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Theory of the intersurfacedband plasmon
Solid State Communications, Printed in Great Britain.
Vo1.50,No.5,
pp.409-412,
THEORY OF THE INTERSURFACEBAND
College
0038-1098/84 Y3.00 + .OO Pergamon Press Ltd.
1984.
PLASMQN
M. Nakayama of General Education, Kyushu University Ropponmatsu, Fukuoka 810, Japan
01,
and T. Kato and K. Ohtomi Fukuoka Institute of Technology, Shimowajiro, Fukuoka 811-02, Japan (Received
31 January
1984 by H.Kamimura)
The dispersion relation of the intersurfaceband plasmon, i.e., collective excitation of the surface electrons of the clean or adsorbed surfaces, is investigated by the self-consistent field The dispersion is predicted to start with the linear term approach. in the wave number parallel to the surface, which will be helpful to The sign of distinguish the plasmon from the individual excitations. the dispersion and the depolarization shift depend on the polarizaThe contribution of the plasmon to tion of the interband transition. the electron energy loss cross section is given.
electrons are confined between two infinite The depolarization shift and the walls[61. negative dispersion were derived from the Later, more nonlocal self-consistent theory. detailed theories have been developed for MOS All these theories dealt with a layers[7]. special kind of the surface states, i.e., the separable case where the wave function is the product of the plane wave parallel to the surface and a function of the perpendicular coordinate z. In the present work we investigate ISBP for a general kind of the surface electronic state. We shall see that ISBP is distinguished from the individual excitations by the q-linear dispersion and the dispersion can be either positive or negative, depending on the polarization of the transition. Let us consider the clean or the The elecperiodically adsorbed surface layer. tronic state is specified by 2D-wave vector& and the surfaceband index x with the eigenfunction $A k and energy EA(~). We take the z-axis as the &itward normal of the surface and denote the cordinate parallel to the surface by f: We expand all the quantities in terms of exp i(_qetit)] and consider only the (q,w)-component, neglecting the Umklapp proce;ses for small qregion. By the standard self-consistent field approach[8,9] with a few simplifying assumptions, we obtain the coupled equations for the induced charge density n of the surface electrons and the electrostatic potential I$.
The characteristic of the plasmon, the normal mode of the charge density fluctuation, of the surface and adsorbate layers differs from that of the bulk solids, reflecting the difference in the electronic structure and the Stern showed that the fredimensionality. quency of the two dimensional(2D)-plasmon of the 2D-electron gas is proportional to the square root of the wave number q parallel to The dispersion was confirmed the surface[l]. for the electrons trapped at the surface of liquid helium and MOS inversion layer[2]. The 2D-plasmon is due to the intraband excitations in terms of the surfaceband structure. The intersurfaceband excitations cause another collective mode called the intersurfaceband The purpose of the present paplasmon(ISBP). per is to investigate the characteristics of ISBP. ISBP has been studied by a few authors mostly based on the phenomenological model in which the surface region is simulated by a thin layer of the classical dielectrics. Gadzuk treated the alkaline adsorbate layer on metals as a layer with the Drude type dielectric function and showed the existence of a normal mode which starts fromo (of the surface layer) at q = 0 and goes downward with the increase of q(negative dispersion)[3]. Similar models with the isotropic dielectric function incorporating the interband transitions were employed to analyze the electron energy loss spectroscopy (EELS) of the clean and adsorbed Chen et al. took the layer model surfaces[4]. with an anisotropic dielectric tensor to take into account the surface quantization of MOS structure and showed that the mode starts from the zero of E [5]. The characteristic frefrom the intersubband quency 9, is %fted frequency wo due to the depolarization effect. Microscopic theory of ISBP was initiated by Newns for the box model in which the surface
n(q,z,w)
= - $$(q,z)Pa(i(jll) x jdz'50(r?,z')~(q,z',w).
