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Effects of finite layer thickness on superlattice plasmons
594
Surface
EFFECTS OF FINITE LAYER THICKNESS PLASMONS
Science 136 (1984) 594-600 North-Holland, Amsterdam
ON SUPERLATTICE
W.L. BLOSS GTE Laboratories, Received
Waltham, Massachusetts
20 May 1983; accepted
02254, USA
for publication
29 August
1983
In recent publications, the dispersion relation for the plasmon oscillations of a semiconductor superlattice, consisting of a periodic array of quantum wells, doped with electrons, was calculated in the limit that the well size approached zero. Here, we extend this theory to include wells of finite thickness. In most situations of physical interest, it is shown that the superlattice plasmon dispersion is modified by only a few percent becauseof the finitewell size.
1. Introduction Semiconductor superlattices, with precise control of layer dimensions, are now a reality as a result of recent advancements in the molecular beam epitaxy (MBE) growth process [1,2]. The MBE process of crystal growth consists of directing thermal beams of atoms (or molecules) onto a heated substrate. As a result of this unique growth process, new structures with different and unusual physical and electronic properties, that were heretofore impossible to make, can now be fabricated. GaAs-GaAlAs superlattices, being the most extensively characterized system so far, can be grown with layer thicknesses as small as tens of angstroms. This type 1 superlattice system forms a periodic array of wells due to the band gap discontinuities between the conduction and valence bands of GaAs and GaAlAs. In fig. 1 is depicted a GaAs-GaAlAs superlattice showing the resultant band structure; here d is the superlattice periodicity and 1 is the well size. One can modulate dope this superlattice system in the MBE growth process by placing the dopants in the GaAlAs barrier regions [3]. Because of the band gap discontinuities, electrons will accumulate in the GaAs wells. This results in a separation of the impurity scattering centers from the carriers giving rise to large mobilities at low temperatures. Consequently, this system is also of prime technological importance [4] for high speed GaAs logic circuits. If the well size I is less than the electron’s de Broglie wavelength, then the electron will be confined to a series of quantum levels (the so-called subband structure) [5]. In fig. 1 we show two subband energies E, and E, for 0039-6028/84/$03.00 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
W. L.. Bless /
Effects of finite layer thickness
595
4V* P
P+l
Fig. 1. A GaAs-GaAlAs superlattice consisting of a periodic array of quantum wells: d is the well periodicity, I is the well width; I?, is the ground subband energy, E, is the first excited subband, and E, is the Fermi level.
each well. Here, we are interested only in the situation where the wells are widely separated in space so that electron wave function overlap between wells is negligible. Thus, electrons cannot move along the superlattice axis and are confined to the well. However, electron motion within the well (perpendicular to the superlattice axis) will not be confined and is free-electron-like. The net result is that one has a system consisting of a periodic array of wells of quasi-two-dimensional electron gases arranged in a superlattice structure. As first shown by Ritchie [6], a two-dimensional electron gas has a plasmon oscillation which varies as q ‘n where q is the in-plane wave vector (in the layer (well) perpendicular to the superlattice axis). Although here we are neglecting interaction between wells by direct electron wave function overlap, the 2D plasmon oscillations of each well can still couple via the long-range Coulomb interaction. In fact, the 2D plasmons of each well will couple into bands of oscillations (the so-called superlattice plasmons) that can propagate along the superlattice axis. In some recent work, a number of researchers have investigated the superlattice plasmon modes of such a periodic system of quasi-2D electron gases. A rather complete analysis of the electrodynamics of the quasi-2D system can be found in the work of Dahl and Sham [7]. Fetter [8], using a hydrodynamic approximation, calculated the plasmon dispersion relation for the superlattice within the limit that the well thickness is zero. We call this the sheet model in what follows. More recent work by Das Sarma and Quinn [9], and by the present author have investigated the problem for a number of different situations. In particular, we have presented a theory [lo] that allows one to include the effects of subband structure and finite layer widths into the calculation for the wave-vector frequency-dependent dielectric constant for the
596
W. L. Bless / Effects of finite layer thickness
superlattice in a straightforward way. Using this model, we have studied the superlattice plasmon modes and also the interaction of these modes with LO optic phonons and magnetic field. In recent experimental work, Olego et al. [ll] have measured the superlattice plasmon dispersion via inelastic light scattering and have verified the prediction that these oscillations are acoustic (o - q) for the in-plane component of the momentum. We have also presented a theory for the plasmon modes of a superlattice [12] in which the periodicity of the superlattice is doubled; such as a superlattice with alternating densities n, and n2. In analogy, with the diatomic situation in phonon dynamics, optic and acoustic-like plasmon bands were found. Finally, excitations between quantum subbands have also been investigated [13]. These excitations are accompanied by a charge response called the depolarization effect [14]. It was shown that these excitations also couple Coulombically when arranged in a superlattice giving rise to intersubband superlattice plasmons. In this short paper, we extend our previous work to include the effect of finite layer thickness (well size) on the superlattice plasmon dispersion relation.
