure. The dipole-moment density is normalized to the bulk moment density ~b for the uniformly polarized infinite lattice [cf. equation (7)]. (b) The dipolar contribution to the local electric field on a lattice plane p, in the simple cubic case, normalized to the dipolar field in the uniformly polarized bulk system, plotted against the plane index for selected values of the volume polarizability F. (.6
(.6
(a)
~ ~F0.25
)b)
r~-O.25
~
O2~
Layer Number (~)
Layer Number (p.)
Fig. 2. (a) Plots of the dipole-moment density on a lattice plane p vs the plane index, for selected values of the volume polarizability F, for the face-centered cubic structure. The dipole-moment density is normalized to the bulk moment density ~b for the uniformly polarized infinite lattice [cf. equation (7)]. (b) the dipolar contribution to the local electric field on a lattice plane p, in the f.c.c. case, normalized to the dipolar field in the uniformly polarized bulk system, plotted against the plane index for selected values of the volume polarizability F. this range [which, for the s.c. system, is (~ + 2~i)_1 ~ F ~ (~— 2~~)’} our method gives well-behaved results. Figure 1(a) shows a plot of ~ vs the lattice-plane index p for selected values of F in the s.c. case. We emphasize that only points corresponding to integral values of p are meaningful, and that we have drawn lines through them as an aid to the eye only. For most values off, the local dipole moment density remains close to the bulk value, differing from it by only a few per cent on the surface layer. However, as F approaches the critical value for the excitation of the uniform polarization mode, Fl/Pb differs dramatically from unity,
becoming as low as 20%. When 0 > F>
2~ 1, 1j 1, exhibits an oscillatory behavior. As Fig. 1(a) shows, ~1/~b can be substantially larger than I when F is close to the upper edge for the band of polarization modes. To obtain a better perspective of our results, we have computed the dipolar contribution to the local electric field on a lattice plane, i.e. ~ = ~ ~ and compared it to the bulk result for a uniform distribution of dipole moments, viz. ~ = (~o+ Figure 1(b) shows the ratio of the two for the s.c. structure as a function of the plane index, again with selected Fl/Pb is greater than unity and P,2/P
(~
—
648
LOCAL-FIELD EFFECT NEAR THE SURFACE OF DIPOLAR LATTICES
values of F. The ratio is seen to be close to unity, except when I” is close to the edges of the polarizationmode band, on either side of which the dipolar field on the surface layer greatly diverges from the bulk value. It is interesting to note, however, that the curves of Figs. 1(a) and (b) corresponding to the same value of F are quite similar, leading us to the approximate identification ~ (81r/3)P,2. This would tend to justify the model used by some investigators [15] studying the propagation of light near a semiconductor surface that the local field near the surface may be written as E + 4irP(z)/3, in terms of a z-dependent dipole.moment density. Unfortunately, though, the situation is found to be quite different for f.c.c. and b.c.c. structures. As representative examples, we have plotted ~M/~b and E~jP.Z/E~iP,Zas functions of p, in the f.c.c. case with certain choices of F, in Figs. 2(a) and (b) respectively. The general features are the same as for the s.c. lattice, viz, the ratios are close to unity except when F approaches its critical range for the excitation of spontaneous polarization modes. On one side of the range, the ratios oscillate around unity, but on the other side they change monotonically with p, the changes being most noticeable near the edges of the polarization band. For this system, however, the identification ~ —(87T/3)P,2 cannot be made without entailing significant errors (‘~20%). This calls into question the prescription for using a local-dipole-moment-density model for treating the local field effect near a surface. We note that our results near the surface are not affected qualitatively on going to a 31-layer slab. In summary, then, we have presented the first results of the local electric field near the surface of a dipolar lattice, in the classical, point-dipole approximation, under the influence of an external electric field. Although the external field was taken to lie along the —
—
—
surface extension toona field an arbitrary directionnormal, is straightforward usinginequation (4b). Our calculation predicts large differences of the local field at the surface from the bulk result for certain values of the volume polarizability F. It will be of interest to see if the effect can be observed experimentally, either in photoemission or in reflectance studies, where [‘(w) can be varied continuously and often rapidly by changing the frequency of light. Our method may be used to
Vol. 33, No.6
define a local dielectric constant on a plane-by-plane basis near the surface, and this may, in turn, be used in a numerical study of light propagation. The method may also be extended to investigate the effect of local fields on photoemission from an ordered overlayer of absorbates on a metal surface, within some simple model of the dynamical response of the metal to change fluctuations. Suggestions about the importance of the local electric field in the chemisorption problem have indeed been offered [16, 17] in connection with surface enhanced Raman spectroscopy. Acknowledgement This work was partially supported by NSF under Grant No. DMR 76-82128. One of us (A.B.) wishes to acknowledge the support of a summer grant the Graduate of the University offrom Maryland. Thanks Research are due toBoard the Computer Center of the University of Maryland for allowing a generous use of their facilities. —
REFERENCES 1. 2. 3. 4. S. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.
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