- Email: [email protected]
Plasmons in doped semiconductor superlattice with charge depletion and a capping layer
~
Solid State Communications, Vol. 91, No. 3, pp. 251-254, 1994 Elsevier Science Ltd Printed in Great Britain 0038-1098/94 $7.00 + .00
t Pergamon
0038-1098(94)E0251-6 PLASMONS IN DOPED SEMICONDUCTOR SUPERLATTICE WITH CHARGE DEPLETION AND A CAPPING LAYER E.L. Albuquerque Departamento de Fisica, Universidade Federal do Rio Grande do Norte, 59072-970-Natal, RN-Brazil (Received 15 November 1993; in revised form 14 March 1994 by A.A. Maradudin) We present a theoretical analysis for the collective plasmon-polariton modes of a doped superlattice consisting of two-dimensional electron gas and two-dimensional hole gas layers separated by media of alternating thickness and dielectric constant. Our theory takes into account the effects due to a capping layer, as well as charge depletion near the superlattice surface. We use a theoretical model based on linear response function together with a transfer matrix approach to determine the plasmons dispersion relation and Green functions. Numerical applications are made to semiconductor superlattices, with interesting features for the plasmon surface modes.
WITH the development in recent years of modulationdoped semiconductor superlattices, many investigations have been turned out to determine the collective excitations of periodic arrays of two-dimensional electron (hole) gas layers. Theories for the spectra of bulk and surface plasmon in these structures have been given, using various different models, by a number of authors [1-9]. On the experimental side, the Raman scattering measurements of Olego et al. [10] have provided confirmation of the theoretical predictions for the plasmon dispersion relation, with subsequent Raman scattering experiments for quantized plasmon modes of finite superlattices [11, 12]. More sophisticated structure, taking into account charge depletion in the electron gas layer nearest to the surface as well as the presence of a capping layer at the surface, has been recently reported [13], and these effects have been proved to give strong influence on the surface plasmon-polariton properties. It is the aim of this letter to present theoretical considerations about surface plasmon-polariton modes in doped semiconductor superlattice made up of two-dimensional electron (2DEG) and hole (2DHG) gas, respectively, separated by dielectric media. Our investigation takes also into account the effects of charge depletion and the presence of a capping layer. Our purpose is two-fold: First, we want to extend the work of Constantinou and Cottam [13], by considering a more complete
structure, as well as our previous work [14], by including surface effects. Second, we would like to explore the existence of novel aspects in the surface plasmon-polariton spectra. Figure 1 shows our superlattiee model, where there is a 2DEG at z = c, L + c, 2L + c, etc., with L=a+b, a and b being the thickness of the dielectric media A and B that fills the space between the charge layers. Also, a 2DHG is found at z = c + a, L + c + a, 2L + c + a, etc. We denote the carrier concentration in each layer by n and p, except near the surface, where it has a different value ns and Ps. There is also a capping layer of dielectric medium C and thickness c on the superlattice surface. The medium outside the superlattice, at z < 0, is supposed to be filled with vacuum. The dielectric function in each medium may be frequency dependent, having the form ej = eooj[1-
2
2],
(1)
where O3pj is the plasma's frequency and ~oojis the background dielectric constant of medium j = a, b, or c. Assuming a p-polarization for the electromagnetic mode, the solutions of Maxwell's equations inside each media can be written as
Exj =
AI# exp(-ajz)
+
A2#exp(ajz),
Hyy = - iwe°eJ [AIyexp(-ajz) + A21 exp(ajz)], ~j
251
(2) (3)
PLASMONS IN DOPED SEMICONDUCTOR SUPERLATTICE
252
LX Free S~face
Z
z=O c
ns
z=c+a
b
T s = Ng'McN
B z =c+L
I
ndh cell
P
!
(11)
=I
fj-1
,
(12)
i
Nj, =
B
Fig. 1. The superlattice coordinate axes.