2. Theory Our model is exactly the same as presented in our earlier work where a theory was developed that included subband structure in the calculation of the dielectric constant for a superlattice [lo]. However, in the calculation for the ground superlattice plasmon, we essentially made the approximation that the wave function in each well was a b-function; hence, our results were equivalent to the sheet model of Fetter. Here, we relax this approximation and take as our wave functions those of an infinite square well and calculate the Coulomb interaction between wells using these wave functions. Since details of the theory can be found in our earlier papers [10,13], we shall be brief here. The plasmon modes of a uniform system are determined from the zeros of the wave-vector frequency-dependent dielectric constant. For the superlattice system under investigation here, the plasmon modes are determined from the zeroes of the determinant of the dielectric constant which is now an n x n matrix defined on well indices i andj.
(1) where l’$(q, w) is the ground subband polarization. Since we are only interested in the ground superlattice plasmons in this paper, we restrict ourselves to the lowest quantum subband in eq. (1). The electron energy dispersion in each well is two-dimensional, E = E, + h2q2/2m,
(2)
W. L. Bless / Effects o/finite layer thickness
where q is the in-plane momentum. the ground polarization, I$(q,
w) = nq*/mw*,
for
Using this dispersion
relation,
0 > qV,,
597
one finds for
(3)
to order q* and where n is the 2D well density, m the effective mass, and V, the Fermi velocity. The Coulomb interaction between wells occurring in eq. (1) is given in terms of subband (ground) wave functions by y,,Cq)=T//dz
dz’e-qlZ-Z”+:(z)
where c is the dielectric
constant.
+(z)=\/2/1
sin(7rz/l),
for
+T(z’) +,(z’)+,(z).
Evaluating
eq. (4) for ground
wave functions
OSzli,
we find
Y,,,(q)= (2qe*/qe)F(q), for i =j, Y:,,(q)= (2ne*/qc)I(q), for i #_i, factors F(q)
where structure
and I(q)
are given by
’ 1 IcqJ =2(W4bh(4) -11 q2P[
q*P
+(297)*]*
*
(54 (5b)
(64 (6’4
F(q)
is the intrawell form factor and I(q) is the interwell form factor. In fig. 2, we plot F(q) and I(q) as a function of r$. We see that I(q) varies little with ql and remains essentially unity, while F(qj”shows a much stronger variation. We now assess the effect of finite layer width on the 2D plasmon energy of a single well (layer). The plasmon dispersion is given by the zero of dielectric constant (eq. (1)) which is now a scalar equation. It is straightforward to show that the plasmon dispersion is
w = 6$“(q)
F( q)l’*,
0)
where wzD(q) = (2lrne*q/mc)‘/*. As 414 0, F(q) -+ 1, and eq. (7) reduces to aiD( the 2D sheet plasmon mode. In fig. 3 we plot eq. (7) taking into account finite layer width via the F(q) structure factor. Writing eq. (7) as w = (w,/JT)(q1)“’
F(q)“‘,
(8)
where wi = 4sne*/mtl is the 3D plasmon, we plot eq. (8) in units of w,/& in fig. 3. The dashed line is for F(q) = 1, the sheet case; solid line includes F(q),
0.6 -
0.4 -
0.2 -
I
I
1.0
0.5
0.6
qp
qf
Fig. 2. Plot of structure factors F(q) and I(q) (defined by eq. (6)) versus
ql.