@(_q,z,uw)= jdz'V(z,z')n(~.z',W). 409
(1) (2)
410
THEORY OF THE INTERSLJRFACEBAND PLASMON
Relevant notations and assumptions are as follows: (I) The index c( represents the pair of the surfaceband (A,u) involved in the transition (X>p). In the actual calculation we shall consider the case where only the ground state surfaceband (denoted by 0) is occupied. In that case c1 denotes the pair (a, 0). (2) 5, is the form factor defined by k+q
‘..-
CX
= Fhh(q,w),
= /dzidz'Scx(4,z)V(z,z')5~(q,z').
aB
equation
(12)
is given by
det[ & + V& ] = 0.
if A=p
(4)
= FXu(z,~) + Fhu(q,-&
v
(3)
k.
We neglect the k-dependence of 5 to simplify the calculationThe approximation holds exactly in two limiting cases, i.e., the case of the separable wave function where electrons move freely along the surface and the case of the tight binding approximation where the overlap between the cells is completely neglected. (3) PC1 is the polarization function defined by P
and
The dispersion
/d_pllr; exp(iqp_)$,, ‘- _
5, =
Vol. 50, No. 5
where
if '+u.
(5)
(13)
where the matrix element of g is to be understood as given in eq.(lO). Since Re Pa diverges logarithmically with plus and minus signs at the lower and upper edges of the individual excitation band, respectively, eq.(l3) has at least a discrete solution in each gap region between the individual excitation bands. The discrete solution corresponds to the surfaceband plasmon. In order to investigate the dispersion in the small q limit, we expand V in terms of qd and qz and employ the relation
where
fdz5&,d
= 6a ,6(s) ,-
to obtain the matrix with the Fermi distribution function f. (4) V(z,z') denotes the electrostatic potential induced by the density exp(iAQ)6(z-2'). The potential is obtained by solving the Poisson equation in the background dielectric medium with the dielectric function E . Here we assume a three layer model forbthe background; Cb
=
EO’
z>d,
=
ER’
d>z>-d,
(7)
=
27ie* {exp(-qlz-z’
1) +
D-l[k,-Es)(Ee+Eo)
+ (ER+‘ppO)e
q(z+z’)
+ 2(ER-Es)(ER-EO)e -2qd cosh(q(z-z'))]},
6cY,06$3,0 + %Bt - ssz&*q2]].
(15)
In the above equation, p and z are the matrix element of the c%-dinatE between the nigenfunctions integrated over the unit cell, R is given by a6 R
(16)
fdz/dz’5n(~,z)l”-z’1Sa*(cl,z’),
DO
=
2kR(EO+Es)
+
(Es+EL2@qd].
1 + VooPo = 0. (8)
(17)
The surfaceband plasmon is classified into the intrasurfaceband and intersurfaceband plasmons by the nature of the transitions mainly involved in the excitation. The intrasurfaceband plasmon is governed by the OO-element of the matrix in eq.(13)
Putting
eq.(15)
where
-
(14)
and
ERs
x e-q(z+z’)
D =
)A!&
element V
+ -?-- [E&$PB*~) EQDO
0.6 = -
- ES)
&
o
-d>z.
We are mainly concerned with the transitions that are nearly resonant with W and describe the irrelevant part by EL. We assume that 5, hence n, take appreciable values only in the The most important part of V region d>z>-d. is the one in the region d>z,z'>-d given by
lJ
2Tie2 V { aB = 7
+ (1-6a )
(18) into eq.(l8)
and noting that (19)
(E~+E~)(E~+E~)
exp(2qd)
(E~-E~)(E~-E~)
exp(-2qd).
(9)
in the lowest order in q, we obtain the dispersion 4?re2N uz = -.---L.
(Eo+Es)
The dipersion equation is obtained as the condition that eqs.(l) and (2) have nontrivial solution of @. After inserting es.(l) into eq.(2), we multiply both the sides by E,(z,z) and integrate over z to obtain
:< = a
v PX 8 a13 B B
(10)
In the above equation mx and mY are the principal values of the conductivity effective mass tensor. The off-diagonal terms contribute only to the higher order terms in q. The dispersion (PO) is nothing but the dispersion of 2%plasmon.