geometry
showing
the
(13)
Here, J) = exp ( - a j j ) , and ~j = ey/ay. On the other hand, inside the bulk of the superlattice we have [15] Tli - T22 + T12A - '1"21A-1 = 0,
where we have defined
(14)
where Tiy are elements of the T-matrix defined by [15]
(k2x - - £jW2/C2) i/2
OLj
iMc,
with (j = a, b, or c)
I
f
b
=
(10)
where the transfer-matrix T s is here defined by
A
P,
n
boundary conditions at z = c and z = c + a, we can find, after a bit of algebra, the following matrix equation lab) = TSlAc),
Capping Layer
a
Vol. 91, No. 3
i[ei(W2/C 2) - k 2 ] 1/2
if kx > f-ji/2 O d./ C , t i f k x < £ji/2 OJ/C,
T = NaiMbN~'lMa,
(15)
(4)
and j = a, b, or c. Outside the superlattice, the electromagnetic fields are Exo = Eo exp (aoZ),
(5)
Hy0 = (iwe0~0)E0 exp (a0z),
(6)
and A : A2b/Alb. The matrix Nj(j = a, b) is identical to the matrix N)~ defined in equation (13), provided we replace aj~ by aj = (rlj/rlj~)aj~, where r/j = n or p for j = a or b, respectively. At the free surface z = 0, Maxwell's boundary conditions yield
E o = Alc + A2c,
where
(16)
C~o = [k2 - w2/c2] 1/2,
for kx > w/c, and ~0 = l / a 0 .
~oEo = - ~ c ( A l c - A2c),
and therefore, At the interface z = 0, the x-component of the electrical field and the y-component of the magnetic field is continuous. On the other hand, at z = c and at z = c + a, the y-component of the magnetic field is discontinuous due to the presence of a current density given by
Jig = iw~oajsEj x,
(7)
where O'js ~ "
~]jse2 / m*w2eo
(8)
and r/1s = ns or p~ for j = a or b, respectively. Defining in each medium the vectors
iaj) = Alj
A2c _ ~c +~0 7 = Alc ~c -- GO N o w using equation (10) we can prove that -
T~I + T~27
+ T
fr
A2j
for j = a,b, and c, and using the electromagnetic
(18)
Equation (18) together with equation (14) represents an implicit dispersion relation for the surface plasmonpolaritons in this structure. The bulk dispersion relation for the infinitely extended superlattice is given in the usual way by [15] cos (QL) = (1/2)Tr(T),
(9)
(17)
(19)
where Q is the Bloch wave number, L = a + b is the size of the superlattice unit cell, and the transfer matrix T is defined in equation (15).
Vol. 91, No. 3
PLASMONS IN DOPED SEMICONDUCTOR SUPERLATTICE
We turn now our attention to the determination of the plasmon Green functions, necessary to calculate the power spectra of the bulk and surface plasmon modes, by using a macroscopic linear response formalism together with Maxwell's equations. This linear response procedure involves solving Maxwell's equations for the electrical field E in the presence of an externally applied harmonic polarization P, given by:
P = Po exp [i(Kzz + kxx
-
wt)].
(20)
Here, Kz is the z-component of the wavevector
K = (kx, O, Kz). Omitting the overall exp [i(kxx- wt)] factor, the solutions of Maxwell's equations inside each media can be given by equations (2) and (3), with the additional term Sj exp (iKz), where [15]
Sj = (w2~j/c2)po - (K. Po)K (r2 +c~2)eoej , j=a,b,c.
(21)
253
0.8
o~
0.6
la_ "~
0.4
g 0.2
0
0,5
I
1.5
2 Kxa
Outside the superlattice the electric field reads
E~o = S~o exp(iKzz) + Eo exp (a0z),
(22)
with S0 defined in a similar way of Sj. Using the electromagnetic boundary conditions described earlier, we can find, after a tedious but straightforward calculation, all Aij's coefficients (i = 1,2; j = a,b,c) defined in equation (9), and therefore the x-component of the electrical field in the j t h layer Exj, for j = a , b , c . The positiondependent Green function Gij= ((Ei(z);Ej(zt)*)} can now be obtained from the linear response of the electric field to the external polarization (20). Considering that in the formal expression of the light scattering cross section, is the Gzz(Z,Z') Green function that has the dominant weighting [16], we restrict our concern to this Green function, which is defined by
Gzz(Z,Z') = ((Ez(z);Ez(z'))} = Ez(z)/Pz(z),
(23)
where the z-component of the electrical field in the j t h layer is related to its x-component by
The Green function (23) has two main parts: one is due to the bulk sample and has the simple form G b=u l k , tz,
z ' ) -- - ~ ~o~j sgn(z - z') exp (-
jlz - z'l),
(25)
whereas the remaining terms have a much more complicated spatial dependence due to the multiple interfaces and the charge layers of the superlattice.
