Theory of acoustic surface plasmons

1Materia Kondentsatuaren Fisika Saila, Zientzi Fakultatea, Euskal Herriko Unibertsitatea, 644 Posta kutxatila, E-48080 Bilbo, Basque Country, Spain 2Donostia International Physics Center (DIPC) and Centro Mixto CSIC-UPV/EHU, Manuel de Lardizabal Pasealekua, E-20018 Donostia, Basque Country, Spain 3Department of Physics and Institute for Condensed Matter Theory, Chonnam National University, Gwangju 500-757, Korea 4Materialen Fisika Saila, Kimika Fakultatea, Euskal Herriko Unibertsitatea, 1072 Posta kutxatila, E-20080 Donostia, Basque Country, Spain 5Department of Physics, Queen’s University, Kingston, Ontario, Canada K7L 3N6 (Received 2 June 2004; published 4 November 2004 )


I. INTRODUCTION
Since the early suggestion of Pines 1 that low-energy plasmons with soundlike long-wavelength dispersion could be realized in the collective motion of a system of two types of electronic carriers, these modes have spurred over the years a remarkable interest and research activity. 2 The possibility of having a longitudinal acoustic mode in a metal-insulatorsemiconductor ͑MIS͒ structure was anticipated by Chaplik. 3 Chaplik considered a simplified model in which a twodimensional (2D) electron gas is separated from a semiinfinite three-dimensional (3D) metal. He found that the screening of valence electrons in the metal changes the 2D plasmon energy from its characteristic square-root wave-vector dependence to a linear dispersion, which was also discussed by Gumhalter 4 in his study of transient interactions of surfacestate electron-hole ͑e-h͒ pairs at surfaces.
Nevertheless, acoustic plasmons were only expected to exist for spatially separated plasmas, as pointed out by Das Sarma and Madhukar. 5 The experimental realization of twodimensionally confined and spatially separated multicomponent structures, such as quantum wells and heterostructures, provided suitable solid-state systems for the observation of acoustic plasmons. 6 Acoustic plasma oscillations were then proposed as possible candidates to mediate the attractive interaction leading to the formation of Cooper pairs in high-T c superconductors. 7,8 Recently, Silkin et al. 9 have shown that metal surfaces where a partially occupied quasi-2D surface-state band coexists in the same region of space with the underlying 3D continuum support a well-defined acoustic surface plasmon, which could not be explained within the original local model of Chaplik. 3 This low-energy collective excitation exhibits linear dispersion at low wave vectors, and might therefore affect e-h and phonon dynamics near the Fermi level. 10 In this paper, we present a model in which the screening of a semiinfinite 3D metal is incorporated into the description of electronic excitations in a 2D electron gas through the introduction of an effective 2D dielectric function. We find that the dynamical screening of valence electrons in the metal changes the 2D plasmon energy from its characteristic square-root behavior to a linear dispersion, not only in the case of a 2D sheet spatially separated from the semiinfinite metal, as anticipated by Chaplik, 3 but also when the 2D sheet coexists in the same region of space with the underlying metal, as occurs in the real situation of surface states at a metal surface. Furthermore, our results indicate that it is the nonlocality of the 3D dynamical response which allows the formation of 2D electron-density acoustic oscillations at metal surfaces, since these oscillations would otherwise be completely screened by the surrounding 3D substrate. Unless stated otherwise, atomic units are used throughout, i.e., e 2 = ប = m e =1.

II. THEORY
A variety of metal surfaces, such as Be(0001) and the (111) surfaces of the noble metals Cu, Ag, and Au, are known to support a partially occupied band of Shockley surface states with energies near the Fermi level. 11 Since the wave function of these states is strongly localized near the surface and decays exponentially into the solid, they can be considered to form a 2D electron gas.

͑7͒
and ͑z ; q , ͒ being the 2D Fourier transform of the total potential at z in the absence of the 2D sheet ͑z;q,͒ = ͵ dzЉ ͫ ␦͑z − zЉ͒ + ͵ dzЈv͑z,zЈ;q͒ ϫ 3D ͑zЈ,zЉ;q,͒ ͬ ext ͑zЉ;q,͒. ͑8͒ Equation (5) suggests that the screening of the 3D subsystem can be incorporated into the description of the electrondensity response at the 2D electron gas through the introduction of the effective density-response function of Eq. (6), whose poles should correspond to 2D electron-density oscillations.

