- Email: [email protected]
GaAs multiple quantum wells
Solid State Communications, Printed in Great Britain.
Vol. 87, No. 5, pp. 481-485, 1993.
0038- 1098193 $6.00 + .OO Pergamon Press Ltd
GAUSSIAN
LINE-SHAPE ANALYSIS OF THE ROOM TEMPERATURE PHOTOREFLECTANCE OF GaAs/AlGaAs AND InAs/GaAs MULTIPLE QUANTUM WELLS
M. Lomascolo*, R. Cingolani, L. Calcagnile, M. Di Giulio, L. Vasanelli Dipartimento
di Scienza dei Materiali, Universiti, * di Lecce, Via per Arnesano, 73100 Lecce, Italy and 0. Brandt+ and K. Ploogt
Max-Planck-Institut
fur Festkijrperforschung, (Received
Heisenbergstr
14 January
1, D-7000 Stuttgart 80, FRG
1993 by E. Molinari)
The Gaussian dielectric function and its derivatives with respect to the main physical parameters have been studied to obtain a simplified lineshape formula for the high temperature photoreflectance spectra. The model has been used to fit the experimental data obtained from InAs/ GaAs and GaAs/AlGaAs Multiple Quantum Wells.
THE MODULATED reflectance spectroscopy (photoreflectance, PR) is a powerful technique to study electronic states in crystals. Its great advantage is the high sensitivity, also at room temperature, and the wide applicability to different semiconductors. Unfortunately, even though this technique is more sensitive than the photoluminescence and the photoluminescence excitation at high temperature, the computational complexity of the room temperature PR line-shape models makes the interpretation of the experimental spectra rather difficult. In this work we present a simplified first-derivative Gaussian analysis of the room temperature PR. The results of this approximated model are compared to the full calculation for the interpretation of the experimental PR spectra measured on GaAs/AlGaAs and InAs/ GaAs Multiple Quantum Wells (MQWs). The normalized change of reflectivity in crystals is related to the modulation of the complex dielectric function E(E) = el (E) + &(E) by:
At low temperature the exciton-phonon coupling is weak and the resonance of the dielectric function can be modeled by a simple Lorentzian shape [2]. On the other hand, at high temperature the exciton-phonon coupling is strong and the line-shape of the optical resonance becomes Gaussian-like [3,4]. For a semiconductor quantum well at room temperature, the dielectric function around the excitonic resonance can be written as [5]:
AR/R = o(~t,
In the above expressions E. is the energy of the excitonic transition, P is the Gaussian full-width at half -maximum of the resonance and f is the oscillator strength. Near the fundamental absorption edge p N 0 [6], thus AR/R is mostly determined by the changes of the the dielectric function real part of AE, = Ae,(Eo, I’, j), which is a function of the three parameters Eo, I’,f. Furthermore, in a small energy range around the resonance, cr is constant. Unfortunately, the presence of the hypergeometric function @(x2) makes difficult the reduction of equation (2) to
~2Pk
+ P(EI,
~2P2,
(1)
where CX,,f?are the Seraphin coefficients [l] and AE,, A62 the changes of the real and complex part of the dielectric function caused by the external modulation.
* Centro Nazionale Ricerca Sviluppo Materiali, SS 7 Km 7, 72100 Brindisi, Italy. t Mitsubishi Electric Corp., Central Research Laboratory, Amagasaki, Hyogo 661, Japan. $ Paul-Drude-Institut fur Festkorperelektronik, Hausvogteiplatz 5-7 O-1086 Berlin, FRG.
1 -f
0 [
6=61+z&=;
7
where
e
-x2
-x@(x2)
J/2 + i2
1, (4
E - E, x=77
(3) J2
@(x2) =
&
f
e*‘dt.