Fig. 2. Plasmon-polariton spectrum for the bulk and several surface modes denoted by r = 0.0, 0.3, 0.7, and 1.0, as explained in the main text. The dotted (full) line means the case where there is (there is not) a capping layer of thickness c = 0 . 5 L . The bulk continuum, shown shaded, is bounded by QL = 0 and 7r, respectively. We intend to show in this letter only its numerical computation in the low-frequency regime, which means no retardation effects. As it is well known, these terms have poles, in terms of the frequency w, corresponding to the dispersion relation for the surface plasmons described by equation (18). The spectral intensity can now be determined through the use of the fluctuation-dissipation theorem [17]
(IEz(z)l 2) =
(h/~r)[n(~)
+ 1] Im
((Ez(z); Ez(z')*)), (26)
where n(w) is the Bose-Einstein thermal factor. At low temperatures this thermal factor is not important. Next, we present some numerical examples to illustrate our results, stressing their dependence with the ratios ns/n and Ps/P. In what follows, we consider parameters values appropriate to GaAs/AlxGal _xAs superlattices [14], i.e., a = 40nm, b = 2a, c = 0.5L, e~a = 12.9, £oob = 12.3, n = p = 6 x 1015 m -2 and m* = 6.4 x 10-32 kg. Figure 2 shows the plasmon-polariton spectrum, where we have plotted a reduced frequency w/Wpa, with wm being the plama's frequency of medium A, against kxa. The bulk plasmon modes form a
PLASMONS IN DOPED S E M I C O N D U C T O R SUPERLATTICE
254
Vol. 91, No. 3
dotted (full) line. The curves show, qualitatively speaking, essentially similar behavior, although they have different strengths, indicating their dependence with the choice of r. The integrated intensity of the bulk modes is, of course, quite independent of r, and only weakly dependent of the capping layer. We do not intend to show it here, since it was already the object of our previous work [15]. In summary, we have presented a concise theoretical analysis of the plasmon-polariton spectrum in superlattices with the further effects of surface charge depletion and a capping layer. Our motivation has been primarily addressed to experimentalists working with Raman scattering from surface plasmon-polaritons, and we hope that the additional features of the plasmons surface mode, discussed here, might be tested by them.
I
0.8
o.6
_c 0.4
0.2
O O
O.5
I
1.5 Kxa
Fig. 3. Surface plasmon integrated intensity for r = 0.3 and 0.7. The dotted (full) line means the case where there is (there is not) a capping layer of thickness c = 0.5L. continuum with its boundaries defined by QL = 0 and 7r, and they are shown shaded here. Although, as it is well known, the spectrum has two bulk bands, we have presented here, for simplicity, only the lower bulk band. There are several surface plasmon branches corresponding to the cases where there is (dotted lines) and there is not (full lines) a capping layer, and for different values of r = ns/n =Ps/P. From this spectrum, we can infer some interesting results: First, note that the surface modes for r different of one emerge from the bulk continuum at a non-zero wavevector due to the surface charge depletion effect. For r equal to one the surface mode merges at kxa = 0, as expected. Second, the surface modes are strongly influenced by the values of r, and lie above or below the bulk continuum, the key factor being the value of r. For instance, in our case we notice that the critical value of r, from which we have surface modes lying above the bulk modes is around 0.55 without a capping layer, and around 0.45 with a capping layer. Third, the capping layer is responsible for a shift in the surface lines, as we can see comparing the dotted and full lines for r = 0.3 and r = 0.7, respectively. Figure 3 shows the integrated intensity, which is given apart of a constant factor by Im Gzz(Z,z') with z = z ' = c/2, against kxa, for r - - 0.3 and r =0.7, respectively~ Again, we consider the case with (without) a capping layer here represented by a
Acknowledgement. - - We thank the Brazilian Research Council CNPq for partial financial support. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.