͑14͒
where 2D 0 ͑q , ͒ and 3D 0 ͑z , zЈ ; q , ͒ represent their noninteracting counterparts. An explicit expression for the 2D noninteracting density-response function 2D 0 ͑q , ͒ was reported by Stern. 13 In order to derive explicit expressions for the 3D noninteracting density-response function 3D 0 ͑z , zЈ ; q , ͒ one needs to rely on simple models, such as the hydrodynamic or infinite-barrier model, but accurate numerical calculations have been carried out 14,15 from the knowledge of the eigen-functions and eigenvalues of the Kohn-Sham Hamiltonian of density-functional theory (DFT). 16 Combining Eqs. (6), (12), and (13), one finds the RPA effective 2D dielectric function The longitudinal modes of the 2D subsystem, or plasmons, are solutions of In the absence of the 3D subsystem, the 3D screened interaction W͑z , zЈ ; q , ͒ reduces to the bare Coulomb interaction v͑z , zЈ ; q͒, and the solution of Eq. (16) leads at long wavelengths to the well-known square-root wave-vector dependence of the 2D plasmon energy 13 ͑17͒ q F and m being the 2D Fermi momentum and 2D effective mass, respectively. The 2D Fermi velocity is simply v F = q F / m. In the presence of the 3D subsystem, the long-wavelength limit of the effective 2D dielectric function of Eq. (15) is found to have two zeros. One zero corresponds to a highfrequency ͑ ӷ v F q͒ oscillation in which 2D and 3D electrons oscillate in phase with one another. The other mode corresponds to a low-frequency acoustic oscillation in which both 2D and 3D electrons oscillate out of phase.
At high frequencies, where ӷ v F q, the long-wavelength limit of the 2D density-response function 2D 0 ͑q , ͒ is known to be On the other hand, when the 2D sheet is located either far inside or far outside the metal surface, the long-wavelength limit of the 3D screened interaction W͑z d , z d ; q , ͒ takes the form where p,s represents either the bulk-plasmon frequency p = ͱ 4n or the conventional surface-plasmon energy s = p / ͱ 2, 17 depending on whether the 2D sheet is located inside or outside the solid. Introduction of Eqs. (18) and (19) into Eqs. (15) and (16) yields a high-frequency mode at At low frequencies, we seek for an acoustic 2D plasmon energy that in the long-wavelength limit takes the form A careful analysis of the 2D density-response function 2D 0 ͑q , ͒ and the 3D screened interaction W͑z d , z d ; q , ͒ shows that at = ␣v F q the long-wavelength limits of these quantities take the form An inspection of Eqs. (15), (22), and (23) indicates that for a low-energy acoustic oscillation to occur the quantity I͑z d ͒ must be different from zero. In that case, the longwavelength limit of the effective 2D dielectric function of Eq. (15) has indeed a zero corresponding to a low-frequency oscillation of energy given by Eq. (21) with In the following, we investigate the impact of the 3D screening on the actual wave-vector dependence of the lowenergy 2D collective excitation. We first consider the two limiting cases in which the 2D sheet is located far inside and far outside the metal surface, and we then carry out selfconsistent calculations of the 3D screened interaction W͑z , zЈ ; q , ͒, which will allow us to obtain plasmon dispersions for arbitrary locations of the 2D sheet.

A. 2D sheet far inside the metal surface
In the case of a 2D sheet that is located far inside the metal surface, the 3D subsystem can safely be assumed to exhibit translational invariance in all directions, i.e., the screened interaction W͑z d , z d ; q , ͒ entering Eq. (15) can be easily obtained from the knowledge of the interacting density-response function 3D ͑k , ͒ of a uniform 3D electron gas, as follows: where k = ͱ q 2 + q z 2 is the magnitude of a 3D wave vector and ⑀ 3D −1 ͑k , ͒ is the inverse dielectric function of a uniform 3D electron gas In the RPA, 3D 0 ͑k , ͒ being the noninteracting density-response function first obtained by Lindhard. 18

Local 3D response
If one characterizes the 3D uniform electron gas by a local dielectric function ⑀ 3D ͑͒, then Eq. (25) yields In a 3D gas of free electrons, ⑀ 3D ͑͒ takes the Drude form This means that in a local picture of the 3D response the characteristic collective oscillations of the 2D electron gas would be completely screened by the sorrounding 3D substrate and no low-energy acoustic mode would exist. 19