THEORY
Vol. 50, No. 5
OF THE INTERSURFACEBAND
The dispersion of ISBP can be obtained by solving eq.(13) for the interband part of the matrix. Calculation is to be performed after When we can single the system is specified. out a particular interband transition CX, an associated plasmon mode is roughly estimated by taking only the aa-element 1 + VolclPc, = 0.
(21)
In the frequency region far from the central frequency of the interband excitation wo, we express P, as 2N w
p =---ELa w 2-uJ2 0.
(22)
and obtain the plasmon linear term
+
2 E&+Es)
[ '!$
frequency
up to the q-
I_P,s~'- ESIzC112ql). (23)
We notice two distinct features of the disperFirst, the frequency is shifted from wa sion. by the term Lao. This is the depolarization out by several authors. shift pointed Second, the dispersion is linear in q for The sig,l of the dispersion is small values. related to the polarization of the interband When p -term dominates over the transition. is mainly zo-term, i.e. the??ransition invoked by the parallel component of the elecincreases with q(positive tric field, w dispersion). oThe dispersion reflects the In the oppoanisotropy of the surface band. site case where the transition is mainly due to the perpendicular field, Go descreases with the surface q(negative dispersion). If system has a certain high symmetry, it may happen that either the parallel or the perpendicular transition is allowed by symmetry. When only the perpendicular transition is allowed, the frequency has the depolarization shift and the negative dispersion. The box model and the MOS inversion layer belong to The dispersion is always isotropic this case. ir:,espective of the anisotropy of the When only the parallel surfaceband structure. transition is allowed, R,, also vanishes by symmetry so that no depolarization shift The ISBP frequency starts from the occurs. upper edge of the individual excitation band and increases with q. Thus, we can obtain about the symmetry of the surface information system from the existence of the depolarizashift and the sense of the dispersion. tion Most effective experimental method to investigate the dispersion of ISBP is the angular resolved electron energy loss So far, however, the spectroscopy(AREELS). cross section formula is given only for the The microscopic classical layer model[k]. formula is derived along the line of the present theory by adding the external potential due to the electron beam to the right hand side If the classical trajectory of eq.(2). approximation is employed, the contribution of the single pair of the surfaceband c1 and the
substrate
411
PLASMON is proportional
to (24)
with A, = jdzV(z,z')exp(qz')5a(q,z)
In the above formula, z'-dependence of V(z,z') is cancelled out by the factor exP(qz'). The cross section has the peak at ISBP frequency Ga besides the structure due to the individual excitation (Im Po) and the surface Plasmon excitation (ECI+ES = 0). The ISBP peak can be distinguished by its q-linear dispersion from other excitations which vary as a function of a? Detailed experiments on the dispersion of the surface transition were reported only for a few cases. Jostell observed both the positive and negative dispersions for alkaline overlayers on Ni[lO]. Thou& he claimed that the dispersion is linear in q, the interpretation is open to question as he did not stttdy extensively the dispersion in the small qregion. Recently, Tochihara et al. exhaustively investigated the dispersion for K overlayer on the 2x1 reconstructed Si(100) surface[ll]. They showed that the dispersion is positive and is fittedfairly well to a linear function of q. In this system electrons are at least partly transferred from K to Si, as is evidenced by the decrease of the work function upon adsorption. ISBP is associated with either the back transfer excitation or the s-p excitation in K. In either case the sharp ISBP peak is expected if the system is in low symmetry. A recent study favours the asymmetric dimer model for the 2x1 reconstructed Si(100) surface[l2]. If the asymmetry is retained by the adsorption of K, the transition by the parallel field would dominate even for the back transfer to the s-orbital of K. Very recently, Tsukada et al. simulated the alkali adatomchains by the parallel rods confining electrons and investigated the collective mode including ISBP by taking into account the Umklapp processes[l3]. The model belongs to the case of the separable wave function and their result conforms to the present theory in the small qlimit, though in their case the positive q2-term predominates over the negative q-term for the perpendicular polarization even for rather small values of q. Further study is needed to identify the nature of the transition. Since the ISBP excitation generally exists and predominates over the individual excitations, we suspect that some of the peaks so far observed in EELS experiments in several surface systems and identified as individual excitations between the surfacebands or levels should actually be ascribed to the ISBP excitation. We hope that AREELS is reinvestigated in the light of the present theory and suggest that the observation of the q-linear dispersion is very helpful in the identification of the peak.
