- Email: [email protected]
A dielectric matrix calculation of the surface-plasmon energy for the silicon (100) surface
i~i~
~~''~i:#~::~
surface science ELSEVIER
Surface Science 357-358 (1996) 270-273
A dielectric matrix calculation of the surface-plasmon energy for the silicon (100) surface A.J. Forsyth a,1, A.E. Smith a,* , T.W. Josefsson b,2 a Department of Physics, Monash University, Clayton, Victoria 3168, Australia b School of Physics, University of Melbourne, Parkoille, Victoria 3052, Australia
Received 15 August 1995; accepted for publication 25 September 1995
Abstract As an extension of previous work, we present preliminary calculations for the dielectric properties of the silicon (100) surface. In particular, the Iql = 2~-/a(0,0,0) and Iql= 27r/a(1,0,0) surface loss function, and corresponding plasmon energies have been calculated within a simple model for the silicon surface. Results are compared and contrasted with volume plasmon calculations and experiment. Keywords: Low index single crystal surfaces; Plasmons; Semi-empirical models and model calculations; Silicon
1. Introduction In a previous paper [1] we discussed the volumeplasmon dispersion relation for silicon in various symmetry directions. This type of calculation may be carried over to a simple model for silicon surfaces. The dispersion relation for plasmons can be obtained directly from dielectric response theory. Due to the discrete translational symmetry of a crystal lattice, an external field of frequency to and wave vector q gives rise to many rapidly oscillating microscopic fields of frequency to and wave vector q + g
* Corresponding author. Fax: + 61 3 9905 3637. I AJF was supported by an Australian Postgraduate Award (APA). 2 TWJ acknowledges financial support from an Australian Postgraduate Research Award (APRA) and an Australian Research Council (ARC) Research Fellowship.
where g is a reciprocal lattice vector. These local field effects (LFE), result in a matrix formulation of the dielectric function [the dielectric matrix (DM), e~.h(q, to)] from which the plasmon dispersion relation m a y be obtained. Previous calculations on volume-plasmons [ 1 - 3 ] show a strong dependence on the particular electronic band structure used as a basis for the calculation. For the present work, we have used a nonlocal empirical pseudopotential method [4] (EPM) for the bulk silicon band structure. The D M is sensitive to detail in the electronic band structure topography, especially at large Iql values, and it is well known that a simple local potential does not accurately reproduce the electronic band structure of Si. In Section 2 we outline the method for obtaining the surface-plasmon energy from the DM, and briefly discuss the calculation of the DM. Accurate calculation of the frequency and wavevector dependent D M is required to obtain the plasmon dispersion relation
0039-6028/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved PH S0039-6028(96)00106-9
A.J. Forsyth et aL / Surface Science 357~-358 (1996)270-273
and these are numerically difficult to perform. Full details of this calculation can be found in Ref. [1]. Comparison of results with experiment and theory comprise Section 3.
2. Surface plasmon energy and the dielectric matrix The dielectric function s ( q , to) relates an external displacement field D(q, to) with the induced field in the solid E(q,to) according to the following equation:
O( q,to) = ~( q,to)E( q,to),
(1)
where q is the wave vector of the field and to is the frequency. The discrete symmetry of the crystal lattice forces us to consider a matrix formulation of the dielectric function, relating the displacement field and the microscopic fields of wave vector q + g where g is a reciprocal lattice vector.
D(q+g,to) = Eo°,,h(q,to)E(q+h,to).
(2)
h
These rapidly varying local "microscopic" fields are to a large extent "averaged out" when one considers many real space unit cells, such as one does on the macroscopic scale [5]. The experimentally more significant "macroscopic" dielectric function is related to the "microscopic" dielectric function by [6] 1
8(q,to) = [ oOg,h(q,to)]o_; .