Fig. 3. Plot of the quasi tw~d~ensional plasmon dispersion equation (7). including structure factor F(q) in reduced energy units (see text). Dashed line is the sheet limit, F(q) = 1.
the finite well case. For q/a~0.5, the correction to the 2D plasmon dispersion is about 5%. We note here that at large wave vectors there are also other contributions besides finite size corrections that have not been included in this paper. In particular, one could easily include the RPA polarization to order q4. In addition, exchange-correlation effects become very important at large wave vectors as shown by Beck and Kumar [15], Rajagopal[16], and Jonson 1173. We point out that for the single well plasmon and the plasmon modes of the superlattice to be discussed in what follows that both kinds of contributions must be included if direct comparison with experiment is made at large wave vectors. We now assess the effects of finite layer thickness on the superlattice plasmons. As shown in ref. [lo], to find the plasmon dispersion, we transform eq. (1) to collective coordinates. After some simple algebra, the superlattice plasmon modes are found to be
l/2 0=~2D(4) Jw +m Ic” 1 e-qlnld
[
elknd
9
R==-CC tz#O
where k is the wave vector along the superlattice axis. Rewriting eq. (9) as &J =
W2D(4) CS(q,w’2,
(9)
W.L. Bless / Effects of finite layer thickness
where S(q, k), the structure be
‘(q, k, = %) +‘(d
factor for the superlattice cos( kd) - ewqd
cosh(qd)
_
cos(kd)
f
plasmon,
599
is evaluated
to
(11)
and F(q) and I(q) are defined by eq. (6). In particular, for F(q) and I(q) equal to unity, this reduces to the result for the superlattice plasmons in the sheet approximation [S]. Thus, the finite size does not affect the plasmon dispersion to lowest order in wave vector. And to next order in ql, we find the correction term to be w=
w,J q, k) - O.l54C$D( q) ql
cosh( qd
) - cos( kd )
02)
where
w,,,(q,k) = azD
(q)(
is the sheet superlattice
sinh( qd) cosh(qd) plasmon
- cos( kd)
(13)
as ql + 0. We plot eq. (10) versus qd, keeping
/ /’
w
(mev)
0.5
1.0
qd Fig. 4. Plot of superlattice plasmon dispersion including finite layer widths. Bottom curves are for @3D= 11.4 meV and I/d = 0.3. Top curves for wp 3D = 23 meV and I/d = 0.67. Solid lines include stkcture factors F(q) and I(q). Dashed line is the two-dimensional sheet limit, F(q) and I(q) equal to one.
600
W.L. Bless / Effects offmite layer thrckness
kd
fixed, in fig. 4 for values of parameters similar to those used in the Olego et al. [ll] light scattering experiment (the lower curves) from a modulated doped GaAs-GaAlAs superlattice. Here, wP= 11.4 meV, l/d = 0.3, and kd is fixed at 4.9. The solid line includes the effect of finite layer thickness via F(q) and i(q); the dashed line is for F(q) and I(q) equal to unity. For qd - 1.O, about the highest value in the Olego et al. experiments, corrections are small, again about 5%. However, for larger i/d ratios; l/d = 0.67 for the upper curves and taking wP = 23 meV, we see that corrections can be as large as 10%. For qd - 1, this corresponds to a lowering in energy of about 1 meV. In conclusion, we have presented a theory that allows one to include finite wave functions into the calculation of the superlattice dielectric constant. A extension of this theory to include subband structure has been recently presented. In that case, we find that coupled intersubband modes can propagate along the superlattice axis [13]. Here, we have considered the effects of finite layer width on the ground superlattice plasmon modes. We find that in most cases and regions of interest (qd or ql small) the effects on the superlattice plasmon dispersion are very small (less than a few percent). However, if the l/d ratio (layer width to superlattice periodicity) is large and qd large, corrections to the superlattice plasmon may be as much as 10%.