412
THEORY OF THE INTERSURFACEBAND
Acknowledgements - The authors would like to thank Prof. Y. Murata and Dr. H. Tochihara, and Prof. M. Tsukada for informing of their experimental and theoretical results,
PLASMON
Vol. 50, No. 5
respectively, prior to the publication and discussion. The work is partly supported by the Grant-in-Aid of Japanese Ministry of Education, Science, and Culture.
REFERENCES
[3] [4]
[51 [6]
[71
F. Stern, Phys.Rev.Lett. 18(1967)546. For a review see sec.IIC and D of the review article, T. Ando, A. B. Fowler, and F. Stern, Rev.Mod.Phys. 54(1982)437. J. W. Gadzuk, Phys.Rev. B1(1970)1267. H. Froitzheim, H. Ibach, and D. L.Mills, Phys. Rev. B11(1975)4980. W. P. Chen, Y. J. Chen, and E.Burstein, Surf.Sci. 58(1976)263. D. M. Newns, Phys.Lett. 38A(1972)341. For a review see sec.IIIC of the article cited in [2].
[al [91 [lo] [ll] [12] [l?]
M. Nakayama, J.Phys.Soc. Japan, 39(1975) 265. M. Nakayama, Proc. 7th Int. Vacuum Congress and 3rd Int. Conf. Solid Surfaces, Vienna, 1977, p.395. U. Jostell, Surf.Sci. 82(1979)333. T. Aruga, H. Tochihara, and Y. Murata, preprint. R. M. Tromp, R. G. Smeenk, F. W. Saris, and D. J. Chadi, Surf.Sci. 133(1983)137. M. Tsukada, H. Ishida, and N. Shima, preprint.
Vo1.50,No.5,
pp.409-412,
THEORY OF THE INTERSURFACEBAND
College
0038-1098/84 Y3.00 + .OO Pergamon Press Ltd.
1984.
PLASMQN
M. Nakayama of General Education, Kyushu University Ropponmatsu, Fukuoka 810, Japan
01,
and T. Kato and K. Ohtomi Fukuoka Institute of Technology, Shimowajiro, Fukuoka 811-02, Japan (Received
31 January
1984 by H.Kamimura)
The dispersion relation of the intersurfaceband plasmon, i.e., collective excitation of the surface electrons of the clean or adsorbed surfaces, is investigated by the self-consistent field The dispersion is predicted to start with the linear term approach. in the wave number parallel to the surface, which will be helpful to The sign of distinguish the plasmon from the individual excitations. the dispersion and the depolarization shift depend on the polarizaThe contribution of the plasmon to tion of the interband transition. the electron energy loss cross section is given.
electrons are confined between two infinite The depolarization shift and the walls[61. negative dispersion were derived from the Later, more nonlocal self-consistent theory. detailed theories have been developed for MOS All these theories dealt with a layers[7]. special kind of the surface states, i.e., the separable case where the wave function is the product of the plane wave parallel to the surface and a function of the perpendicular coordinate z. In the present work we investigate ISBP for a general kind of the surface electronic state. We shall see that ISBP is distinguished from the individual excitations by the q-linear dispersion and the dispersion can be either positive or negative, depending on the polarization of the transition. Let us consider the clean or the The elecperiodically adsorbed surface layer. tronic state is specified by 2D-wave vector& and the surfaceband index x with the eigenfunction $A k and energy EA(~). We take the z-axis as the &itward normal of the surface and denote the cordinate parallel to the surface by f: We expand all the quantities in terms of exp i(_qetit)] and consider only the (q,w)-component, neglecting the Umklapp proce;ses for small qregion. By the standard self-consistent field approach[8,9] with a few simplifying assumptions, we obtain the coupled equations for the induced charge density n of the surface electrons and the electrostatic potential I$.
