- Email: [email protected]
Optical response of saddle point excitons
Solid State Communications, Printed in Great Britam.
Vol. 52, No. 3, pp. 351-354,
0038-1098/84 $3.00 + .OO Pergamon Press Ltd.
1984.
OPTICAL RESPONSE OF SADDLE POINT EXCITONS I. Balslev Fysisk Institut,
Odense University,
DK-5230 Odense M, Denmark
(Received 15 April 1984 by M. Gzrdomz) The linear optical response of interband transitions near saddle points is studied within the framework of the effective mass approximation. The method is based on Stahl’s configuration space theory. Unlike previous works the present treatment allows an unrestricted range of effective mass ratios, and it generates both real and imaginary part of the dielectric function. Spectra are calculated by numerical integration of a two dimensional second order partial differential equation. The results obtained for Ml saddle noints agree well with the observed lineshape of the El peak in ZnTe. * -
1. INTRODUCTION THE UNDERSTANDING of linear optical properties associated with electronic interband transitions in semiconductors involves three important steps. First, the characteristic features of the spectrum of ez [the imaginary part of the dielectric function E(W)] is assigned to singularities in the joint density of one electron states [ 1, 21. Next, the modifications caused by electron-hole interaction are taken into account [3,4]. Finally, a Kramers-Kronig analysis or an oscillator model is incorporated in the comparison with experiments, because these usually involve both e1 and e2. For MOcritical points [5] these steps are fully explored. Here Elliott’s theory [6] on the excitonic resonances and the Coulomb enhanced absorption continuum agree in great detail with experiments [7,8]. More uncertain is our knowledge on saddle point excitons, i.e. Coulomb correlated pair states near MI and M2 saddle points in the dependence of interband energy on wavevector. Some insight is gained by considering a short range electron-hole potential [9-l 11, but comparison with experiments shows that this truncation of the Coulomb potential is a poor approximation. As shown by Velicky and Sak [ 121 and by Kane [ 131 an untruncated Coulomb potential can be treated in a restricted range of mass ratios by applying an adiabatic approximation. This method is accurate if one of the principal reduced masses is numerically much larger than the two others. The purpose of the present communication is to demonstrate that the optical response of saddle point excitons can be calculated for any mass ratio by means of Stahl’s configuration space theory [ 14, 151. An additional advantage of this new method is that both ei and e2 come out of the calculations, and so the
optical observable quantities such as the reflectivity are directly available. As an example, the calculated spectra for MI critical points are fitted to experimental reflectivity spectra near the El peak of ZnTe measured by Laugier ef al. [ 161. 2. M,, CRITICAL POINTS for a spherical MC,
Stahl’s polariton equations critical point are [ 151
[
-Ev;-Ev;+
V(r)+h(w,-cd)
Y(F,R) 1
= M(@?@), [w’ + c’V;]E(E)
(1) = - c
1 M(?)Y(F, R) d3r,
(2)
where P and i? are relative and center of mass coordinates, 1-1and m are reduced and total effective masses, V(r) is the electron-hole (e-h) potential, M(F) is the dipole transition density, Y(i;, R) is the e-h amplitude, .!?(I?) is the electric field, ho+ is the energy gap, and w is the polariton frequency. Bulk solutions are Y(?, R) = u(r) exp (q * I?) which implies that the dielectric function given by
[
+v2+gq2+V@)+h(~,-~)
e(q, w) is
u(F)
1
W(F) j- M(L’)u(i;‘) d3+
= eo(e(q, 0) - 1) We shall neglect non-local response which corresponds to taking the 4 = 0 limit of equation (3). M(F) is essentially the Fourier transform of the 351
(3)
352
OPTICAL RESPONSE OF SADDLE POINT EXCITONS
Vol. 52, No. 3
momentum interband matrix element p(E). Therefore the usual assumption of little variation of p(x) near the critical point implies that M(T) is non-zero only in a few lattice cells near P = 0. On the other hand, the validity of the effective mass approximation is restricted to slowly varying amplitudes in the scale of a lattice constant. Consequently, the solutions to equation (3) for r = IFI less than a few lattice constants can be considered as an analytical continuation with little importance for the spectral structure of E(W) near wg. We are free to chose any function M(F) as long as it has the correct symmetry and it is reasonably localized about P = 0. With MO and r. as constants, [ 171 uses M(r) = M&r
- ro)/(47&),
(4)
in which case E(W) can be calculated from -$vz
+ V(r) + h(w,-
w) z&(r) = 0, 1
[
(5)
and E(W) = l-p
E07f$?