Hydrodynamic 3D response
Dispersion effects of the 3D subsystem can be incorporated approximately in a hydrodynamic model. In this approximation, the dielectric function of a 3D uniform electron gas is found to be 18 where ␤ = ͱ 1/3k F represents the speed of propagation of hydrodynamic disturbances in the electron system, 20 and k F is the 3D Fermi momentum.
which yields the following simple expression for the acoustic coefficient of Eq. (24):

Full 3D response
We have carried out numerical calculations of the RPA effective dielectric function of Eq. (15), by using the full 2D 0 ͑q , ͒ and 3D 0 ͑k , ͒ density-response functions, and choosing the electron-density parameters r s 2D = 3.14 and r s 3D = 1.87 corresponding to the (0001) surface of Be. 21 The results we have obtained with q = 0.01a 0 −1 and q = 0.1a 0 −1 are displayed in Figs. 1(a) and 1(b), respectively. We observe that at energies below the upper edge u = v F q + q 2 / ͑2m͒ (vertical dashed line) of the 2D e-h pair continuum (where 2D e-h pairs can be excited) the real part of the effective dielectric function is nearly constant and the imaginary part is large, as would occur in the absence of the 3D susbtrate. At energies above u , momentum and energy conservation prevents 2D e-h pairs from being produced, and Im ⑀ eff ͑q , ͒ is very small.
Collective excitations are related to a zero of Re ⑀ eff ͑q , ͒ in a region where Im ⑀ eff ͑q , ͒ is small and lead, therefore, to a maximum in the energy-loss function Im͓−⑀ eff −1 ͑q , ͔͒. 22 In the absence of the 3D substrate, a 2D plasmon would occur at 2D = 1.22 eV for q = 0.01a 0 −1 and 2D = 3.99 eV for q = 0.1a 0 −1 . However, Fig. 1 shows that in the presence of the 3D substrate a well-defined low-energy acoustic plasmon occurs, the sound velocity being just over the 2D Fermi velocity v F . The small width of the plasmon peak is entirely due to plasmon decay into e-h pairs of the 3D substrate. We have carried out calculations of the effective 2D dielectric function of Eq. (15) for a variety of 2D and 3D electron densities, and we have found that a well-defined acoustic plasmon of energy = ␣v F q is always present for 2D wave vectors up to a maximum value of q ϳ q F where the acoustic-plasmon dispersion merges with u . The coefficient ␣ that we have obtained from the zeros in Eq. (16) is represented by stars in Fig. 2 versus the 3D Wigner radius r s 3D , together with the prediction of Eq. (24) as obtained with the computed RPA value of I͑z d → −ϱ͒ (solid line) and the hydrodynamic prediction of Eq. (33) (dotted line). Figure 2 shows that Eq. (33) is a good representation of the linear dispersion of this low-energy plasmon, especially at the highest 3D electron densities. Figure 2 also shows that for low electron densities the hydrodynamic prediction is too small, which is due to the fact that at low densities the longwavelength limit of the 3D screened interaction is underestimated in this approximation.

B. 2D sheet far outside the metal surface
In the case of a 2D sheet that is located far outside the metal surface, where the 3D electron density is negligible, the 3D screened interaction of Eq. (7) at z = zЈ = z d takes the form where g͑q , ͒ is the so-called surface-response function of the 3D subsystem g͑q,͒ = − v q ͵ dz 1 ͵ dz 2 e q͑z 1 +z 2 ͒ 3D ͑z 1 ,z 2 ;q,͒. ͑35͒

Local 3D response
In the simplest possible model of a metal surface, one characterizes the 3D substrate at z ഛ 0 by a local dielectric function which jumps discontinuously at the surface from ⑀ 3D ͑͒ inside the metal ͑z ഛ 0͒ to zero outside ͑z Ͼ 0͒. Witin this model, 23 which is precisely the long-wavelength limit of the actual surface-response function.
For large values of the distance z d between the 2D sheet and the metal surface, one can write which is the result first obtained by Chaplik 3 by using the Drude-like 2D density-response function of Eq. (18).

Nonlocal 3D response
An inspection of Eq. (34) shows that the long-wavelength limit of the screened interaction W͑z d , z d ; q , ͒ is dictated not only by the local ͑q =0͒ surface-response function g local ͑q , ͒ but also by the leading correction in q of the actual nonlocal g͑q , ͒. Feibelman showed that up to first order in an expansion in powers of q, the surface-response function of a jellium surface can be written as 24 which at low frequencies yields

͑41͒
The frequency-dependent d Ќ ͑͒ function occurring in Eq.
(40) represents the centroid of the induced 3D charge density, which in the static limit ͑ =0͒ reduces to the image plane of an external point charge. Using Eq. (41), we find the actual long-wavelength limit of Eq. (34): which combined with Eq. (24) yields This shows that the acoustic-plasmon sound velocity derived from the local model [see Eq. (38)] remains unchanged, as long as z d is replaced by the coordinate of the 2D sheet relative to the position of the image plane.