The characteristic of the plasmon, the normal mode of the charge density fluctuation, of the surface and adsorbate layers differs from that of the bulk solids, reflecting the difference in the electronic structure and the Stern showed that the fredimensionality. quency of the two dimensional(2D)-plasmon of the 2D-electron gas is proportional to the square root of the wave number q parallel to The dispersion was confirmed the surface[l]. for the electrons trapped at the surface of liquid helium and MOS inversion layer[2]. The 2D-plasmon is due to the intraband excitations in terms of the surfaceband structure. The intersurfaceband excitations cause another collective mode called the intersurfaceband The purpose of the present paplasmon(ISBP). per is to investigate the characteristics of ISBP. ISBP has been studied by a few authors mostly based on the phenomenological model in which the surface region is simulated by a thin layer of the classical dielectrics. Gadzuk treated the alkaline adsorbate layer on metals as a layer with the Drude type dielectric function and showed the existence of a normal mode which starts fromo (of the surface layer) at q = 0 and goes downward with the increase of q(negative dispersion)[3]. Similar models with the isotropic dielectric function incorporating the interband transitions were employed to analyze the electron energy loss spectroscopy (EELS) of the clean and adsorbed Chen et al. took the layer model surfaces[4]. with an anisotropic dielectric tensor to take into account the surface quantization of MOS structure and showed that the mode starts from the zero of E [5]. The characteristic frefrom the intersubband quency 9, is %fted frequency wo due to the depolarization effect. Microscopic theory of ISBP was initiated by Newns for the box model in which the surface
n(q,z,w)
= - $$(q,z)Pa(i(jll) x jdz'50(r?,z')~(q,z',w).
@(_q,z,uw)= jdz'V(z,z')n(~.z',W). 409
(1) (2)
410
THEORY OF THE INTERSLJRFACEBAND PLASMON
Relevant notations and assumptions are as follows: (I) The index c( represents the pair of the surfaceband (A,u) involved in the transition (X>p). In the actual calculation we shall consider the case where only the ground state surfaceband (denoted by 0) is occupied. In that case c1 denotes the pair (a, 0). (2) 5, is the form factor defined by k+q
‘..-
CX
= Fhh(q,w),
= /dzidz'Scx(4,z)V(z,z')5~(q,z').
aB
equation
(12)
is given by
det[ & + V& ] = 0.
if A=p
(4)
= FXu(z,~) + Fhu(q,-&
v
(3)
k.
We neglect the k-dependence of 5 to simplify the calculationThe approximation holds exactly in two limiting cases, i.e., the case of the separable wave function where electrons move freely along the surface and the case of the tight binding approximation where the overlap between the cells is completely neglected. (3) PC1 is the polarization function defined by P
and
The dispersion
/d_pllr; exp(iqp_)$,, ‘- _
5, =
Vol. 50, No. 5
where
if '+u.
(5)
(13)
where the matrix element of g is to be understood as given in eq.(lO). Since Re Pa diverges logarithmically with plus and minus signs at the lower and upper edges of the individual excitation band, respectively, eq.(l3) has at least a discrete solution in each gap region between the individual excitation bands. The discrete solution corresponds to the surfaceband plasmon. In order to investigate the dispersion in the small q limit, we expand V in terms of qd and qz and employ the relation
where
fdz5&,d
= 6a ,6(s) ,-
to obtain the matrix with the Fermi distribution function f. (4) V(z,z') denotes the electrostatic potential induced by the density exp(iAQ)6(z-2'). The potential is obtained by solving the Poisson equation in the background dielectric medium with the dielectric function E . Here we assume a three layer model forbthe background; Cb
=
EO’
z>d,
=
ER’
d>z>-d,
(7)
=
27ie* {exp(-qlz-z’
1) +
D-l[k,-Es)(Ee+Eo)
+ (ER+‘ppO)e
q(z+z’)
+ 2(ER-Es)(ER-EO)e -2qd cosh(q(z-z'))]},
6cY,06$3,0 + %Bt - ssz&*q2]].