u+(ro) a24+(ro)/ar - au_(ro)/ar ’
(6)
where u+(ro) = u_(ro) and f stands for r 2 ro. It is instructive to apply equations (5) and (6) to the simple case of no e-h interaction. With k = (21.((w c+)/h)” one finds u+(r) a exp (ikr)/r and u_(r) a sin (kr)/r. Then, from equation (6), f(w)=l-
fl; 2EoAirghzk
11 - exp
(2ikdl.
(7)
I 36
I
Photon
I
I 38
energy
(eV)
I
I 4.0
Fig. 1. Calculated and measured spectra of ZnTe. Curves A, B, and D are calculated with Coulomb interaction, while curve C is e2 calculated without e-h interaction and otherwise unchanged parameters. The experimental curve E is taken from [ 16,201 as explained in the text.
In the spectral range given by Iklro < 1, E(W)
=I+$0 f+ik , 0
( 1
(8)
in agreement with a conventional treatment. Note that e2 is independent of r. while the “background dielectric constant” near wg is determined by ro. In the validity range of equation (8) we may write u+(r0) E(w)=1-A ~~7~r;h~ a2d+p,yar
(9)
.
3. Ml CRITICAL POINTS Turning to Ml saddle points we assume axial symmetry about the z-axis. Then equation (5) should be replaced by
There are severe difficulties in treating equation (10). Thus, the combination of spherical symmetry [V(r)] and cylindrical symmetry of the differential operators seems to rule out an analytical treatment. Furthermore, the behaviour of u_(r, 0) cannot be established, because non-parabolicity of bands needs to be included in order to stabilize the limit I -+ 0. A conventional calculation of the joint density of states near strictly parabolic saddle points involves cutting away distant parts of z space. Therefore it is reasonable to consider r. as a cut off radius and solve equation (10) only for r > ro. There are two further approximations necessary for a numerical treatment. First, u(r, 0) is assumed for simplicity to be isotropic at r = ro, while M(T) is allowed to have a free angular dependence u(ro,e)
= 0, where ml is positive, ml1 is negative, and z/(x2
+ y2)“2.
tg 6 =
(IO)
Mr,
0)
= =
u(r0), f(e
)Mo 6 (r -
(11) rO)/(4nri),
(12)
with M(r, O)/Mo normalized in P space. This is allowed because the details of the continuation of u(r, 0) to
OPTICAL RESPONSE OF SADDLE POINT EXCITONS
Vol. 52, No. 3
small values of r are unimportant. equation (9)
Then, analogous to
where G(8) = &sin20 1
+2cos2B. 2mll
(14)
Secondly, the radiation condition at large values of r can only be stated approximately. As the leading differential operator at large distances is G(o)aZ/arZ, it is reasonable to assume that
iar I 1 au
---
u
= [h(wg -
w>/G(e)]“2,
r’max
at some large distance r,, . The above two approximations depend critically on the existence of damping. Without damping there is an ambiguity in taking the square root in equation (15), and since ml1 is negative there is a singularity on the cone given by G(B) = 0. It turns out that a convenient damping is obtained by adding to the left hand side of equation (10) an imaginary contribution given by iyV*u(r, 0) corresponding to a non-local damping mechanism. Then G(0) in equation (15) should be replaced by G(0) - iy, and the square root in equation (15) should be chosen to have a positive real part. It is interesting to clarify in detail the asymptotic behaviour given by equation (15) for a small, but finite damping constant y. For w > og, u(r, 0) is an outgoing wave which is damped in its propagation direction. It is weakly damped for 0 = n/2, and almost evanescent for 0 = 0 (or n). For w < og, u(r, f3) is an ingoing wave which is amplified in its propagation direction, weakly amplified for 0 = 0, and almost exponential for t9 = n/2. Thus, if the waves are considered electron-like for o > wg and hole-like otherwise (sign change in the definition of i;), then they are always outgoing and damped.