(3)
Eq. (3) demonstrates how the LFE modify the experimentally observable macroscopic dielectric function [7,8]. The macroscopic dielectric function with the inclusion of LFE can be obtained from the Adler [9] and Wiser [6] (AW) dielectric function, which is given by 41re 2 8g'h(q'to) = 6g'h -- 12
q+gl]q+h[
fo[ e,,( k + q)] -fo[ E/,( k)] a~'O* k,n,nZ"~' e,,( k + q) - e,( k ) + hw + ihot
x lim
K"
(4)
X(k + X
271
q,n'lei(q+g)rlk,n >
,
(5)
where 12 is the crystal volume, fo(E) is the Fermi fimction and n,n' are the band indices, labelling the states Ik,n> of energy E,(k) in the solid, a is a small finite inverse lifetime, which is necessary for convergence. The calculation of Sg h(q, to) has been discussed in detail elsewhere [1,7], hence we shall only outline the computational procedure here. The band structure used to calculate the E,(k) is obtained by a nonlocal empirical pseudopotential method [4] with the wavefunction expanded over 59 reciprocal lattice vectors. Full details of this calculation can be found in Ref. [10]. The summation over k is performed using the special points scheme of Monkhurst and Pack. [11] We consider at least 182 special points in the irreducible Brillouin zone (IBZ) summation, and to eliminate redundant calculations we select only those sets of independent k c and k c + q points that are related by symmetry. If the incident frequency to is close to a possible energy band transition level, [Wc,v(kc, q) + hw[ < 1 eV, we use the analytic continuation integral of Dalton and Gilat [12], otherwise root sampling techniques suffice. T h e ~summation over valence and conduction bands is performed over the 4 highest valence bands and 41 lowest conduction bands. This large number of conduction bands is necessary for accurate values of 8g,h(q, to) at large values of q. Finally, we have found it necessary to invert dielectric matrices of the order of 59 × 59. Dielectric matrices smaller than this were found to be less accurate at high ]q[ magnitudes, for volume plasm0ns [1]. The resonance condition for a volume-plasmon is given by e(q,w)= 0. The presence of a surface means that new boundary conditions must now be satisfied. Our surface model consists of a Silicon slab which is infinite in the x- and y-directions, and in the half-space z < 0. The remaining half space is vacuum ( s o = 1). This semi-infinite slab allows us to ignore the coupling of plasmons on opposite surfaces of the slab, which leads to an experimentally observable splitting of the surface loss, function. For nonradiative plasmon modes, the boundary conditions give the surface resonance Condition as e ( q , t o ) =
AJ. Forsyth et al./ Surface Science 357-358 (1996) 270-273
272 4.0
I
i
I
I
1.5
i
I
I
q=(o,o o)
<
/i f i
2s
g o la_
,? 2.0 o .J
1.0
1.5
3
~\
-~
0.5
0.5 0.0
0.0
0
,0
5
10
15
20
Frequency (eV)
25 •
Fig. 1. The calculated loss function for bulk silicon (dotted line),
and the silicon surface (solid line) for
(
-1 / Im e ( q , w ) + 1 "
(6)
Throughout this work we shall take the peak of the surface loss function to be the surface-plasmon energy.
3. Results In Fig. 1 we present a comparison of the loss functions for bulk silicon (dotted line) and the silicon surface (solid line) for Iql = 0. While not particularly evident in the q = 2~r/a(0,0,0) loss function, at higher values of Iql there is often a lot of fine structure in the calculated loss function, due to the electronic structure of the solid [1,7,8]. This is clearly shown in the q = 2qr/a(1,0,0) loss function of Fig. 2. The fine structure can make it difficult to accurately determine the position of the loss function maxima. For this reason we fit a series of smooth Drude functions to the calculated loss function [7] - 1
+ 1
= E
j=l
10
15
20
25
30
Frequency ( o r )
Fig. 2. Comparison of the surface loss ~ncfion for q= 2 ~ / a ( 0 , 0 , 0 ) and q = 2 ~ / ~ 1 , 0 , 0 ) .
Iql = 0.