(4)
0
481
GAUSSIAN
482
LINE-SHAPE
a simple analytical form, suitable for the line-shape fitting to the experimental spectra. This imposes the same limitation to the interpretation of the room temperature PR. In the following we present the results of a theoretical and experimental study of the dependence of the PR line-shape on the physical parameters involved in the modulation of the dielectric function. In our treatment we follow the first derivative model to interpret the resonance shape [7,8]. Deriving the real part of the dielectric function with respect to the modulation field F we obtain [6]:
Vol. 87, No. 5
ANALYSIS
0
The I, and Z2 integrals are plotted in Fig 1 as a function of energy, for different values of I’(SmeV < r < lSmeV), characteristics of excitonic resonance in Quantum Wells at room temperature. Around the resonance energy (Eo), Ii and Z2 approach the values 1 and 0, respectively, whereas they diverge away from Eo. As a general trend, both curves indicate that Ii and Z2 are almost constant in the energy interval fl? around the resonance energy Eo. In a first approximation we can therefore assume Ii N 1 and Z,zO for Eo--l?
which clearly evidences the individual contributions of Es, r, and f to the total modulation of the dielectric function. The shape of the resonance is thus proportional to the linear combination of the derivatives of e1 with respect to Es, I’, f, weighted (6) by the respective (d&/%‘), (dI’/dF) and (af/lW) coefficients. In order to carry out the three derivatives and the line-shape fit of the experimental data, we In Fig.(2) we compare the el (E) function and its three have to evaluate numerically the following two derivatives appearing in equation (5) calculated by using either equation (6) (dashed line) or the full form integrals, which appear in the dielectric function equation (2) (continuous line). These curves clearly and in its derivatives.
6
~“““““““‘~
I
-611, ,,
I , , , , , , , , , ,
E,-10
E,+lO
E0
Energy
[mev]
E,-10
E.+lO
E0
Energy
,I
[meV]
Fig. 1. Plots of the I, and Z2functions in a limited range around the resonance energy and for various values of the ?J parameter (5 < r < 15 meV with 1 meV step).
No. 5
GAUSSIAN
b-10
b
E.+lO
b-10
b Energy [meV]
E,+lO
LINE-SHAPE
483
ANALYSIS
b Energy [meV]
b+lO
Fig. 2. Dielectric function and its derivatives with respect to I’,f, Et,, calculated by equation (2) (continuous line) and equation (6) (dashed line) for different l? values (a = 5 meV, b = 8 meV, c = 11 meV). demonstrate that the two expressions do not differ much, being almost coincident around the resonance energy. This allows us to calculate the photoreflectance spectrum in a simplified form, by using in equation (5) the derivatives of the approximated dielectric function [equation (6)]. The global lineshape of the PR-spectrum is obtained by adjusting the three parameters (d&J@), (dr/dF) and (afl@‘), which weight the derivatives of cl(E) plotted in Fig 2. These parameters are fitted to the experimental spectra by standard least-squares minimization. The best-fit [9] of the full model [equation (2)] to the experimental spectra takes about 1 hour, whereas it shortens dramatically (by a factor of 30) with the simplified equation (6) where no computational complications due to the adjustment of the free parameters contained in the hypergeometric functions Z, and Z2 are needed. The PR spectra at room temperature were measured on MBE-grown GaAs/Alt,3,,Ga,,,0As MQWs with well and barrier widths of 100 A and 2OOA, respectively, and on InAs/GaAs MQWs with well and barrier widths of 3 A and 120 A, respectively, using the experimental configuration reported in the
literature [IO]. A 2mW He-Ne laser mechanically chopped at 15 Hz was used to modulate the dielectric function. The investigated samples were previously studied by low-temperature photoreflectance and photoluminescence excitation spectroscopy, in order to have detailed information on quantized states [ll, 121. The penetration depth of the modulating He-Ne laser radiation amounts to about 5000 A, just enough to guarantee a fairly homogenous excitation of the investigated samples [13]. For the GaAs/AlGaAs MQWs, the conduction and valence subbands were calculated according to the usual envelope function model in the effective mass approximation [14]. Taking into account the exciton binding energy [15], we found the fundamental heavy-hole and light-hole resonances, El Ih and Elll respectively, are split by about 15meV. This value compares very well with the experimental photoreflectance spectrum measured at room temperature [dots in Fig.3(a)]. The spectrum exhibits a distinct resonance around 1.44 eV and a second, more structured feature, around 1.455 eV. These are compared with the best-fit calculated line-shapes obtained by the full model equation (2) (continuous