4. THE NUMERICAL TREATMENT The numerical treatment is based on solving difference equations derived from equation (10) in the domain 0 < 8 < n/2 and r. < r < rmax . The boundary conditions are: au(r, 0)/h = 0, aup, n/2)/& = 0, u(ro) = 0, and equation (15) (with damping included) atr=r,,. In order to allow small steps near r = r. the difference equations have ln (r) and 8 as independent variables with constant step sizes A0 and A ln (r). The parameters in the calculations are ml/ml,, 7, ro, and the computational constants A8, A In (r), and
353
rmax. Typical spectra with and without Coulomb potential [V(r) = - e’/e,,r in equation (lo)] are shown in Fig. 1. The following parameters are used: A0 = 0.15, = 10aB,ro = O.la,, y = A In (r) = 0.023, r,, 0.2 aiE%, and ml/mII = - 0.2, where aB and E, are Bohr radius and binding energy, respectively, of an exciton with an isotropic effective mass of ml. For the sake of comparison with experiments on ZnTe, E, is chosen to 0.045 eV. Then with es = 7.3 [18] one finds: aB = 218, ml = 0.18mo and ml1 = - 0.9mo. Spectra withro = 0.05-O.l5an, y = 0.1-0.5&E%, and lml/mrr I = 0.1-0.5 are also calculated (not shown) in order to explore the influence of parameter changes.
5. DISCUSSION AND CONCLUSION Curve C in Fig. 1 is e2 calculated without e-h interaction. The behaviour of this agrees well with the corresponding analytical result for Mi critical points (a negative square root contribution below ug [ 1,2]). The e2 spectrum with Coulomb interaction (curve A) shows a considerable Coulomb enhancement giving rise to a peak about 2J?, below wg. In the limit mll/rnL + 0 and y + 0, the peak is 4E, below wg [12, 131. The magnitude of the Coulomb enhancement (a factor of - 3) is roughly the same as found for Si at 3 eV for many body calculations [ 191. The calculated spectra with Coulomb interaction are fitted to experiments on ZnTe for which Laugier ef al. [ 161 have recorded very accurate derivative spectra of the normal incidence reflectivity. Integrating their 82 K spectrum and adopting from [20] a peak reflectivity of 48% at the El peak, one obtains curve E in Fig. 1. The calculated curve D is a result of a crude fit in which Mi, E, and ho, are varied. (7 is deliberately kept at a very low value). The strength of e1 and e2 in Fig. 1 corresponds to this choice of Mi. Note the oscillatory behaviour just below wg. This corresponds to excited states of pseudo two-dimensional excitons treated in [12, 131. It is probable that the measured satellite peak at 3.90 eV is the optical response of such states (the much weaker er + A, structure is at 4.00eV [ 161). A similar assignment of extra structure in the spectrum of CdTe is published by Cardona [21] and by Kane [13]. A full discussion of the influence on the spectra of changing the parameters shall be postponed to a later publication. A few conclusions from the study of parameter variation shall be given here. The fit is rather sensitive to the value of ml, and the best value is ml = 0.18mo. This agrees well with Kane’s estimate [13], namely ml = 0.15mo for an El gap of 3.6eV. Another well determined quantity is the gap energy hw, = 3.84 eV. This is 60 meV higher than the very sharp
354
OPTICAL RESPONSE OF SADDLE POINT EXCITONS
negative peak in the derivative reflectivity spectrum of ZnTe [ 161. By changing simultaneously Mi and r,, the quality of the fit can be retained with altered spectra of e1 and e2. The present value of el/eZ at hw = 3.6 eV is about 2, while Kramer+Kronig analyses of experiments yield about one for this ratio [22]. So far, it is not clear if this discrepancy is solely due to a wrong choice of Mi and ro, or to some extent due to inaccurate asymptotic assumptions in the Kramers-Kronig analysis. The present work has shown that Stahl’s configuration space theory is a valuable tool for clarifying the linear optical response of saddle point excitons. In the further exploration of this application of configuration space theory there is a strong need for incorporating damping terms derived from realistic scattering models, and for more refined computational methods. Acknowledgement - The author has benefitted from discussions with A. Stahl.
greatly
10. 11. 12. 13. 14. 15. 16.