References [l] A.Y. Cho and J.R. Arthur, Progr. Solid State Chem. 10 (1975) 157. [2] L.L. Chang and L. Esaki, Progr. Crystal Growth Characterization 2 (1979) 3; Surface Sci. 98 (1980) 70. [3] R. Dingle, H.L. Stormer, AC. Gossard and W. Wiegmann, Appl. Phys. Letters 33 (1978) 665. [4] T. Mimura, S. Hiyamizu, T. Fuju and K. Nassbu, Japan. J. Appt. Phys. 19 (1980) L225; T. Mimura, S. Hiyamizu, K, Joshin and K. Hikosaka, Japan. 3. Appl. Phys. 20 (1981) L317. [5] T. Ando, A.B. Fowler and F. Stern, Rev. Mod. Phys. 54 (1982) 437. [6] R.H. Ritchie, Phys. Rev. 106 (1957) 874. [7] D.A. Dahl and L.J. Sham, Phys. Rev. B16 (1977) 651. [8] A.L. Fetter, Ann. Phys. 88 (1974) 1. [9] S. Das Sarma and J.J. Quinn, Phys. Rev. B25 (1982) 7603. [lo] W.L. Bloss and E.M. Brody, Solid State Commun. 43 (1982) 523. [ll] D. Olego, A. Pinczuk, A.C. Gossard and W. Wiegmann, Phys. Rev. B25 (1982) 7867. [12] W.L. Bless, Solid State Commun. 44 (1982) 363. [13] W.L. Bless, Solid State Commun. 46 (1982) 143. [14] S.J. Allen, Jr., D.C. Tsui and B. Vinter, Solid State Commun. 20 (1976) 425. 1151 D.E. Beck and P. Kumar, Phys. Rev. B13 (1976) 2859. [16] A.K. Rajagopal, Phys. Rev. BlS (1977) 4264. [17] M. Jonson, J. Phys. C9 (1976) 3055.
Surface
EFFECTS OF FINITE LAYER THICKNESS PLASMONS
Science 136 (1984) 594-600 North-Holland, Amsterdam
ON SUPERLATTICE
W.L. BLOSS GTE Laboratories, Received
Waltham, Massachusetts
20 May 1983; accepted
02254, USA
for publication
29 August
1983
In recent publications, the dispersion relation for the plasmon oscillations of a semiconductor superlattice, consisting of a periodic array of quantum wells, doped with electrons, was calculated in the limit that the well size approached zero. Here, we extend this theory to include wells of finite thickness. In most situations of physical interest, it is shown that the superlattice plasmon dispersion is modified by only a few percent becauseof the finitewell size.
1. Introduction Semiconductor superlattices, with precise control of layer dimensions, are now a reality as a result of recent advancements in the molecular beam epitaxy (MBE) growth process [1,2]. The MBE process of crystal growth consists of directing thermal beams of atoms (or molecules) onto a heated substrate. As a result of this unique growth process, new structures with different and unusual physical and electronic properties, that were heretofore impossible to make, can now be fabricated. GaAs-GaAlAs superlattices, being the most extensively characterized system so far, can be grown with layer thicknesses as small as tens of angstroms. This type 1 superlattice system forms a periodic array of wells due to the band gap discontinuities between the conduction and valence bands of GaAs and GaAlAs. In fig. 1 is depicted a GaAs-GaAlAs superlattice showing the resultant band structure; here d is the superlattice periodicity and 1 is the well size. One can modulate dope this superlattice system in the MBE growth process by placing the dopants in the GaAlAs barrier regions [3]. Because of the band gap discontinuities, electrons will accumulate in the GaAs wells. This results in a separation of the impurity scattering centers from the carriers giving rise to large mobilities at low temperatures. Consequently, this system is also of prime technological importance [4] for high speed GaAs logic circuits. If the well size I is less than the electron’s de Broglie wavelength, then the electron will be confined to a series of quantum levels (the so-called subband structure) [5]. In fig. 1 we show two subband energies E, and E, for 0039-6028/84/$03.00 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
W. L.. Bless /
Effects of finite layer thickness
595
4V* P
P+l
Fig. 1. A GaAs-GaAlAs superlattice consisting of a periodic array of quantum wells: d is the well periodicity, I is the well width; I?, is the ground subband energy, E, is the first excited subband, and E, is the Fermi level.