G.F. Giuliani & J.J. Quinn, Phys. Rev. Lett. 51, 919 (1983). S. Das Sarma & J.J. Quinn, Phys. Rev. B12, 7603 (1982); Errata 27, 6516 (1983). A . Tselis & J.J. Quinn, Phys. Rev. B29, 3318 (1984). R. Mayanovic, G.F. Giuliani & J.J. Quinn, Phys. Rev. B33, 8390 (1986). G.A. Farias, M.M. Auto & E.L. Albuquerque, Phys. Rev. B38, 12540 (1988). B.L. Johnson & R.E. Camley, Phys. Rev. B38, 3311 (1988). M.G. Cottam & N.C. Constantinou, Solid State Commun. 69, 321 (1989). G. Fasol, R.D. King-Smith, D. Richards, U. Ekenberg, N. Mestres & K. Ploog, Phys. Rev. B39, 12695 (1989). G.A. Farias, M.M. Auto, E.L. Albuquerque & P. Fulco, Z. Phys. 1180, 207 (1990). D. Olego, A. Pinczuk, A.C. Gossard & W. Wiegmann, Phys. Rev. B25, 7867 (1982). A. Pinczuk, M.G. Lamont & A.C. Gossard, Phys. Rev. Lett. 56, 2092 (1986). G. Fasol, N. Mestres, H.P. Hughes, A. Fisher & K. Ploog, Phys. Rev. Lett. 56, 2517 (1986). N.C. Constantinou & M.G. Cottam, SolidState Commun. 81, 321 (1992). E.L. Albuquerque & M.G. Cottam, Solid State Commun. 81, 383 (1992). E.L. Albuquerque, Phys. Lett. A150, 417 (1990). M. Babiker, N.C. Constantinou & M.G. Cottam, J. Phys. C19, 5849 (1986). W. Hayes & R. Loudon, Scattering of Light by Crystals, Wiley, New York (1978).
Solid State Communications, Vol. 91, No. 3, pp. 251-254, 1994 Elsevier Science Ltd Printed in Great Britain 0038-1098/94 $7.00 + .00
t Pergamon
0038-1098(94)E0251-6 PLASMONS IN DOPED SEMICONDUCTOR SUPERLATTICE WITH CHARGE DEPLETION AND A CAPPING LAYER E.L. Albuquerque Departamento de Fisica, Universidade Federal do Rio Grande do Norte, 59072-970-Natal, RN-Brazil (Received 15 November 1993; in revised form 14 March 1994 by A.A. Maradudin) We present a theoretical analysis for the collective plasmon-polariton modes of a doped superlattice consisting of two-dimensional electron gas and two-dimensional hole gas layers separated by media of alternating thickness and dielectric constant. Our theory takes into account the effects due to a capping layer, as well as charge depletion near the superlattice surface. We use a theoretical model based on linear response function together with a transfer matrix approach to determine the plasmons dispersion relation and Green functions. Numerical applications are made to semiconductor superlattices, with interesting features for the plasmon surface modes.
WITH the development in recent years of modulationdoped semiconductor superlattices, many investigations have been turned out to determine the collective excitations of periodic arrays of two-dimensional electron (hole) gas layers. Theories for the spectra of bulk and surface plasmon in these structures have been given, using various different models, by a number of authors [1-9]. On the experimental side, the Raman scattering measurements of Olego et al. [10] have provided confirmation of the theoretical predictions for the plasmon dispersion relation, with subsequent Raman scattering experiments for quantized plasmon modes of finite superlattices [11, 12]. More sophisticated structure, taking into account charge depletion in the electron gas layer nearest to the surface as well as the presence of a capping layer at the surface, has been recently reported [13], and these effects have been proved to give strong influence on the surface plasmon-polariton properties. It is the aim of this letter to present theoretical considerations about surface plasmon-polariton modes in doped semiconductor superlattice made up of two-dimensional electron (2DEG) and hole (2DHG) gas, respectively, separated by dielectric media. Our investigation takes also into account the effects of charge depletion and the presence of a capping layer. Our purpose is two-fold: First, we want to extend the work of Constantinou and Cottam [13], by considering a more complete
structure, as well as our previous work [14], by including surface effects. Second, we would like to explore the existence of novel aspects in the surface plasmon-polariton spectra. Figure 1 shows our superlattiee model, where there is a 2DEG at z = c, L + c, 2L + c, etc., with L=a+b, a and b being the thickness of the dielectric media A and B that fills the space between the charge layers. Also, a 2DHG is found at z = c + a, L + c + a, 2L + c + a, etc. We denote the carrier concentration in each layer by n and p, except near the surface, where it has a different value ns and Ps. There is also a capping layer of dielectric medium C and thickness c on the superlattice surface. The medium outside the superlattice, at z < 0, is supposed to be filled with vacuum. The dielectric function in each medium may be frequency dependent, having the form ej = eooj[1-
2
2],
(1)
where O3pj is the plasma's frequency and ~oojis the background dielectric constant of medium j = a, b, or c. Assuming a p-polarization for the electromagnetic mode, the solutions of Maxwell's equations inside each media can be written as
Exj =
AI# exp(-ajz)
+
A2#exp(ajz),
Hyy = - iwe°eJ [AIyexp(-ajz) + A21 exp(ajz)], ~j
251
(2) (3)
PLASMONS IN DOPED SEMICONDUCTOR SUPERLATTICE
252
LX Free S~face
Z
z=O c
ns
z=c+a
b
T s = Ng'McN
B z =c+L
I
ndh cell
P
!