17. 18.
REFERENCES 1. 2. 3. 4.
J.C. Phillips, Solid State Phys. 18, 56 (1966). M. Cardona, Modulation Spectroscopy, Solid State Phys. Suppl. 11 (1969). G.H. Wannier, Phys. Rev. 52,91 (1937). R.S. Knox, l?reory of Excitons, Solid State Phys. SuppZ. 5 (1963).
19. 20. 21. 22.
Vol. 52, No. 3
L. Van Hove, Phys. Rev. 89, 189 (1953). R.J. Elliott,Phys. Rev. 108, 1384 (1957). S. Nikitine, Phil. Msg. 4, 1 (1959). G.G. Macfarlane, T.P. McLean, J.E. Quarrington & V. Roberts, Phys. Rev. 108, 1377 (1957). Y. Toyozawa, M. moue, T. Inui, M. Okazaki & E. Hanamura, J. Phys. Sot. Japan SuppZ. 21, 133 (1967). J. Hermanson, Phys. Rev. 150,660 (1966); 166, 893 (1968). R.M. Martin, J.A. Van Vechten, J.E. Row & D.E. Aspnes, Phys. Rev. B6, 2500 (1972). B. Velicky & Sak, Phys. Status Solidi 16, 147 (1966). E.O. Kane,Phys. Rev. 180,852 (1969). A. Stahl, P ys. Status Solidi (b) 94,221 (1979). A. Stahl, P3s. Status Solidi (b) 106,575 (1981). A. Laugier, B. Montegu, D. Barbier, J. Chevallier, J.C. Guillaume & K. Somogyi, Phys. Status Solidi d (b) 99,319 (1980). A. Stahl & I. Balslev, Phys. Status Solidi (b) 113, 583 (1982). A.G. Fisher, J.M. Carides & J.N. Dresner, Solid State Commun. 2, 157 (1964). W. Hanke & L.J. Sham, Phys. Rev. B21,4656 (1980). M. Cardona & D.L. Greenaway, Phys. Rev. 131,98 (1963). PI,P. 38. M. Cardona, J. Appl. Phys. 36,2 18 1 (1965) (private communication).
Vol. 52, No. 3, pp. 351-354,
0038-1098/84 $3.00 + .OO Pergamon Press Ltd.
1984.
OPTICAL RESPONSE OF SADDLE POINT EXCITONS I. Balslev Fysisk Institut,
Odense University,
DK-5230 Odense M, Denmark
(Received 15 April 1984 by M. Gzrdomz) The linear optical response of interband transitions near saddle points is studied within the framework of the effective mass approximation. The method is based on Stahl’s configuration space theory. Unlike previous works the present treatment allows an unrestricted range of effective mass ratios, and it generates both real and imaginary part of the dielectric function. Spectra are calculated by numerical integration of a two dimensional second order partial differential equation. The results obtained for Ml saddle noints agree well with the observed lineshape of the El peak in ZnTe. * -
1. INTRODUCTION THE UNDERSTANDING of linear optical properties associated with electronic interband transitions in semiconductors involves three important steps. First, the characteristic features of the spectrum of ez [the imaginary part of the dielectric function E(W)] is assigned to singularities in the joint density of one electron states [ 1, 21. Next, the modifications caused by electron-hole interaction are taken into account [3,4]. Finally, a Kramers-Kronig analysis or an oscillator model is incorporated in the comparison with experiments, because these usually involve both e1 and e2. For MOcritical points [5] these steps are fully explored. Here Elliott’s theory [6] on the excitonic resonances and the Coulomb enhanced absorption continuum agree in great detail with experiments [7,8]. More uncertain is our knowledge on saddle point excitons, i.e. Coulomb correlated pair states near MI and M2 saddle points in the dependence of interband energy on wavevector. Some insight is gained by considering a short range electron-hole potential [9-l 11, but comparison with experiments shows that this truncation of the Coulomb potential is a poor approximation. As shown by Velicky and Sak [ 121 and by Kane [ 131 an untruncated Coulomb potential can be treated in a restricted range of mass ratios by applying an adiabatic approximation. This method is accurate if one of the principal reduced masses is numerically much larger than the two others. The purpose of the present communication is to demonstrate that the optical response of saddle point excitons can be calculated for any mass ratio by means of Stahl’s configuration space theory [ 14, 151. An additional advantage of this new method is that both ei and e2 come out of the calculations, and so the