each well. Here, we are interested only in the situation where the wells are widely separated in space so that electron wave function overlap between wells is negligible. Thus, electrons cannot move along the superlattice axis and are confined to the well. However, electron motion within the well (perpendicular to the superlattice axis) will not be confined and is free-electron-like. The net result is that one has a system consisting of a periodic array of wells of quasi-two-dimensional electron gases arranged in a superlattice structure. As first shown by Ritchie [6], a two-dimensional electron gas has a plasmon oscillation which varies as q ‘n where q is the in-plane wave vector (in the layer (well) perpendicular to the superlattice axis). Although here we are neglecting interaction between wells by direct electron wave function overlap, the 2D plasmon oscillations of each well can still couple via the long-range Coulomb interaction. In fact, the 2D plasmons of each well will couple into bands of oscillations (the so-called superlattice plasmons) that can propagate along the superlattice axis. In some recent work, a number of researchers have investigated the superlattice plasmon modes of such a periodic system of quasi-2D electron gases. A rather complete analysis of the electrodynamics of the quasi-2D system can be found in the work of Dahl and Sham [7]. Fetter [8], using a hydrodynamic approximation, calculated the plasmon dispersion relation for the superlattice within the limit that the well thickness is zero. We call this the sheet model in what follows. More recent work by Das Sarma and Quinn [9], and by the present author have investigated the problem for a number of different situations. In particular, we have presented a theory [lo] that allows one to include the effects of subband structure and finite layer widths into the calculation for the wave-vector frequency-dependent dielectric constant for the
596
W. L. Bless / Effects of finite layer thickness
superlattice in a straightforward way. Using this model, we have studied the superlattice plasmon modes and also the interaction of these modes with LO optic phonons and magnetic field. In recent experimental work, Olego et al. [ll] have measured the superlattice plasmon dispersion via inelastic light scattering and have verified the prediction that these oscillations are acoustic (o - q) for the in-plane component of the momentum. We have also presented a theory for the plasmon modes of a superlattice [12] in which the periodicity of the superlattice is doubled; such as a superlattice with alternating densities n, and n2. In analogy, with the diatomic situation in phonon dynamics, optic and acoustic-like plasmon bands were found. Finally, excitations between quantum subbands have also been investigated [13]. These excitations are accompanied by a charge response called the depolarization effect [14]. It was shown that these excitations also couple Coulombically when arranged in a superlattice giving rise to intersubband superlattice plasmons. In this short paper, we extend our previous work to include the effect of finite layer thickness (well size) on the superlattice plasmon dispersion relation.
2. Theory Our model is exactly the same as presented in our earlier work where a theory was developed that included subband structure in the calculation of the dielectric constant for a superlattice [lo]. However, in the calculation for the ground superlattice plasmon, we essentially made the approximation that the wave function in each well was a b-function; hence, our results were equivalent to the sheet model of Fetter. Here, we relax this approximation and take as our wave functions those of an infinite square well and calculate the Coulomb interaction between wells using these wave functions. Since details of the theory can be found in our earlier papers [10,13], we shall be brief here. The plasmon modes of a uniform system are determined from the zeros of the wave-vector frequency-dependent dielectric constant. For the superlattice system under investigation here, the plasmon modes are determined from the zeroes of the determinant of the dielectric constant which is now an n x n matrix defined on well indices i andj.