484
GAUSSIAN LINE-SHAPE
-
b)
1.42
InAs/GaAs
1.44
I .48
I 46
Energy
[eV]
Fig. 3. (a) Experimental room temperature PR spectrum of a GaAs/AlGaAs MQWs heterostructures (symbols). The continous and dashed lines are best-fit PR spectra calculated by equation (2) and equation (6), respectively. The dot-dashed line is PR spectrum calculated by the simplified equation (6) and using the best-fit parameters obtained by the fitting of the full model equation (2). (b) The same as in Fig. 3(a) but for the InAs/GaAs MQWs. line) and by the simplified model equation (6) (dashed line). Both equations (2) and (6) reproduce the experimental spectrum well giving almost indistinguishable traces. The best-fit values of the excitonic resonance energies and of the oscillator strengths are compared and are reported in Table. 1. The approximated formula equation (6) slightly overestimates the broadening parameter I’ by about
ANALYSIS
Vol. 87, No. 5
1 meV. This is consistent with the assumption of constant values for the I, and Z, integrals which holds only in a limited energy range around the resonance. It is interesting to note that the determination of the exciton energies obtained shown by the PR line-shape calculated by equation (6) using the best-fit parameters obtained from the full model equation (2) [dot-dashed line in Fig 3(a)]. In this case the calculated curve well reproduces the PR spectrum around the resonance and falls away from that due to the need of a larger I? value. In Fig. 3(b) we show the experimental and calculated PR spectra of the InAs/GaAs MQWs at room temperature. As already shown in [12], excitons in ultrathin InAs MQWs are strongly localized around the isoelectronic InAs planes, resulting in sharp resonances in the dielectric response of the heterostructure. The heavy and light hole exciton localization energies were calculated by a linear-chain tight-binding model [12] as well as by the envelope function model [l 11, giving a splitting of 30 meV between the Ellh and Et,, eigenstates. This is consistent with the two structures observed in the experimental PR spectrum at room temperature. Also in this case the best-fit regression of the full model equation (2) (continuous line) and the approximated model equation (6) (dashed line) give almost identical results and well reproduce the experimental spectrum (see Table 1). As for the GaAs case, the broadening parameters in the approximated model result slightly larger, but the determination of the eigenenergies is coincident. The use of the best-fit parameters of full model equation (2) in equation (6) without any adjustment, resulting in a poorer fit [dot-dashed line in Fig. 3(b)]. Again the I parameters should be increased by about 20% to get a good fit. These results are confirmed by systematic PR experiments performed on various III-V QW heterostructures group. In conclusion, equation (6) gives a simple and reliable means to calculate the room temperature
Table 1. Best-fit parameters obtained by using equation (2) or equation (6) for GaAslAlGaAs and ZnAslGaAs MQ Ws rllh(meV)
Equation (2) Equation (6)
1.443f0.001 1.443f0.001
3.72f0.09 4.08f0.08
InAs/GaAs
Elth(eV)
IXmeV)
Equation (2) Equation (6)
1.368f0.002 1.368f0.003
5f2 6f2
fllh
x
1f.P
1.74f0.06 1.85f0.05
fllh
x
1r4
1.6f0.8 1.8f0.9
El deV)
WmW
1.458f0.001 1.458f0.001
4.6&O.1 5.2fO. 1
ElldeV
bdmeV)
l.400f0.004 1.400f0.004
8f2 lOf2
fill
x
lop6
2.6f0.2 2.6f0.2 &I
x
w4
0.9f0.4 l.lf0.4
GAUSSIAN
Vol. 87, No. 5
LINE-SHAPE
photoreflectance line-shape. Besides a small correction to the phenomenolocical broadening parameter this model provides correct values for the exciton eigenenergy, with great advantages for the study of confined exciton states in quantum wells at high temperature. Acknowledgements
One of the GaAs/AlGaAs samples used for this study were grown by the TASC Laboratory of Trieste’-Italy.
REFERENCES 1. 2.
:: 5. 6. 7.