optical observable quantities such as the reflectivity are directly available. As an example, the calculated spectra for MI critical points are fitted to experimental reflectivity spectra near the El peak of ZnTe measured by Laugier ef al. [ 161. 2. M,, CRITICAL POINTS for a spherical MC,
Stahl’s polariton equations critical point are [ 151
[
-Ev;-Ev;+
V(r)+h(w,-cd)
Y(F,R) 1
= M(@?@), [w’ + c’V;]E(E)
(1) = - c
1 M(?)Y(F, R) d3r,
(2)
where P and i? are relative and center of mass coordinates, 1-1and m are reduced and total effective masses, V(r) is the electron-hole (e-h) potential, M(F) is the dipole transition density, Y(i;, R) is the e-h amplitude, .!?(I?) is the electric field, ho+ is the energy gap, and w is the polariton frequency. Bulk solutions are Y(?, R) = u(r) exp (q * I?) which implies that the dielectric function given by
[
+v2+gq2+V@)+h(~,-~)
e(q, w) is
u(F)
1
W(F) j- M(L’)u(i;‘) d3+
= eo(e(q, 0) - 1) We shall neglect non-local response which corresponds to taking the 4 = 0 limit of equation (3). M(F) is essentially the Fourier transform of the 351
(3)
352
OPTICAL RESPONSE OF SADDLE POINT EXCITONS
Vol. 52, No. 3
momentum interband matrix element p(E). Therefore the usual assumption of little variation of p(x) near the critical point implies that M(T) is non-zero only in a few lattice cells near P = 0. On the other hand, the validity of the effective mass approximation is restricted to slowly varying amplitudes in the scale of a lattice constant. Consequently, the solutions to equation (3) for r = IFI less than a few lattice constants can be considered as an analytical continuation with little importance for the spectral structure of E(W) near wg. We are free to chose any function M(F) as long as it has the correct symmetry and it is reasonably localized about P = 0. With MO and r. as constants, [ 171 uses M(r) = M&r
- ro)/(47&),
(4)
in which case E(W) can be calculated from -$vz
+ V(r) + h(w,-
w) z&(r) = 0, 1
[
(5)
and E(W) = l-p
E07f$?
u+(ro) a24+(ro)/ar - au_(ro)/ar ’
(6)
where u+(ro) = u_(ro) and f stands for r 2 ro. It is instructive to apply equations (5) and (6) to the simple case of no e-h interaction. With k = (21.((w c+)/h)” one finds u+(r) a exp (ikr)/r and u_(r) a sin (kr)/r. Then, from equation (6), f(w)=l-
fl; 2EoAirghzk
11 - exp
(2ikdl.
(7)
I 36
I
Photon
I
I 38
energy
(eV)
I
I 4.0
Fig. 1. Calculated and measured spectra of ZnTe. Curves A, B, and D are calculated with Coulomb interaction, while curve C is e2 calculated without e-h interaction and otherwise unchanged parameters. The experimental curve E is taken from [ 16,201 as explained in the text.
In the spectral range given by Iklro < 1, E(W)
=I+$0 f+ik , 0
( 1
(8)
in agreement with a conventional treatment. Note that e2 is independent of r. while the “background dielectric constant” near wg is determined by ro. In the validity range of equation (8) we may write u+(r0) E(w)=1-A ~~7~r;h~ a2d+p,yar
(9)
.
3. Ml CRITICAL POINTS Turning to Ml saddle points we assume axial symmetry about the z-axis. Then equation (5) should be replaced by
There are severe difficulties in treating equation (10). Thus, the combination of spherical symmetry [V(r)] and cylindrical symmetry of the differential operators seems to rule out an analytical treatment. Furthermore, the behaviour of u_(r, 0) cannot be established, because non-parabolicity of bands needs to be included in order to stabilize the limit I -+ 0. A conventional calculation of the joint density of states near strictly parabolic saddle points involves cutting away distant parts of z space. Therefore it is reasonable to consider r. as a cut off radius and solve equation (10) only for r > ro. There are two further approximations necessary for a numerical treatment. First, u(r, 0) is assumed for simplicity to be isotropic at r = ro, while M(T) is allowed to have a free angular dependence u(ro,e)
= 0, where ml is positive, ml1 is negative, and z/(x2
+ y2)“2.