Full 3D response
In order to compute the full RPA surface-response function of Eq. (35), we follow the method described in Ref. 14 for a jellium slab. We first assume that the 3D electron density vanishes at a distance z 0 from either jellium edge, 25 and compute the noninteracting density-response function 3D 0 ͑z , zЈ ; q , ͒ from the knowledge of the self-consistent Kohn-Sham wave functions and energies of DFT, 16 which we obtain in the local-density approximation (LDA). 26 We then introduce a double-cosine Fourier representation for both the noninteracting and the interacting density-response functions, and find explicit expressions for the surfaceresponse function in terms of the Fourier coefficients of the density-response function. 27 To ensure that our slab calculations are a faithful representation of the actual surfaceresponse function of a semiinfinite 3D system, we follow the extrapolation procedure described in Ref. 28.
We have carried out numerical calculations of the effective dielectric function of Eq. (15), by using the full 2D noninteracting density-response function, 2D 0 ͑q , ͒, and the self-consistent RPA surface-response function, g͑q , ͒, with electron-density parameters r s 2D = 3.14 and r s 3D = 1.87 corresponding to Be(0001).
The results we have obtained for a 2D sheet located at z d = F are displayed in Figs. 3(a) (with q = 0.01a 0 −1 ) and 3(b) (with q = 0.1a 0 −1 ), F =2 / k F being the 3D Fermi wavelength. Figure 3 clearly shows that in the presence of the 3D substrate a well-defined low-energy acoustic plasmon occurs, the sound velocity being close to that predicted by Eq. (43) with d Ќ ͑0͒ = 0.2 F (vertical long-dashed lines). The actual plasmon energy is smaller than predicted by Eq. (43), especially at the shortest wavelengths ͑q = 0.1a 0 −1 ͒, simply due to the bending of the plasmon dispersion as a function of q (see Fig. 7 below).

Hydrodynamic 3D response
An explicit expression for the screened interaction W͑z , zЈ ; q , ͒ of Eq. (7) can be obtained in a hydrodynamic model in which the 3D electron density is assumed to change abruptly at the surface from n inside the metal to zero outside. After writing and linearizing the basic hydrodynamic equations, i.e., the continuity and the Bernouilli equation, we find which combined with Eq. (24) yields an explicit expression for the acoustic coefficient ␣. We note that in a local description of the electronic response of the solid surface ͑␤ =0͒ the 3D screened interaction W͑z d , z d ; q , ␣v F q͒ is zero inside the solid ͑z d ഛ 0͒ and 4z d outside ͑z d Ͼ 0͒. This shows that in the 2D long-wavelength limit ͑q → 0͒ the nonlocality of the 3D response is only present inside the solid ͑z d ഛ 0͒, where finite values of the 3D momentum k are possible.
Alternatively, the screened interaction W͑z , zЈ ; q , ͒ can be obtained within a specular-reflection model (SRM) 29 or, equivalently, a classical infinite-barrier model (CIBM) 30,31 of the surface, which have the virtue of describing the 3D screened interaction in terms of the bulk dielectric function ⑀ 3D ͑k , ͒ of a 3D uniform (and infinite) electron gas (see Appendix). If this bulk dielectric function is chosen to be the hydrodynamic dielectric function of Eq. (31), then these models yield Eq. (44). A more accurate description of the bulk dielectric function ⑀ 3D ͑k , ͒ yields a result that still co-incides with that of Eq. (44) outside the surface ͑z d Ͼ 0͒, though small differences may arise at z d Ͻ 0.
When the 2D sheet is located far inside the metal ͑z d Ӷ 0͒, Eq. (44) yields the hydrodynamic asymptotic behavior dictated by Eq. (32), and the SRM combined with the RPA bulk dielectric function yields the correct RPA asymptotic behavior. However, these hydrodynamic and specularreflection models, which are both based on the assumption that the 3D electron density drops abruptly to zero at the surface, fail to reproduce the correct asymptotic behavior outside the surface [see Eq. (42)]. This is due to the fact that the leading correction in q of the surface-response function g͑q , ͒ is governed by the spill out of the electron density into the vacuum, which is not present in these models. Fig. 1, but now for a 2D sheet that is located at one 3D Fermi wavelength outside the metal surface ͑z d = F ͒. The long-dashed vertical lines here represent the plasmon energy = ␣v F q predicted by Eq. (43) with d Ќ = 0.2 F . For real frequencies, a 2D sheet that is located at z d = F exhibits a plasmon peak that at q = 0.01a 0 −1 is extremely sharp (as z d → ϱ the plasmon peak becomes a delta function);