(15)
In the above equation, p and z are the matrix element of the c%-dinatE between the nigenfunctions integrated over the unit cell, R is given by a6 R
(16)
fdz/dz’5n(~,z)l”-z’1Sa*(cl,z’),
DO
=
2kR(EO+Es)
+
(Es+EL2@qd].
1 + VooPo = 0. (8)
(17)
The surfaceband plasmon is classified into the intrasurfaceband and intersurfaceband plasmons by the nature of the transitions mainly involved in the excitation. The intrasurfaceband plasmon is governed by the OO-element of the matrix in eq.(13)
Putting
eq.(15)
where
-
(14)
and
ERs
x e-q(z+z’)
D =
)A!&
element V
+ -?-- [E&$PB*~) EQDO
0.6 = -
- ES)
&
o
-d>z.
We are mainly concerned with the transitions that are nearly resonant with W and describe the irrelevant part by EL. We assume that 5, hence n, take appreciable values only in the The most important part of V region d>z>-d. is the one in the region d>z,z'>-d given by
lJ
2Tie2 V { aB = 7
+ (1-6a )
(18) into eq.(l8)
and noting that (19)
(E~+E~)(E~+E~)
exp(2qd)
(E~-E~)(E~-E~)
exp(-2qd).
(9)
in the lowest order in q, we obtain the dispersion 4?re2N uz = -.---L.
(Eo+Es)
The dipersion equation is obtained as the condition that eqs.(l) and (2) have nontrivial solution of @. After inserting es.(l) into eq.(2), we multiply both the sides by E,(z,z) and integrate over z to obtain
:< = a
v PX 8 a13 B B
(10)
In the above equation mx and mY are the principal values of the conductivity effective mass tensor. The off-diagonal terms contribute only to the higher order terms in q. The dispersion (PO) is nothing but the dispersion of 2%plasmon.
THEORY
Vol. 50, No. 5
OF THE INTERSURFACEBAND
The dispersion of ISBP can be obtained by solving eq.(13) for the interband part of the matrix. Calculation is to be performed after When we can single the system is specified. out a particular interband transition CX, an associated plasmon mode is roughly estimated by taking only the aa-element 1 + VolclPc, = 0.
(21)
In the frequency region far from the central frequency of the interband excitation wo, we express P, as 2N w
p =---ELa w 2-uJ2 0.
(22)
and obtain the plasmon linear term
+
2 E&+Es)
[ '!$
frequency
up to the q-
I_P,s~'- ESIzC112ql). (23)
We notice two distinct features of the disperFirst, the frequency is shifted from wa sion. by the term Lao. This is the depolarization out by several authors. shift pointed Second, the dispersion is linear in q for The sig,l of the dispersion is small values. related to the polarization of the interband When p -term dominates over the transition. is mainly zo-term, i.e. the??ransition invoked by the parallel component of the elecincreases with q(positive tric field, w dispersion). oThe dispersion reflects the In the oppoanisotropy of the surface band. site case where the transition is mainly due to the perpendicular field, Go descreases with the surface q(negative dispersion). If system has a certain high symmetry, it may happen that either the parallel or the perpendicular transition is allowed by symmetry. When only the perpendicular transition is allowed, the frequency has the depolarization shift and the negative dispersion. The box model and the MOS inversion layer belong to The dispersion is always isotropic this case. ir:,espective of the anisotropy of the When only the parallel surfaceband structure. transition is allowed, R,, also vanishes by symmetry so that no depolarization shift The ISBP frequency starts from the occurs. upper edge of the individual excitation band and increases with q. Thus, we can obtain about the symmetry of the surface information system from the existence of the depolarizashift and the sense of the dispersion. tion Most effective experimental method to investigate the dispersion of ISBP is the angular resolved electron energy loss So far, however, the spectroscopy(AREELS). cross section formula is given only for the The microscopic classical layer model[k]. formula is derived along the line of the present theory by adding the external potential due to the electron beam to the right hand side If the classical trajectory of eq.(2). approximation is employed, the contribution of the single pair of the surfaceband c1 and the