tg 6 =
(IO)
Mr,
0)
= =
u(r0), f(e
)Mo 6 (r -
(11) rO)/(4nri),
(12)
with M(r, O)/Mo normalized in P space. This is allowed because the details of the continuation of u(r, 0) to
OPTICAL RESPONSE OF SADDLE POINT EXCITONS
Vol. 52, No. 3
small values of r are unimportant. equation (9)
Then, analogous to
where G(8) = &sin20 1
+2cos2B. 2mll
(14)
Secondly, the radiation condition at large values of r can only be stated approximately. As the leading differential operator at large distances is G(o)aZ/arZ, it is reasonable to assume that
iar I 1 au
---
u
= [h(wg -
w>/G(e)]“2,
r’max
at some large distance r,, . The above two approximations depend critically on the existence of damping. Without damping there is an ambiguity in taking the square root in equation (15), and since ml1 is negative there is a singularity on the cone given by G(B) = 0. It turns out that a convenient damping is obtained by adding to the left hand side of equation (10) an imaginary contribution given by iyV*u(r, 0) corresponding to a non-local damping mechanism. Then G(0) in equation (15) should be replaced by G(0) - iy, and the square root in equation (15) should be chosen to have a positive real part. It is interesting to clarify in detail the asymptotic behaviour given by equation (15) for a small, but finite damping constant y. For w > og, u(r, 0) is an outgoing wave which is damped in its propagation direction. It is weakly damped for 0 = n/2, and almost evanescent for 0 = 0 (or n). For w < og, u(r, f3) is an ingoing wave which is amplified in its propagation direction, weakly amplified for 0 = 0, and almost exponential for t9 = n/2. Thus, if the waves are considered electron-like for o > wg and hole-like otherwise (sign change in the definition of i;), then they are always outgoing and damped.
4. THE NUMERICAL TREATMENT The numerical treatment is based on solving difference equations derived from equation (10) in the domain 0 < 8 < n/2 and r. < r < rmax . The boundary conditions are: au(r, 0)/h = 0, aup, n/2)/& = 0, u(ro) = 0, and equation (15) (with damping included) atr=r,,. In order to allow small steps near r = r. the difference equations have ln (r) and 8 as independent variables with constant step sizes A0 and A ln (r). The parameters in the calculations are ml/ml,, 7, ro, and the computational constants A8, A In (r), and
353
rmax. Typical spectra with and without Coulomb potential [V(r) = - e’/e,,r in equation (lo)] are shown in Fig. 1. The following parameters are used: A0 = 0.15, = 10aB,ro = O.la,, y = A In (r) = 0.023, r,, 0.2 aiE%, and ml/mII = - 0.2, where aB and E, are Bohr radius and binding energy, respectively, of an exciton with an isotropic effective mass of ml. For the sake of comparison with experiments on ZnTe, E, is chosen to 0.045 eV. Then with es = 7.3 [18] one finds: aB = 218, ml = 0.18mo and ml1 = - 0.9mo. Spectra withro = 0.05-O.l5an, y = 0.1-0.5&E%, and lml/mrr I = 0.1-0.5 are also calculated (not shown) in order to explore the influence of parameter changes.