B.O. Seraphin & N. Bottka, Phys. Rev. 145,628 (1966). D.E. Aspnes, Handbook on Semiconductors, Vo1.2, p. 109, (North-Holland, Amsterdam (1980). Y. Toyozawa, Prog. Theor. Phys. 27, 53 (1958). Y. Toyozawa, Prog. Theor. Phys. 27,89 (1962). W.M. Theis, G.D. Sanders, C.E. Leak, K.K. Bajaj & H. Morkoc, Phys. Rev. B37, 3042 (1988).
B.V. Shanabrook, O.J. Glembocki & W.T. Beard, Phys. Rev. B35, 2540 (1987). R. Enderlein, D. Jiang & Y. Tang, Phys. Status Solidi. (b) 145, 167 (1988).
8.
ANALYSIS
485
H. Shen, S.H. Pan, Fred H. Pollak, R.N. Sacks, Phys. Rev. B37, 10 919 (1988).
9% F. James & M. Roos, Programma MINUIT 10 Cern Computer Centre (1989). J.L. Shay, Phys. Rev. B2, 803 (1970). 11’ 0. Brandt, L. Tapfer, R. Cingolani, K. Ploog, M. Hohenstein & F. Phillips Phys. Rev. B41, 12 599 (1991). 12. R. Cingolani, 0. Brandt, L. Tapfer, G.C. La Rocca & K. Ploog, Phys. Rev. B42,3209 (1990). 13. The absorption coefficient of GaAs at 2eV and T = 300K amounts to about 2 x 104cm-‘, resulting in a penetration depth of N 5OOOA. The GaAs MQWs consists of 20 periods, 16 of which are modulated within the top 50008, of the sample. For the 10 periods InAs/GaAs heterostructure, the whole MQWs region is well within the penetration depth of the modulating light. Photoreflectance experiments as a function of the modulating wavelength are in progress in our laboratories to establish any dependence on the homogeneity of the modulating field across the heterostructure. 14. G. Bastard, Wave Mechanics Applied to Semiconductor Heterostructures. Les Ulis, Paris (1975). 15. E.S. Koteles & J.C. Chi, Phys. Rev. B37, 6332 (1988).
Vol. 87, No. 5, pp. 481-485, 1993.
0038- 1098193 $6.00 + .OO Pergamon Press Ltd
GAUSSIAN
LINE-SHAPE ANALYSIS OF THE ROOM TEMPERATURE PHOTOREFLECTANCE OF GaAs/AlGaAs AND InAs/GaAs MULTIPLE QUANTUM WELLS
M. Lomascolo*, R. Cingolani, L. Calcagnile, M. Di Giulio, L. Vasanelli Dipartimento
di Scienza dei Materiali, Universiti, * di Lecce, Via per Arnesano, 73100 Lecce, Italy and 0. Brandt+ and K. Ploogt
Max-Planck-Institut
fur Festkijrperforschung, (Received
Heisenbergstr
14 January
1, D-7000 Stuttgart 80, FRG
1993 by E. Molinari)
The Gaussian dielectric function and its derivatives with respect to the main physical parameters have been studied to obtain a simplified lineshape formula for the high temperature photoreflectance spectra. The model has been used to fit the experimental data obtained from InAs/ GaAs and GaAs/AlGaAs Multiple Quantum Wells.