- 2 0 [13]. This resonance condition leads to the surface loss function [13,14]
Im e ( q ~ 3
5
30
AjDj(Fj,aj,~o),
(7)
where the Drude functions are of the form
oj(r.aj,o,) =
o,2)2 + r ? o ;
(8)
We have found that truncating the series at n = 3 gives a sufficiently accurate fit to the calculated loss function. There is a discrepancy of approximately 1.5 eV between the calculated surface plasmon energy and the experimental value for q = 0, as can be seen in Table 1. It has been shown in volume plasmon calculations [1-3] that calculations of this type tend to over estimate the IqL = 0 plasmon energy by 1-2 Table 1 Surface and volume plasmon energies q = 21r / a(0,0,0) Calculated Experiment [13,15] E s (eV) htOp (bulk) (eV) htOp/xl'2 (eV)
12.7 17.75 12.6
10.8+0.2 16.9 12.0
q = 2~-/a(1,0,0) Calculated Experiment Es h tOp (bulk) (eV) htOp/vr2 (eV)
16.35 22.35 15.8
A.I. Forsyth et a l . / Surface ScienCe 357-358 (1996) 270-273
eV. While the use of an LDA bandstructure incorporating exchange-correlation effects in Ref. [3] did reduce the plasmon energy for large [q[, it had little to no effect on the Iq[ = 0 plasmon energy. We believe that this effect is due to the use of the RPA rather than any bandstmcture effect. The slight discrepancy between E s and h % / ~ - is due to deviations of the dielectric function from that of a flee electron gas. This often results in the surface plasmon appearing at a higher energy than that determined by hoJp/V"-2 [13], and this trend is reproduced here. From Table 1 it can be seen that these bandstructure effects become more apparent for large values of q. For q = 2~-/a(1,0,0), we calculate a surface plasmon energy of 16.35 eV, compared with the "free electron" inspired (h % / v ~ ' ) result of 15.8 eV. The dispersion of the surface plasmon is clearly visible in Fig. 2. With only one point for [q[ > 0, it is difficult to quantitatively discuss the dispersion parameter for the (100) surface, but we do draw some preliminary comparisons with the bulk case. In a free electron gas, the dispersion relation for plasmons of small (compared with the Fermi wave vector qf) q is quadratic and given by h2 6 hogp(q) = hwp(O) + 2m a q 2 ' a =
Ef
5 h%(0) '
(9) where h % is the [q[ = 0 plasmon energy and E e the Fermi energy. The dispersion relation for a real solid also follows this general low Iql quadratic behaviour, and it is convenient (for the purposes of parameterization) to consider a more general form of Eq. (9) htop(q) = a p + apq 2,
(10)
where /2p is the measured Iql = 0 plasmon energy and ap the dispersion coefficient. For the volume plasmon, ap = 4.49 eV (2or/a) -2 for Iq] < X and ap = 3.58 eV (2or/a) -2 for Iql > X [1], compared with an experimentally determined value of o~p = 4.18 eV (2or/a) -2 [13], where X is the point 2~-/a(1,0,0). If we apply the same analysis to the surface, preliminary results would indicate a surface dispersion parameter % = 3.65 eV (2or/a) -2, though more work is required to establish this with certainty.
273
4. Conclusions and further work
We have calculated the surface plasmon energy for a plasmon on the (100) surface of silicon, and obtained a reasonable fit to the experimentally determined values. The discrepancy between the calculated and experimental values cannot be attributed to the type of bandstructure used for the calculation, and is more likely an artifact of the RPA dielectric matrix. Initial calculations give a surface dispersion parameter of the order of a s = 3.65 eV (27r/a) -2 , which is close to the value obtained for the volume plasmon in the (100) direction. Work is proceeding on a calculation of the loss function for Iql > 0, and hence a full dispersion relation for the surface plasmort.
A more realistic model is being developed, incorporating a bona fide surface bandstructure which will include the effects of relaxation and reconstruction of the silicon surface.
References [1] A.J. Forsyth, T.W. Josefsson and A.E. Smith, A Dielectric Matrix Calculation of the Volume-plasmon Dispersion Relation in Silicon, Phys. Rev. B, submitted. [2] R. Daling, W. Van Haeringen and B. Farid, Phys. Rev. B 44 (1991) 2952. [3] R. Daling, W. Van Haeringen and B. Farid, Phys. Rev. B 45 (1992) 8970. [4] M.L. Cohen and J.R. Chelikovsky, Electronic Structure and Optical Properties of Semiconductors, 2nd ed. (Springer, Berlin, 1989). [5] N.W. Ashcrofl and N.D. Mermin, Solid State Physics, (HRW International Editions, Hong Kong, 1976). [6] N. Wiser, Phys. Rev. 129 (1963) 62. [7] T.W. Josefsson and A.E. Smith, Phys. Rev. B 50 (1994) 7322. [8] T.W. Josefsson and A.E. Smith, Phys. Lett. A 180 (1993) 174. [9] S. Adler, Phys. Rev. 126 (1962) 413. [10] T.W. Josefsson and A.E. Smith, Aust. J. Phys. 50 (1993) 7322. [11] H.J. Monkhurst and J.D. Pack, Phys. Rev. B 13 (1976) 518. [12] N.W. Dalton and G. Gilat, Solid State Commun. 10 (1972) 287. [13] H. Raether, Springer Tracts in Modem Physics, Vol. 88, (Springer, New York, 1980). [14] H. Ibach and D.L. Mills, Electron Energy Loss Spectroscopy and Surface Vibrations (Academic Press, New York, 1982). [15] J. Schilling, Z. Phys. B 25 (1976) 61.