5. DISCUSSION AND CONCLUSION Curve C in Fig. 1 is e2 calculated without e-h interaction. The behaviour of this agrees well with the corresponding analytical result for Mi critical points (a negative square root contribution below ug [ 1,2]). The e2 spectrum with Coulomb interaction (curve A) shows a considerable Coulomb enhancement giving rise to a peak about 2J?, below wg. In the limit mll/rnL + 0 and y + 0, the peak is 4E, below wg [12, 131. The magnitude of the Coulomb enhancement (a factor of - 3) is roughly the same as found for Si at 3 eV for many body calculations [ 191. The calculated spectra with Coulomb interaction are fitted to experiments on ZnTe for which Laugier ef al. [ 161 have recorded very accurate derivative spectra of the normal incidence reflectivity. Integrating their 82 K spectrum and adopting from [20] a peak reflectivity of 48% at the El peak, one obtains curve E in Fig. 1. The calculated curve D is a result of a crude fit in which Mi, E, and ho, are varied. (7 is deliberately kept at a very low value). The strength of e1 and e2 in Fig. 1 corresponds to this choice of Mi. Note the oscillatory behaviour just below wg. This corresponds to excited states of pseudo two-dimensional excitons treated in [12, 131. It is probable that the measured satellite peak at 3.90 eV is the optical response of such states (the much weaker er + A, structure is at 4.00eV [ 161). A similar assignment of extra structure in the spectrum of CdTe is published by Cardona [21] and by Kane [13]. A full discussion of the influence on the spectra of changing the parameters shall be postponed to a later publication. A few conclusions from the study of parameter variation shall be given here. The fit is rather sensitive to the value of ml, and the best value is ml = 0.18mo. This agrees well with Kane’s estimate [13], namely ml = 0.15mo for an El gap of 3.6eV. Another well determined quantity is the gap energy hw, = 3.84 eV. This is 60 meV higher than the very sharp
354
OPTICAL RESPONSE OF SADDLE POINT EXCITONS
negative peak in the derivative reflectivity spectrum of ZnTe [ 161. By changing simultaneously Mi and r,, the quality of the fit can be retained with altered spectra of e1 and e2. The present value of el/eZ at hw = 3.6 eV is about 2, while Kramer+Kronig analyses of experiments yield about one for this ratio [22]. So far, it is not clear if this discrepancy is solely due to a wrong choice of Mi and ro, or to some extent due to inaccurate asymptotic assumptions in the Kramers-Kronig analysis. The present work has shown that Stahl’s configuration space theory is a valuable tool for clarifying the linear optical response of saddle point excitons. In the further exploration of this application of configuration space theory there is a strong need for incorporating damping terms derived from realistic scattering models, and for more refined computational methods. Acknowledgement - The author has benefitted from discussions with A. Stahl.
greatly
10. 11. 12. 13. 14. 15. 16.
17. 18.
REFERENCES 1. 2. 3. 4.
J.C. Phillips, Solid State Phys. 18, 56 (1966). M. Cardona, Modulation Spectroscopy, Solid State Phys. Suppl. 11 (1969). G.H. Wannier, Phys. Rev. 52,91 (1937). R.S. Knox, l?reory of Excitons, Solid State Phys. SuppZ. 5 (1963).
19. 20. 21. 22.
Vol. 52, No. 3
L. Van Hove, Phys. Rev. 89, 189 (1953). R.J. Elliott,Phys. Rev. 108, 1384 (1957). S. Nikitine, Phil. Msg. 4, 1 (1959). G.G. Macfarlane, T.P. McLean, J.E. Quarrington & V. Roberts, Phys. Rev. 108, 1377 (1957). Y. Toyozawa, M. moue, T. Inui, M. Okazaki & E. Hanamura, J. Phys. Sot. Japan SuppZ. 21, 133 (1967). J. Hermanson, Phys. Rev. 150,660 (1966); 166, 893 (1968). R.M. Martin, J.A. Van Vechten, J.E. Row & D.E. Aspnes, Phys. Rev. B6, 2500 (1972). B. Velicky & Sak, Phys. Status Solidi 16, 147 (1966). E.O. Kane,Phys. Rev. 180,852 (1969). A. Stahl, P ys. Status Solidi (b) 94,221 (1979). A. Stahl, P3s. Status Solidi (b) 106,575 (1981). A. Laugier, B. Montegu, D. Barbier, J. Chevallier, J.C. Guillaume & K. Somogyi, Phys. Status Solidi d (b) 99,319 (1980). A. Stahl & I. Balslev, Phys. Status Solidi (b) 113, 583 (1982). A.G. Fisher, J.M. Carides & J.N. Dresner, Solid State Commun. 2, 157 (1964). W. Hanke & L.J. Sham, Phys. Rev. B21,4656 (1980). M. Cardona & D.L. Greenaway, Phys. Rev. 131,98 (1963). PI,P. 38. M. Cardona, J. Appl. Phys. 36,2 18 1 (1965) (private communication).