(11)
=I
fj-1
,
(12)
i
Nj, =
B
Fig. 1. The superlattice coordinate axes.
geometry
showing
the
(13)
Here, J) = exp ( - a j j ) , and ~j = ey/ay. On the other hand, inside the bulk of the superlattice we have [15] Tli - T22 + T12A - '1"21A-1 = 0,
where we have defined
(14)
where Tiy are elements of the T-matrix defined by [15]
(k2x - - £jW2/C2) i/2
OLj
iMc,
with (j = a, b, or c)
I
f
b
=
(10)
where the transfer-matrix T s is here defined by
A
P,
n
boundary conditions at z = c and z = c + a, we can find, after a bit of algebra, the following matrix equation lab) = TSlAc),
Capping Layer
a
Vol. 91, No. 3
i[ei(W2/C 2) - k 2 ] 1/2
if kx > f-ji/2 O d./ C , t i f k x < £ji/2 OJ/C,
T = NaiMbN~'lMa,
(15)
(4)
and j = a, b, or c. Outside the superlattice, the electromagnetic fields are Exo = Eo exp (aoZ),
(5)
Hy0 = (iwe0~0)E0 exp (a0z),
(6)
and A : A2b/Alb. The matrix Nj(j = a, b) is identical to the matrix N)~ defined in equation (13), provided we replace aj~ by aj = (rlj/rlj~)aj~, where r/j = n or p for j = a or b, respectively. At the free surface z = 0, Maxwell's boundary conditions yield
E o = Alc + A2c,
where
(16)
C~o = [k2 - w2/c2] 1/2,
for kx > w/c, and ~0 = l / a 0 .
~oEo = - ~ c ( A l c - A2c),
and therefore, At the interface z = 0, the x-component of the electrical field and the y-component of the magnetic field is continuous. On the other hand, at z = c and at z = c + a, the y-component of the magnetic field is discontinuous due to the presence of a current density given by
Jig = iw~oajsEj x,
(7)
where O'js ~ "
~]jse2 / m*w2eo
(8)
and r/1s = ns or p~ for j = a or b, respectively. Defining in each medium the vectors
iaj) = Alj
A2c _ ~c +~0 7 = Alc ~c -- GO N o w using equation (10) we can prove that -
T~I + T~27
+ T
fr
A2j
for j = a,b, and c, and using the electromagnetic
(18)
Equation (18) together with equation (14) represents an implicit dispersion relation for the surface plasmonpolaritons in this structure. The bulk dispersion relation for the infinitely extended superlattice is given in the usual way by [15] cos (QL) = (1/2)Tr(T),
(9)
(17)
(19)
where Q is the Bloch wave number, L = a + b is the size of the superlattice unit cell, and the transfer matrix T is defined in equation (15).
Vol. 91, No. 3
PLASMONS IN DOPED SEMICONDUCTOR SUPERLATTICE
We turn now our attention to the determination of the plasmon Green functions, necessary to calculate the power spectra of the bulk and surface plasmon modes, by using a macroscopic linear response formalism together with Maxwell's equations. This linear response procedure involves solving Maxwell's equations for the electrical field E in the presence of an externally applied harmonic polarization P, given by:
P = Po exp [i(Kzz + kxx
-
wt)].
(20)
Here, Kz is the z-component of the wavevector
K = (kx, O, Kz). Omitting the overall exp [i(kxx- wt)] factor, the solutions of Maxwell's equations inside each media can be given by equations (2) and (3), with the additional term Sj exp (iKz), where [15]
Sj = (w2~j/c2)po - (K. Po)K (r2 +c~2)eoej , j=a,b,c.
(21)
253
0.8
o~
0.6
la_ "~
0.4
g 0.2
0
0,5
I
1.5
2 Kxa
Outside the superlattice the electric field reads
E~o = S~o exp(iKzz) + Eo exp (a0z),
(22)
with S0 defined in a similar way of Sj. Using the electromagnetic boundary conditions described earlier, we can find, after a tedious but straightforward calculation, all Aij's coefficients (i = 1,2; j = a,b,c) defined in equation (9), and therefore the x-component of the electrical field in the j t h layer Exj, for j = a , b , c . The positiondependent Green function Gij= ((Ei(z);Ej(zt)*)} can now be obtained from the linear response of the electric field to the external polarization (20). Considering that in the formal expression of the light scattering cross section, is the Gzz(Z,Z') Green function that has the dominant weighting [16], we restrict our concern to this Green function, which is defined by
Gzz(Z,Z') = ((Ez(z);Ez(z'))} = Ez(z)/Pz(z),
(23)
where the z-component of the electrical field in the j t h layer is related to its x-component by
The Green function (23) has two main parts: one is due to the bulk sample and has the simple form G b=u l k , tz,
z ' ) -- - ~ ~o~j sgn(z - z') exp (-
jlz - z'l),
(25)
whereas the remaining terms have a much more complicated spatial dependence due to the multiple interfaces and the charge layers of the superlattice.