Full 3D response
For an arbitrary location of the 2D sheet we need to compute the full screened interaction W͑z d , z d ; q , ͒ of Eq. (7).
To calculate this quantity we consider a jellium slab, as we did to obtain the surface-response function g͑q , ͒, and we find explicit expressions in terms of the Fourier coefficients of the interacting density-response function, 27  In Fig. 4, the long-wavelength limit I͑z d ͒ of the screened interaction W͑z d , z d ; q , ␣v F q͒ [see Eq. (23)] is displayed versus z d , as obtained with r s 3D = 1.87 from our full selfconsistent RPA calculations (thick solid line) and from the hydrodynamic Eq. (44) (thin solid line). Far inside the solid, our full calculation is close to the hydrodynamic prediction (see also Fig. 2) and coincides with the result one obtains from the bulk screened interaction of Eq. (25) (horizontal dashed line). Near the surface, our full calculation considerably deviates from the hydrodynamic prediction and converges far outside the solid with the asymptotic curve dictated by Eq. (42) with d Ќ ͑0͒ = 0.2 F (dotted line). 32 At this point, it is interesting to note that within a local picture of the 3D response the long-wavelength I͑z d ͒ screened interaction would be zero for all locations of the 2D sheet inside the metal ͑z d ഛ 0͒, showing that the screening of 2D electron-density oscillations would be complete and no acoustic surface plasmon would occur. It is precisely the nonlocality of the 3D response (finite values of the 3D momentum k are still present in the 2D long-wavelength limit) which provides incomplete screening and allows, therefore, the formation of acoustic surface plasmons in the interior of the solid. We also note that within a simple nonlocal picture of the 3D response, such as the hydrodynamic and specularreflection models described above, the screening of 2D electron-density oscillations would still be complete at the jellium edge ͑z d =0͒. Hence, in the real situation where the 2D surface-state band is located very near the jellium edge the occurrence of acoustic surface plasmons is originated by a combination of the nonlocality of the 3D response and the spill out of the 3D electron density into the vacuum.  Fig. 5(b)]) by using the full 2D noninteracting density-response function 2D 0 ͑q , ͒ and the self-consistent RPA screened interaction W͑z d , z d ; q , ͒, with electron-density parameters r s 2D = 3.14 and r s 3D = 1.87 corresponding to Be(0001). In these figures the 2D sheet has been taken to be located at z d =0, as approximately occurs with the quasi-2D surface-state band in Be(0001). For comparison, also shown in these figures are the results we have obtained for the energy-loss function when the 2D sheet is located inside the metal at z d =− F and outside the metal at z d = F / 2 and z d = F .
An inspection of Fig. 5 shows that (i) the results we have obtained for z d =− F and z d = F are exactly reproduced by the limiting Eqs. (25) and (34) appropriate for a 2D sheet far inside and far outside the metal surface, respectively, and (ii) in the actual situation where the 2D surface-state band is located very near the jellium positive background edge ͑z d =0͒, a well-defined low-energy acoustic plasmon occurs, the sound velocity being very close to the limiting case of a 2D sheet far inside the metal surface and being, therefore, just above u . This is in agreement with the recent prediction that in a real metal surface where a partially occupied quasi-2D surface-state band coexists in the same region of space with the underlying 3D continuum an acoustic surface plasmon should occur at energies just above the upper edge of the 2D e-h pair continuum. 9 The sound velocity v s ͑ = v s q͒ of the acoustic plasmon that is visible in Fig. 5 is displayed in Fig. 6 versus the location z d of the 2D sheet relative to the jellium edge, as obtained from our full RPA self-consistent calculation of the effective 2D dielectric function of Eq. The sound velocity of Fig. 6 (open circles) has been obtained from the effective 2D dielectric function at very low 2D momenta, where the energy of the acoustic plasmon is linear in q. The behavior of this plasmon energy as a function of the 2D momentum q is displayed in Fig. 7, with the 2D sheet chosen to be located far inside the solid (thick   Fig. 1, but now for a 2D sheet that is located at the jellium edge ͑z d =0͒. Also shown is the effective 2D energy-loss function Im ͓−⑀ eff −1 ͑q , ͔͒ for z d =− F , z d = F / 2, and z d = F (dotted lines). The open circles represent the effective 2D energy-loss function Im ͓−⑀ eff −1 ͑q , ͔͒ obtained from the limiting Eq. (25) appropriate for a 2D sheet far inside the metal and from the limiting Eq. (34) with z d = F appropriate for a 2D sheet far outside the metal. These calculations are found to coincide with the full calculations for z d =− F and z d = F , respectively. As in Fig. 3(a), the calculations presented here for z d = F and q = 0.01a 0 −1 have been carried out by replacing the energy by a complex quantity + i with = 0.05 eV. All remaining calculations have been carried out for real frequencies, i.e., with =0.