(1) where l’$(q, w) is the ground subband polarization. Since we are only interested in the ground superlattice plasmons in this paper, we restrict ourselves to the lowest quantum subband in eq. (1). The electron energy dispersion in each well is two-dimensional, E = E, + h2q2/2m,
(2)
W. L. Bless / Effects o/finite layer thickness
where q is the in-plane momentum. the ground polarization, I$(q,
w) = nq*/mw*,
for
Using this dispersion
relation,
0 > qV,,
597
one finds for
(3)
to order q* and where n is the 2D well density, m the effective mass, and V, the Fermi velocity. The Coulomb interaction between wells occurring in eq. (1) is given in terms of subband (ground) wave functions by y,,Cq)=T//dz
dz’e-qlZ-Z”+:(z)
where c is the dielectric
constant.
+(z)=\/2/1
sin(7rz/l),
for
+T(z’) +,(z’)+,(z).
Evaluating
eq. (4) for ground
wave functions
OSzli,
we find
Y,,,(q)= (2qe*/qe)F(q), for i =j, Y:,,(q)= (2ne*/qc)I(q), for i #_i, factors F(q)
where structure
and I(q)
are given by
’ 1 IcqJ =2(W4bh(4) -11 q2P[
q*P
+(297)*]*
*
(54 (5b)
(64 (6’4
F(q)
is the intrawell form factor and I(q) is the interwell form factor. In fig. 2, we plot F(q) and I(q) as a function of r$. We see that I(q) varies little with ql and remains essentially unity, while F(qj”shows a much stronger variation. We now assess the effect of finite layer width on the 2D plasmon energy of a single well (layer). The plasmon dispersion is given by the zero of dielectric constant (eq. (1)) which is now a scalar equation. It is straightforward to show that the plasmon dispersion is
w = 6$“(q)
F( q)l’*,
0)
where wzD(q) = (2lrne*q/mc)‘/*. As 414 0, F(q) -+ 1, and eq. (7) reduces to aiD( the 2D sheet plasmon mode. In fig. 3 we plot eq. (7) taking into account finite layer width via the F(q) structure factor. Writing eq. (7) as w = (w,/JT)(q1)“’
F(q)“‘,
(8)
where wi = 4sne*/mtl is the 3D plasmon, we plot eq. (8) in units of w,/& in fig. 3. The dashed line is for F(q) = 1, the sheet case; solid line includes F(q),
0.6 -
0.4 -
0.2 -
I
I
1.0
0.5
0.6
qp
qf
Fig. 2. Plot of structure factors F(q) and I(q) (defined by eq. (6)) versus
ql.
Fig. 3. Plot of the quasi tw~d~ensional plasmon dispersion equation (7). including structure factor F(q) in reduced energy units (see text). Dashed line is the sheet limit, F(q) = 1.
the finite well case. For q/a~0.5, the correction to the 2D plasmon dispersion is about 5%. We note here that at large wave vectors there are also other contributions besides finite size corrections that have not been included in this paper. In particular, one could easily include the RPA polarization to order q4. In addition, exchange-correlation effects become very important at large wave vectors as shown by Beck and Kumar [15], Rajagopal[16], and Jonson 1173. We point out that for the single well plasmon and the plasmon modes of the superlattice to be discussed in what follows that both kinds of contributions must be included if direct comparison with experiment is made at large wave vectors. We now assess the effects of finite layer thickness on the superlattice plasmons. As shown in ref. [lo], to find the plasmon dispersion, we transform eq. (1) to collective coordinates. After some simple algebra, the superlattice plasmon modes are found to be
l/2 0=~2D(4) Jw +m Ic” 1 e-qlnld
[
elknd
9
R==-CC tz#O
where k is the wave vector along the superlattice axis. Rewriting eq. (9) as &J =
W2D(4) CS(q,w’2,
(9)
W.L. Bless / Effects of finite layer thickness
where S(q, k), the structure be
‘(q, k, = %) +‘(d
factor for the superlattice cos( kd) - ewqd
cosh(qd)
_
cos(kd)
f
plasmon,
599
is evaluated
to
(11)
and F(q) and I(q) are defined by eq. (6). In particular, for F(q) and I(q) equal to unity, this reduces to the result for the superlattice plasmons in the sheet approximation [S]. Thus, the finite size does not affect the plasmon dispersion to lowest order in wave vector. And to next order in ql, we find the correction term to be w=
w,J q, k) - O.l54C$D( q) ql
cosh( qd
) - cos( kd )
02)
where
w,,,(q,k) = azD
(q)(
is the sheet superlattice
sinh( qd) cosh(qd) plasmon
- cos( kd)
(13)
as ql + 0. We plot eq. (10) versus qd, keeping
/ /’
w
(mev)
0.5
1.0
qd Fig. 4. Plot of superlattice plasmon dispersion including finite layer widths. Bottom curves are for @3D= 11.4 meV and I/d = 0.3. Top curves for wp 3D = 23 meV and I/d = 0.67. Solid lines include stkcture factors F(q) and I(q). Dashed line is the two-dimensional sheet limit, F(q) and I(q) equal to one.