~~''~i:#~::~
surface science ELSEVIER
Surface Science 357-358 (1996) 270-273
A dielectric matrix calculation of the surface-plasmon energy for the silicon (100) surface A.J. Forsyth a,1, A.E. Smith a,* , T.W. Josefsson b,2 a Department of Physics, Monash University, Clayton, Victoria 3168, Australia b School of Physics, University of Melbourne, Parkoille, Victoria 3052, Australia
Received 15 August 1995; accepted for publication 25 September 1995
Abstract As an extension of previous work, we present preliminary calculations for the dielectric properties of the silicon (100) surface. In particular, the Iql = 2~-/a(0,0,0) and Iql= 27r/a(1,0,0) surface loss function, and corresponding plasmon energies have been calculated within a simple model for the silicon surface. Results are compared and contrasted with volume plasmon calculations and experiment. Keywords: Low index single crystal surfaces; Plasmons; Semi-empirical models and model calculations; Silicon
1. Introduction In a previous paper [1] we discussed the volumeplasmon dispersion relation for silicon in various symmetry directions. This type of calculation may be carried over to a simple model for silicon surfaces. The dispersion relation for plasmons can be obtained directly from dielectric response theory. Due to the discrete translational symmetry of a crystal lattice, an external field of frequency to and wave vector q gives rise to many rapidly oscillating microscopic fields of frequency to and wave vector q + g
* Corresponding author. Fax: + 61 3 9905 3637. I AJF was supported by an Australian Postgraduate Award (APA). 2 TWJ acknowledges financial support from an Australian Postgraduate Research Award (APRA) and an Australian Research Council (ARC) Research Fellowship.
where g is a reciprocal lattice vector. These local field effects (LFE), result in a matrix formulation of the dielectric function [the dielectric matrix (DM), e~.h(q, to)] from which the plasmon dispersion relation m a y be obtained. Previous calculations on volume-plasmons [ 1 - 3 ] show a strong dependence on the particular electronic band structure used as a basis for the calculation. For the present work, we have used a nonlocal empirical pseudopotential method [4] (EPM) for the bulk silicon band structure. The D M is sensitive to detail in the electronic band structure topography, especially at large Iql values, and it is well known that a simple local potential does not accurately reproduce the electronic band structure of Si. In Section 2 we outline the method for obtaining the surface-plasmon energy from the DM, and briefly discuss the calculation of the DM. Accurate calculation of the frequency and wavevector dependent D M is required to obtain the plasmon dispersion relation
0039-6028/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved PH S0039-6028(96)00106-9
A.J. Forsyth et aL / Surface Science 357~-358 (1996)270-273
and these are numerically difficult to perform. Full details of this calculation can be found in Ref. [1]. Comparison of results with experiment and theory comprise Section 3.
2. Surface plasmon energy and the dielectric matrix The dielectric function s ( q , to) relates an external displacement field D(q, to) with the induced field in the solid E(q,to) according to the following equation:
O( q,to) = ~( q,to)E( q,to),
(1)
where q is the wave vector of the field and to is the frequency. The discrete symmetry of the crystal lattice forces us to consider a matrix formulation of the dielectric function, relating the displacement field and the microscopic fields of wave vector q + g where g is a reciprocal lattice vector.
D(q+g,to) = Eo°,,h(q,to)E(q+h,to).
(2)
h
These rapidly varying local "microscopic" fields are to a large extent "averaged out" when one considers many real space unit cells, such as one does on the macroscopic scale [5]. The experimentally more significant "macroscopic" dielectric function is related to the "microscopic" dielectric function by [6] 1
8(q,to) = [ oOg,h(q,to)]o_; .