III. SUMMARY AND CONCLUSIONS
The partially occupied band of Shockley surface states in a variety of metal surfaces is known to form a quasi-2D electron gas that is immersed in a semiinfinite 3D gas of valence electrons. In order to describe the impact of the dynamical screening of the semi-infinite 3D continuum on the electronic excitations at the 2D electron gas of Shockley surface states, we have presented a model in which the dynamical screening of 3D valence electrons is incorporated through the introduction of an effective 2D dielectric function.
We have considered the two limiting cases in which the 2D sheet is located far inside and far outside the metal surface. In both cases, the dynamical screening of the valence electrons in the metal is found to change the 2D plasmon energy from its characteristic square-root behavior to a linear dispersion, the sound velocity being proportional to the Fermi momentum of the 2D gas. As this collective oscilla-tion occurs in a region of 2D momentum space where 2D e -h pairs cannot be produced, this is a well-defined acoustic plasmon. The finite width of the plasmon peak is due to a small probability for the plasmon to decay into e-h pairs of the 3D substrate.
We have shown explicitly that when the 2D sheet coexists in the same region of space with the underlying 3D continuum the origin of acoustic surface plasmons, which have been overlooked over the years, is dictated by a combination of the nonlocality of the 3D response and the spill out of the 3D electron density into the vacuum, both providing incomplete screening of the 2D electron-density oscillations.
We have carried out self-consistent DFT calculations of the dynamical density-response function of the 3D system of valence electrons, and we have found that a well-defined acoustic plasmon exists for all possible locations of the 2D sheet relative to the metal surface. The energy dispersion of this acoustic surface plasmon is slightly higher than the en-FIG. 6. The open circles represent the sound velocity v s ͑ = v s q͒ of the low-energy acoustic plasmon that is visible in Fig. 5 versus the location z d of the 2D sheet with respect to the jellium edge. The horizontal short-dashed line represents the result we have obtained from the limiting Eq. (25) appropriate for a 2D sheet far inside the metal. The dotted line represents the result we have obtained from the limiting Eq. (43) with d Ќ = 0.2 F , which is appropriate for a 2D sheet far outside the metal. The long-wavelength limit v F of the upper edge u / q of the 2D e-h pair continuum is represented by an horizonal longdashed line. The thick and thin solid lines represent the results obtained from Eq. (24) by using the actual RPA I͑z d ͒ and the hydrodynamic Eq. (44), respectively. 2D and 3D electron densities have been taken to be those corresponding to the Wigner radii r s 2D = 3.14 and r s 3D = 1.87, respectively. The 2D effective mass has been taken to be m =1.
FIG. 7. Dispersion of the acoustic plasmon occurring in a 2D sheet that is taken to be located far inside the solid (thick dotted line), at z d =0 (open circles), at z d = F (solid line), and infinitely far outside the metal (solid circles). The thick dashed line represents the upper edge u = v F q + q 2 / ͑2m͒ of the 2D e-h pair continuum. The thin dotted line represents the 2D plasmon energy 2D dictated by Eq. (17), which is accurate at long wavelengths ͑q → 0͒. The thin dashed line represents the 2D plasmon energy ͱ 2D 2 +3v F 2 q 2 / 4 that is obtained after an expansion of 2D 0 ͑q , ͒ in powers of v F q / . 2D and 3D electron densities have been taken to be those corresponding to the Wigner radii r s 2D = 3.14 and r s 3D = 1.87, respectively. The 2D effective mass has been taken to be m =1. ergy of the collective excitation that has recently been predicted to exist at real metal surfaces where a quasi-2D surface-state band coexists with the underlying 3D continuum. 9 Small differences between the plasmon energies obtained here and those reported previously 9 are due to the absence in the present model of transitions between 2D and 3D states. 33