600
W.L. Bless / Effects offmite layer thrckness
kd
fixed, in fig. 4 for values of parameters similar to those used in the Olego et al. [ll] light scattering experiment (the lower curves) from a modulated doped GaAs-GaAlAs superlattice. Here, wP= 11.4 meV, l/d = 0.3, and kd is fixed at 4.9. The solid line includes the effect of finite layer thickness via F(q) and i(q); the dashed line is for F(q) and I(q) equal to unity. For qd - 1.O, about the highest value in the Olego et al. experiments, corrections are small, again about 5%. However, for larger i/d ratios; l/d = 0.67 for the upper curves and taking wP = 23 meV, we see that corrections can be as large as 10%. For qd - 1, this corresponds to a lowering in energy of about 1 meV. In conclusion, we have presented a theory that allows one to include finite wave functions into the calculation of the superlattice dielectric constant. A extension of this theory to include subband structure has been recently presented. In that case, we find that coupled intersubband modes can propagate along the superlattice axis [13]. Here, we have considered the effects of finite layer width on the ground superlattice plasmon modes. We find that in most cases and regions of interest (qd or ql small) the effects on the superlattice plasmon dispersion are very small (less than a few percent). However, if the l/d ratio (layer width to superlattice periodicity) is large and qd large, corrections to the superlattice plasmon may be as much as 10%.
References [l] A.Y. Cho and J.R. Arthur, Progr. Solid State Chem. 10 (1975) 157. [2] L.L. Chang and L. Esaki, Progr. Crystal Growth Characterization 2 (1979) 3; Surface Sci. 98 (1980) 70. [3] R. Dingle, H.L. Stormer, AC. Gossard and W. Wiegmann, Appl. Phys. Letters 33 (1978) 665. [4] T. Mimura, S. Hiyamizu, T. Fuju and K. Nassbu, Japan. J. Appt. Phys. 19 (1980) L225; T. Mimura, S. Hiyamizu, K, Joshin and K. Hikosaka, Japan. 3. Appl. Phys. 20 (1981) L317. [5] T. Ando, A.B. Fowler and F. Stern, Rev. Mod. Phys. 54 (1982) 437. [6] R.H. Ritchie, Phys. Rev. 106 (1957) 874. [7] D.A. Dahl and L.J. Sham, Phys. Rev. B16 (1977) 651. [8] A.L. Fetter, Ann. Phys. 88 (1974) 1. [9] S. Das Sarma and J.J. Quinn, Phys. Rev. B25 (1982) 7603. [lo] W.L. Bloss and E.M. Brody, Solid State Commun. 43 (1982) 523. [ll] D. Olego, A. Pinczuk, A.C. Gossard and W. Wiegmann, Phys. Rev. B25 (1982) 7867. [12] W.L. Bless, Solid State Commun. 44 (1982) 363. [13] W.L. Bless, Solid State Commun. 46 (1982) 143. [14] S.J. Allen, Jr., D.C. Tsui and B. Vinter, Solid State Commun. 20 (1976) 425. 1151 D.E. Beck and P. Kumar, Phys. Rev. B13 (1976) 2859. [16] A.K. Rajagopal, Phys. Rev. BlS (1977) 4264. [17] M. Jonson, J. Phys. C9 (1976) 3055.