(3)
Eq. (3) demonstrates how the LFE modify the experimentally observable macroscopic dielectric function [7,8]. The macroscopic dielectric function with the inclusion of LFE can be obtained from the Adler [9] and Wiser [6] (AW) dielectric function, which is given by 41re 2 8g'h(q'to) = 6g'h -- 12
q+gl]q+h[
fo[ e,,( k + q)] -fo[ E/,( k)] a~'O* k,n,nZ"~' e,,( k + q) - e,( k ) + hw + ihot
x lim
K"
(4)
X(k + X
271
q,n'lei(q+g)rlk,n >
(5)
where 12 is the crystal volume, fo(E) is the Fermi fimction and n,n' are the band indices, labelling the states Ik,n> of energy E,(k) in the solid, a is a small finite inverse lifetime, which is necessary for convergence. The calculation of Sg h(q, to) has been discussed in detail elsewhere [1,7], hence we shall only outline the computational procedure here. The band structure used to calculate the E,(k) is obtained by a nonlocal empirical pseudopotential method [4] with the wavefunction expanded over 59 reciprocal lattice vectors. Full details of this calculation can be found in Ref. [10]. The summation over k is performed using the special points scheme of Monkhurst and Pack. [11] We consider at least 182 special points in the irreducible Brillouin zone (IBZ) summation, and to eliminate redundant calculations we select only those sets of independent k c and k c + q points that are related by symmetry. If the incident frequency to is close to a possible energy band transition level, [Wc,v(kc, q) + hw[ < 1 eV, we use the analytic continuation integral of Dalton and Gilat [12], otherwise root sampling techniques suffice. T h e ~summation over valence and conduction bands is performed over the 4 highest valence bands and 41 lowest conduction bands. This large number of conduction bands is necessary for accurate values of 8g,h(q, to) at large values of q. Finally, we have found it necessary to invert dielectric matrices of the order of 59 × 59. Dielectric matrices smaller than this were found to be less accurate at high ]q[ magnitudes, for volume plasm0ns [1]. The resonance condition for a volume-plasmon is given by e(q,w)= 0. The presence of a surface means that new boundary conditions must now be satisfied. Our surface model consists of a Silicon slab which is infinite in the x- and y-directions, and in the half-space z < 0. The remaining half space is vacuum ( s o = 1). This semi-infinite slab allows us to ignore the coupling of plasmons on opposite surfaces of the slab, which leads to an experimentally observable splitting of the surface loss, function. For nonradiative plasmon modes, the boundary conditions give the surface resonance Condition as e ( q , t o ) =
AJ. Forsyth et al./ Surface Science 357-358 (1996) 270-273
272 4.0
I
i
I
I
1.5
i
I
I
q=(o,o o)
<
/i f i
2s
g o la_
,? 2.0 o .J
1.0
1.5
3
~\
-~
0.5
0.5 0.0
0.0
0
,0
5
10
15
20
Frequency (eV)
25 •
Fig. 1. The calculated loss function for bulk silicon (dotted line),
and the silicon surface (solid line) for
(
-1 / Im e ( q , w ) + 1 "
(6)
Throughout this work we shall take the peak of the surface loss function to be the surface-plasmon energy.
3. Results In Fig. 1 we present a comparison of the loss functions for bulk silicon (dotted line) and the silicon surface (solid line) for Iql = 0. While not particularly evident in the q = 2~r/a(0,0,0) loss function, at higher values of Iql there is often a lot of fine structure in the calculated loss function, due to the electronic structure of the solid [1,7,8]. This is clearly shown in the q = 2qr/a(1,0,0) loss function of Fig. 2. The fine structure can make it difficult to accurately determine the position of the loss function maxima. For this reason we fit a series of smooth Drude functions to the calculated loss function [7] - 1
+ 1
= E
j=l
10
15
20
25
30
Frequency ( o r )
Fig. 2. Comparison of the surface loss ~ncfion for q= 2 ~ / a ( 0 , 0 , 0 ) and q = 2 ~ / ~ 1 , 0 , 0 ) .
Iql = 0.