substrate
411
PLASMON is proportional
to (24)
with A, = jdzV(z,z')exp(qz')5a(q,z)
In the above formula, z'-dependence of V(z,z') is cancelled out by the factor exP(qz'). The cross section has the peak at ISBP frequency Ga besides the structure due to the individual excitation (Im Po) and the surface Plasmon excitation (ECI+ES = 0). The ISBP peak can be distinguished by its q-linear dispersion from other excitations which vary as a function of a? Detailed experiments on the dispersion of the surface transition were reported only for a few cases. Jostell observed both the positive and negative dispersions for alkaline overlayers on Ni[lO]. Thou& he claimed that the dispersion is linear in q, the interpretation is open to question as he did not stttdy extensively the dispersion in the small qregion. Recently, Tochihara et al. exhaustively investigated the dispersion for K overlayer on the 2x1 reconstructed Si(100) surface[ll]. They showed that the dispersion is positive and is fittedfairly well to a linear function of q. In this system electrons are at least partly transferred from K to Si, as is evidenced by the decrease of the work function upon adsorption. ISBP is associated with either the back transfer excitation or the s-p excitation in K. In either case the sharp ISBP peak is expected if the system is in low symmetry. A recent study favours the asymmetric dimer model for the 2x1 reconstructed Si(100) surface[l2]. If the asymmetry is retained by the adsorption of K, the transition by the parallel field would dominate even for the back transfer to the s-orbital of K. Very recently, Tsukada et al. simulated the alkali adatomchains by the parallel rods confining electrons and investigated the collective mode including ISBP by taking into account the Umklapp processes[l3]. The model belongs to the case of the separable wave function and their result conforms to the present theory in the small qlimit, though in their case the positive q2-term predominates over the negative q-term for the perpendicular polarization even for rather small values of q. Further study is needed to identify the nature of the transition. Since the ISBP excitation generally exists and predominates over the individual excitations, we suspect that some of the peaks so far observed in EELS experiments in several surface systems and identified as individual excitations between the surfacebands or levels should actually be ascribed to the ISBP excitation. We hope that AREELS is reinvestigated in the light of the present theory and suggest that the observation of the q-linear dispersion is very helpful in the identification of the peak.
412
THEORY OF THE INTERSURFACEBAND
Acknowledgements - The authors would like to thank Prof. Y. Murata and Dr. H. Tochihara, and Prof. M. Tsukada for informing of their experimental and theoretical results,
PLASMON
Vol. 50, No. 5
respectively, prior to the publication and discussion. The work is partly supported by the Grant-in-Aid of Japanese Ministry of Education, Science, and Culture.
REFERENCES
[3] [4]
[51 [6]
[71
F. Stern, Phys.Rev.Lett. 18(1967)546. For a review see sec.IIC and D of the review article, T. Ando, A. B. Fowler, and F. Stern, Rev.Mod.Phys. 54(1982)437. J. W. Gadzuk, Phys.Rev. B1(1970)1267. H. Froitzheim, H. Ibach, and D. L.Mills, Phys. Rev. B11(1975)4980. W. P. Chen, Y. J. Chen, and E.Burstein, Surf.Sci. 58(1976)263. D. M. Newns, Phys.Lett. 38A(1972)341. For a review see sec.IIIC of the article cited in [2].
[al [91 [lo] [ll] [12] [l?]
M. Nakayama, J.Phys.Soc. Japan, 39(1975) 265. M. Nakayama, Proc. 7th Int. Vacuum Congress and 3rd Int. Conf. Solid Surfaces, Vienna, 1977, p.395. U. Jostell, Surf.Sci. 82(1979)333. T. Aruga, H. Tochihara, and Y. Murata, preprint. R. M. Tromp, R. G. Smeenk, F. W. Saris, and D. J. Chadi, Surf.Sci. 133(1983)137. M. Tsukada, H. Ishida, and N. Shima, preprint.