THE MODULATED reflectance spectroscopy (photoreflectance, PR) is a powerful technique to study electronic states in crystals. Its great advantage is the high sensitivity, also at room temperature, and the wide applicability to different semiconductors. Unfortunately, even though this technique is more sensitive than the photoluminescence and the photoluminescence excitation at high temperature, the computational complexity of the room temperature PR line-shape models makes the interpretation of the experimental spectra rather difficult. In this work we present a simplified first-derivative Gaussian analysis of the room temperature PR. The results of this approximated model are compared to the full calculation for the interpretation of the experimental PR spectra measured on GaAs/AlGaAs and InAs/ GaAs Multiple Quantum Wells (MQWs). The normalized change of reflectivity in crystals is related to the modulation of the complex dielectric function E(E) = el (E) + &(E) by:
At low temperature the exciton-phonon coupling is weak and the resonance of the dielectric function can be modeled by a simple Lorentzian shape [2]. On the other hand, at high temperature the exciton-phonon coupling is strong and the line-shape of the optical resonance becomes Gaussian-like [3,4]. For a semiconductor quantum well at room temperature, the dielectric function around the excitonic resonance can be written as [5]:
AR/R = o(~t,
In the above expressions E. is the energy of the excitonic transition, P is the Gaussian full-width at half -maximum of the resonance and f is the oscillator strength. Near the fundamental absorption edge p N 0 [6], thus AR/R is mostly determined by the changes of the the dielectric function real part of AE, = Ae,(Eo, I’, j), which is a function of the three parameters Eo, I’,f. Furthermore, in a small energy range around the resonance, cr is constant. Unfortunately, the presence of the hypergeometric function @(x2) makes difficult the reduction of equation (2) to
~2Pk
+ P(EI,
~2P2,
(1)
where CX,,f?are the Seraphin coefficients [l] and AE,, A62 the changes of the real and complex part of the dielectric function caused by the external modulation.
* Centro Nazionale Ricerca Sviluppo Materiali, SS 7 Km 7, 72100 Brindisi, Italy. t Mitsubishi Electric Corp., Central Research Laboratory, Amagasaki, Hyogo 661, Japan. $ Paul-Drude-Institut fur Festkorperelektronik, Hausvogteiplatz 5-7 O-1086 Berlin, FRG.
1 -f
0 [
6=61+z&=;
7
where
e
-x2
-x@(x2)
J/2 + i2
1, (4
E - E, x=77
(3) J2
@(x2) =
&
f
e*‘dt.
(4)
0
481
GAUSSIAN
482
LINE-SHAPE
a simple analytical form, suitable for the line-shape fitting to the experimental spectra. This imposes the same limitation to the interpretation of the room temperature PR. In the following we present the results of a theoretical and experimental study of the dependence of the PR line-shape on the physical parameters involved in the modulation of the dielectric function. In our treatment we follow the first derivative model to interpret the resonance shape [7,8]. Deriving the real part of the dielectric function with respect to the modulation field F we obtain [6]:
Vol. 87, No. 5
ANALYSIS
0
The I, and Z2 integrals are plotted in Fig 1 as a function of energy, for different values of I’(SmeV < r < lSmeV), characteristics of excitonic resonance in Quantum Wells at room temperature. Around the resonance energy (Eo), Ii and Z2 approach the values 1 and 0, respectively, whereas they diverge away from Eo. As a general trend, both curves indicate that Ii and Z2 are almost constant in the energy interval fl? around the resonance energy Eo. In a first approximation we can therefore assume Ii N 1 and Z,zO for Eo--l?
which clearly evidences the individual contributions of Es, r, and f to the total modulation of the dielectric function. The shape of the resonance is thus proportional to the linear combination of the derivatives of e1 with respect to Es, I’, f, weighted (6) by the respective (d&/%‘), (dI’/dF) and (af/lW) coefficients. In order to carry out the three derivatives and the line-shape fit of the experimental data, we In Fig.(2) we compare the el (E) function and its three have to evaluate numerically the following two derivatives appearing in equation (5) calculated by using either equation (6) (dashed line) or the full form integrals, which appear in the dielectric function equation (2) (continuous line). These curves clearly and in its derivatives.
6
~“““““““‘~
I
-611, ,,
I , , , , , , , , , ,
E,-10
E,+lO
E0
Energy
[mev]
E,-10
E.+lO
E0
Energy
,I
[meV]
Fig. 1. Plots of the I, and Z2functions in a limited range around the resonance energy and for various values of the ?J parameter (5 < r < 15 meV with 1 meV step).