Fig. 2. Plasmon-polariton spectrum for the bulk and several surface modes denoted by r = 0.0, 0.3, 0.7, and 1.0, as explained in the main text. The dotted (full) line means the case where there is (there is not) a capping layer of thickness c = 0 . 5 L . The bulk continuum, shown shaded, is bounded by QL = 0 and 7r, respectively. We intend to show in this letter only its numerical computation in the low-frequency regime, which means no retardation effects. As it is well known, these terms have poles, in terms of the frequency w, corresponding to the dispersion relation for the surface plasmons described by equation (18). The spectral intensity can now be determined through the use of the fluctuation-dissipation theorem [17]
(IEz(z)l 2) =
(h/~r)[n(~)
+ 1] Im
((Ez(z); Ez(z')*)), (26)
where n(w) is the Bose-Einstein thermal factor. At low temperatures this thermal factor is not important. Next, we present some numerical examples to illustrate our results, stressing their dependence with the ratios ns/n and Ps/P. In what follows, we consider parameters values appropriate to GaAs/AlxGal _xAs superlattices [14], i.e., a = 40nm, b = 2a, c = 0.5L, e~a = 12.9, £oob = 12.3, n = p = 6 x 1015 m -2 and m* = 6.4 x 10-32 kg. Figure 2 shows the plasmon-polariton spectrum, where we have plotted a reduced frequency w/Wpa, with wm being the plama's frequency of medium A, against kxa. The bulk plasmon modes form a
PLASMONS IN DOPED S E M I C O N D U C T O R SUPERLATTICE
254
Vol. 91, No. 3
dotted (full) line. The curves show, qualitatively speaking, essentially similar behavior, although they have different strengths, indicating their dependence with the choice of r. The integrated intensity of the bulk modes is, of course, quite independent of r, and only weakly dependent of the capping layer. We do not intend to show it here, since it was already the object of our previous work [15]. In summary, we have presented a concise theoretical analysis of the plasmon-polariton spectrum in superlattices with the further effects of surface charge depletion and a capping layer. Our motivation has been primarily addressed to experimentalists working with Raman scattering from surface plasmon-polaritons, and we hope that the additional features of the plasmons surface mode, discussed here, might be tested by them.
I
0.8
o.6
_c 0.4
0.2
O O
O.5
I
1.5 Kxa
Fig. 3. Surface plasmon integrated intensity for r = 0.3 and 0.7. The dotted (full) line means the case where there is (there is not) a capping layer of thickness c = 0.5L. continuum with its boundaries defined by QL = 0 and 7r, and they are shown shaded here. Although, as it is well known, the spectrum has two bulk bands, we have presented here, for simplicity, only the lower bulk band. There are several surface plasmon branches corresponding to the cases where there is (dotted lines) and there is not (full lines) a capping layer, and for different values of r = ns/n =Ps/P. From this spectrum, we can infer some interesting results: First, note that the surface modes for r different of one emerge from the bulk continuum at a non-zero wavevector due to the surface charge depletion effect. For r equal to one the surface mode merges at kxa = 0, as expected. Second, the surface modes are strongly influenced by the values of r, and lie above or below the bulk continuum, the key factor being the value of r. For instance, in our case we notice that the critical value of r, from which we have surface modes lying above the bulk modes is around 0.55 without a capping layer, and around 0.45 with a capping layer. Third, the capping layer is responsible for a shift in the surface lines, as we can see comparing the dotted and full lines for r = 0.3 and r = 0.7, respectively. Figure 3 shows the integrated intensity, which is given apart of a constant factor by Im Gzz(Z,z') with z = z ' = c/2, against kxa, for r - - 0.3 and r =0.7, respectively~ Again, we consider the case with (without) a capping layer here represented by a
Acknowledgement. - - We thank the Brazilian Research Council CNPq for partial financial support. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.
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