- 2 0 [13]. This resonance condition leads to the surface loss function [13,14]
Im e ( q ~ 3
5
30
AjDj(Fj,aj,~o),
(7)
where the Drude functions are of the form
oj(r.aj,o,) =
o,2)2 + r ? o ;
(8)
We have found that truncating the series at n = 3 gives a sufficiently accurate fit to the calculated loss function. There is a discrepancy of approximately 1.5 eV between the calculated surface plasmon energy and the experimental value for q = 0, as can be seen in Table 1. It has been shown in volume plasmon calculations [1-3] that calculations of this type tend to over estimate the IqL = 0 plasmon energy by 1-2 Table 1 Surface and volume plasmon energies q = 21r / a(0,0,0) Calculated Experiment [13,15] E s (eV) htOp (bulk) (eV) htOp/xl'2 (eV)
12.7 17.75 12.6
10.8+0.2 16.9 12.0
q = 2~-/a(1,0,0) Calculated Experiment Es h tOp (bulk) (eV) htOp/vr2 (eV)
16.35 22.35 15.8
A.I. Forsyth et a l . / Surface ScienCe 357-358 (1996) 270-273
eV. While the use of an LDA bandstructure incorporating exchange-correlation effects in Ref. [3] did reduce the plasmon energy for large [q[, it had little to no effect on the Iq[ = 0 plasmon energy. We believe that this effect is due to the use of the RPA rather than any bandstmcture effect. The slight discrepancy between E s and h % / ~ - is due to deviations of the dielectric function from that of a flee electron gas. This often results in the surface plasmon appearing at a higher energy than that determined by hoJp/V"-2 [13], and this trend is reproduced here. From Table 1 it can be seen that these bandstructure effects become more apparent for large values of q. For q = 2~-/a(1,0,0), we calculate a surface plasmon energy of 16.35 eV, compared with the "free electron" inspired (h % / v ~ ' ) result of 15.8 eV. The dispersion of the surface plasmon is clearly visible in Fig. 2. With only one point for [q[ > 0, it is difficult to quantitatively discuss the dispersion parameter for the (100) surface, but we do draw some preliminary comparisons with the bulk case. In a free electron gas, the dispersion relation for plasmons of small (compared with the Fermi wave vector qf) q is quadratic and given by h2 6 hogp(q) = hwp(O) + 2m a q 2 ' a =
Ef
5 h%(0) '
(9) where h % is the [q[ = 0 plasmon energy and E e the Fermi energy. The dispersion relation for a real solid also follows this general low Iql quadratic behaviour, and it is convenient (for the purposes of parameterization) to consider a more general form of Eq. (9) htop(q) = a p + apq 2,
(10)
where /2p is the measured Iql = 0 plasmon energy and ap the dispersion coefficient. For the volume plasmon, ap = 4.49 eV (2or/a) -2 for Iq] < X and ap = 3.58 eV (2or/a) -2 for Iql > X [1], compared with an experimentally determined value of o~p = 4.18 eV (2or/a) -2 [13], where X is the point 2~-/a(1,0,0). If we apply the same analysis to the surface, preliminary results would indicate a surface dispersion parameter % = 3.65 eV (2or/a) -2, though more work is required to establish this with certainty.
273
4. Conclusions and further work
We have calculated the surface plasmon energy for a plasmon on the (100) surface of silicon, and obtained a reasonable fit to the experimentally determined values. The discrepancy between the calculated and experimental values cannot be attributed to the type of bandstructure used for the calculation, and is more likely an artifact of the RPA dielectric matrix. Initial calculations give a surface dispersion parameter of the order of a s = 3.65 eV (27r/a) -2 , which is close to the value obtained for the volume plasmon in the (100) direction. Work is proceeding on a calculation of the loss function for Iql > 0, and hence a full dispersion relation for the surface plasmort.
A more realistic model is being developed, incorporating a bona fide surface bandstructure which will include the effects of relaxation and reconstruction of the silicon surface.
References [1] A.J. Forsyth, T.W. Josefsson and A.E. Smith, A Dielectric Matrix Calculation of the Volume-plasmon Dispersion Relation in Silicon, Phys. Rev. B, submitted. [2] R. Daling, W. Van Haeringen and B. Farid, Phys. Rev. B 44 (1991) 2952. [3] R. Daling, W. Van Haeringen and B. Farid, Phys. Rev. B 45 (1992) 8970. [4] M.L. Cohen and J.R. Chelikovsky, Electronic Structure and Optical Properties of Semiconductors, 2nd ed. (Springer, Berlin, 1989). [5] N.W. Ashcrofl and N.D. Mermin, Solid State Physics, (HRW International Editions, Hong Kong, 1976). [6] N. Wiser, Phys. Rev. 129 (1963) 62. [7] T.W. Josefsson and A.E. Smith, Phys. Rev. B 50 (1994) 7322. [8] T.W. Josefsson and A.E. Smith, Phys. Lett. A 180 (1993) 174. [9] S. Adler, Phys. Rev. 126 (1962) 413. [10] T.W. Josefsson and A.E. Smith, Aust. J. Phys. 50 (1993) 7322. [11] H.J. Monkhurst and J.D. Pack, Phys. Rev. B 13 (1976) 518. [12] N.W. Dalton and G. Gilat, Solid State Commun. 10 (1972) 287. [13] H. Raether, Springer Tracts in Modem Physics, Vol. 88, (Springer, New York, 1980). [14] H. Ibach and D.L. Mills, Electron Energy Loss Spectroscopy and Surface Vibrations (Academic Press, New York, 1982). [15] J. Schilling, Z. Phys. B 25 (1976) 61.