No. 5
GAUSSIAN
b-10
b
E.+lO
b-10
b Energy [meV]
E,+lO
LINE-SHAPE
483
ANALYSIS
b Energy [meV]
b+lO
Fig. 2. Dielectric function and its derivatives with respect to I’,f, Et,, calculated by equation (2) (continuous line) and equation (6) (dashed line) for different l? values (a = 5 meV, b = 8 meV, c = 11 meV). demonstrate that the two expressions do not differ much, being almost coincident around the resonance energy. This allows us to calculate the photoreflectance spectrum in a simplified form, by using in equation (5) the derivatives of the approximated dielectric function [equation (6)]. The global lineshape of the PR-spectrum is obtained by adjusting the three parameters (d&J@), (dr/dF) and (afl@‘), which weight the derivatives of cl(E) plotted in Fig 2. These parameters are fitted to the experimental spectra by standard least-squares minimization. The best-fit [9] of the full model [equation (2)] to the experimental spectra takes about 1 hour, whereas it shortens dramatically (by a factor of 30) with the simplified equation (6) where no computational complications due to the adjustment of the free parameters contained in the hypergeometric functions Z, and Z2 are needed. The PR spectra at room temperature were measured on MBE-grown GaAs/Alt,3,,Ga,,,0As MQWs with well and barrier widths of 100 A and 2OOA, respectively, and on InAs/GaAs MQWs with well and barrier widths of 3 A and 120 A, respectively, using the experimental configuration reported in the
literature [IO]. A 2mW He-Ne laser mechanically chopped at 15 Hz was used to modulate the dielectric function. The investigated samples were previously studied by low-temperature photoreflectance and photoluminescence excitation spectroscopy, in order to have detailed information on quantized states [ll, 121. The penetration depth of the modulating He-Ne laser radiation amounts to about 5000 A, just enough to guarantee a fairly homogenous excitation of the investigated samples [13]. For the GaAs/AlGaAs MQWs, the conduction and valence subbands were calculated according to the usual envelope function model in the effective mass approximation [14]. Taking into account the exciton binding energy [15], we found the fundamental heavy-hole and light-hole resonances, El Ih and Elll respectively, are split by about 15meV. This value compares very well with the experimental photoreflectance spectrum measured at room temperature [dots in Fig.3(a)]. The spectrum exhibits a distinct resonance around 1.44 eV and a second, more structured feature, around 1.455 eV. These are compared with the best-fit calculated line-shapes obtained by the full model equation (2) (continuous
484
GAUSSIAN LINE-SHAPE
-
b)
1.42
InAs/GaAs
1.44
I .48
I 46
Energy
[eV]
Fig. 3. (a) Experimental room temperature PR spectrum of a GaAs/AlGaAs MQWs heterostructures (symbols). The continous and dashed lines are best-fit PR spectra calculated by equation (2) and equation (6), respectively. The dot-dashed line is PR spectrum calculated by the simplified equation (6) and using the best-fit parameters obtained by the fitting of the full model equation (2). (b) The same as in Fig. 3(a) but for the InAs/GaAs MQWs. line) and by the simplified model equation (6) (dashed line). Both equations (2) and (6) reproduce the experimental spectrum well giving almost indistinguishable traces. The best-fit values of the excitonic resonance energies and of the oscillator strengths are compared and are reported in Table. 1. The approximated formula equation (6) slightly overestimates the broadening parameter I’ by about
ANALYSIS
Vol. 87, No. 5
1 meV. This is consistent with the assumption of constant values for the I, and Z, integrals which holds only in a limited energy range around the resonance. It is interesting to note that the determination of the exciton energies obtained shown by the PR line-shape calculated by equation (6) using the best-fit parameters obtained from the full model equation (2) [dot-dashed line in Fig 3(a)]. In this case the calculated curve well reproduces the PR spectrum around the resonance and falls away from that due to the need of a larger I? value. In Fig. 3(b) we show the experimental and calculated PR spectra of the InAs/GaAs MQWs at room temperature. As already shown in [12], excitons in ultrathin InAs MQWs are strongly localized around the isoelectronic InAs planes, resulting in sharp resonances in the dielectric response of the heterostructure. The heavy and light hole exciton localization energies were calculated by a linear-chain tight-binding model [12] as well as by the envelope function model [l 11, giving a splitting of 30 meV between the Ellh and Et,, eigenstates. This is consistent with the two structures observed in the experimental PR spectrum at room temperature. Also in this case the best-fit regression of the full model equation (2) (continuous line) and the approximated model equation (6) (dashed line) give almost identical results and well reproduce the experimental spectrum (see Table 1). As for the GaAs case, the broadening parameters in the approximated model result slightly larger, but the determination of the eigenenergies is coincident. The use of the best-fit parameters of full model equation (2) in equation (6) without any adjustment, resulting in a poorer fit [dot-dashed line in Fig. 3(b)]. Again the I parameters should be increased by about 20% to get a good fit. These results are confirmed by systematic PR experiments performed on various III-V QW heterostructures group. In conclusion, equation (6) gives a simple and reliable means to calculate the room temperature
Table 1. Best-fit parameters obtained by using equation (2) or equation (6) for GaAslAlGaAs and ZnAslGaAs MQ Ws rllh(meV)
Equation (2) Equation (6)
1.443f0.001 1.443f0.001
3.72f0.09 4.08f0.08
InAs/GaAs
Elth(eV)
IXmeV)
Equation (2) Equation (6)
1.368f0.002 1.368f0.003
5f2 6f2
fllh
x
1f.P
1.74f0.06 1.85f0.05
fllh
x
1r4
1.6f0.8 1.8f0.9
El deV)
WmW
1.458f0.001 1.458f0.001
4.6&O.1 5.2fO. 1
ElldeV
bdmeV)
l.400f0.004 1.400f0.004
8f2 lOf2
fill
x
lop6
2.6f0.2 2.6f0.2 &I
x
w4
0.9f0.4 l.lf0.4
GAUSSIAN
Vol. 87, No. 5
LINE-SHAPE
photoreflectance line-shape. Besides a small correction to the phenomenolocical broadening parameter this model provides correct values for the exciton eigenenergy, with great advantages for the study of confined exciton states in quantum wells at high temperature. Acknowledgements
One of the GaAs/AlGaAs samples used for this study were grown by the TASC Laboratory of Trieste’-Italy.
REFERENCES 1. 2.
:: 5. 6. 7.
B.O. Seraphin & N. Bottka, Phys. Rev. 145,628 (1966). D.E. Aspnes, Handbook on Semiconductors, Vo1.2, p. 109, (North-Holland, Amsterdam (1980). Y. Toyozawa, Prog. Theor. Phys. 27, 53 (1958). Y. Toyozawa, Prog. Theor. Phys. 27,89 (1962). W.M. Theis, G.D. Sanders, C.E. Leak, K.K. Bajaj & H. Morkoc, Phys. Rev. B37, 3042 (1988).
B.V. Shanabrook, O.J. Glembocki & W.T. Beard, Phys. Rev. B35, 2540 (1987). R. Enderlein, D. Jiang & Y. Tang, Phys. Status Solidi. (b) 145, 167 (1988).
8.
ANALYSIS
485
H. Shen, S.H. Pan, Fred H. Pollak, R.N. Sacks, Phys. Rev. B37, 10 919 (1988).
9% F. James & M. Roos, Programma MINUIT 10 Cern Computer Centre (1989). J.L. Shay, Phys. Rev. B2, 803 (1970). 11’ 0. Brandt, L. Tapfer, R. Cingolani, K. Ploog, M. Hohenstein & F. Phillips Phys. Rev. B41, 12 599 (1991). 12. R. Cingolani, 0. Brandt, L. Tapfer, G.C. La Rocca & K. Ploog, Phys. Rev. B42,3209 (1990). 13. The absorption coefficient of GaAs at 2eV and T = 300K amounts to about 2 x 104cm-‘, resulting in a penetration depth of N 5OOOA. The GaAs MQWs consists of 20 periods, 16 of which are modulated within the top 50008, of the sample. For the 10 periods InAs/GaAs heterostructure, the whole MQWs region is well within the penetration depth of the modulating light. Photoreflectance experiments as a function of the modulating wavelength are in progress in our laboratories to establish any dependence on the homogeneity of the modulating field across the heterostructure. 14. G. Bastard, Wave Mechanics Applied to Semiconductor Heterostructures. Les Ulis, Paris (1975). 15. E.S. Koteles & J.C. Chi, Phys. Rev. B37, 6332 (1988).