- Email: [email protected]
Simulation of excitonic spectra in electric field to characterize the quality of low dimensional structures
Applied Surface Science 166 Ž2000. 273–277 www.elsevier.nlrlocaterapsusc
Simulation of excitonic spectra in electric field to characterize the quality of low dimensional structures O.L. Lazarenkova) , A.N. Pikhtin St. Petersburg Electrotechnical UniÕersity, Professor PopoÕ str. 5, St. Petersburg 197376, Russian Federation
Abstract In order to characterize the real low dimensional structures, the simulation of excitonic spectra has been done. Absorption spectrum of multiple quantum well ŽMQW. in graded electric field, the effect of macroscopic quantum well width and depth fluctuations, and the influence of surface roughness have been considered. The simulated spectra are in a good agreement with experimental data. q 2000 Published by Elsevier Science B.V. PACS: 73.40 and 78.20; 71.35; S7.12; S7.15 Keywords: Quantum well; Electric field; Inhomogeneity; Modulation optical spectroscopy
A lot of modern devices of nanoelectronics and optoelectronics are based on specific properties of low dimensional structures. Their characteristics are very dependent on the quality of ultra thin epitaxial layers. Therefore, it is necessary to develop the techniques for investigation of quantum-size structures after the formation as during the technology process. Photoreflectance ŽPR. and phototransmittance ŽPT. as contactless techniques of modulation spectroscopy are the major tools for quantum well ŽQW. characterization. Their essence is to record the change of a reflected ŽPR. or transmitted ŽPT. probe light beam under the periodic modulation of the sample charac-
) Corresponding author. Tel.: q7-812-234-31-64; fax: q7-812234-31-64. E-mail address: omk.meeeltech.ru ŽO.L. Lazarenkova..
teristics by the second light beam, which changes the built-in electric field. Thus, the relatively sharp features are observed in the spectra, even at room temperature. However, some difficulties are currently encountered in analyzing the observed spectra of QW structures. The excitonic effects have to be taken into account on principle. In this paper, we develop an approach previously proposed by ourselves w1x to simulate PR and PT excitonic spectra for QWs with different kinds of heterogeneity. To solve this problem, one should know the effect of electric field on reflection and absorption of QW structure. In Ref. w1x we have demonstrated that for simulation of optical spectra in electric field in the approximation of weakly interacting quantum states w2x one can use formulas that deal with optical transitions between quasi bound electron and hole states with finite broadening. The energy of the electron Žhole. state corresponds to the real part of the resolvent
0169-4332r00r$ - see front matter q 2000 Published by Elsevier Science B.V. PII: S 0 1 6 9 - 4 3 3 2 Ž 0 0 . 0 0 4 0 5 - 0
274
O.L. LazarenkoÕa, A.N. Pikhtinr Applied Surface Science 166 (2000) 273–277
poles of the one-electron Hamiltonian, and the fieldinduced homogeneous broadening of the states corresponds to the imaginary part of the resolvent poles. The effect of electric field on the energy and homogeneous broadening of electron Žhole. state has been discussed in detail in Ref. w2x. The probabilities of interband optical transitions may have a very intricate field dependence as a consequence of the electron and hole envelope wave function symmetry transformation in an electric field w1x.
1. Absorption spectrum of multiple quantum well (MQW) in graded electric field In real heterostructures, the electric field may be uniform or graded as in optical modulators based on MQW structures placed in the i-area of a p-i-n diode Žsee Fig. 1a.. In MQW structures, it is impossible to neglect the electric field gradient inside the structures as it is possible for an ultra thin layer of a single quantum well ŽSQW.. The interference of signals of QWs influenced by different values of electric field may be the cause of some changes of optical spectra. In Fig. 1 we compare the calculated spectra of GaAsr Ga 0.32 Al 0.68 As SQW Ždotted line. and MQW Žsolid line. structures for different values of the mean electric field F in the active region. The width of the ˚ well was the same in both structures: L s 95 A, ˚ barrier width is L B s 98 A, the quantity of QWs is N s 50. The electric field in the active region differs on about 25 kVrcm. We have not used any fitting parameters in our calculations. One can see that the effect of the electric field gradient varies for different mean values of electric fields. It is not equivalent to the similar broadening of each excitonic resonance because of complicated field-dependence of the parameters that define absorption. The comparison of calculated spectra with experimental data Žcircles in Fig. 1. reported in Ref. w3x has demonstrated a good agreement. One can follow the redistribution in electric field of the different excitonic transitions intensity both in experimental and calculated spectra. The fact that the experimental excitonic peaks are wider than the calculated ones
Fig. 1. The distribution of electric field in real optical modulator Ža. and the effect of gradient electric field on the absorption spectra of GaAsrGa 0.32 Al 0.68 As SQW Ždotted line. and MQW Žsolid line. structures for different mean electric field: Žb. F s10 kVrcm, Žc. F s 47 kVrcm, Žd. F s 73 kVrcm. The width of well is the same in both cases: Ls9.5 nm, barrier width is L B s9.8 nm, the number of QWs is N s 50. The increase of the electric field in the active region is about 25 kVrcm. The experimental data Žcircles. have been taken from Ref. w3x. The measurements and simulations refer to room temperature spectra.
convinces that one should take into account inhomogeneous broadening caused by well width and depth fluctuations.
2. The effect of QW width fluctuation In real structures the island-like fluctuation of the epilayer thickness may exist. It results in QW width fluctuation. The character of exciton-fluctuation interaction depends on the ratio of the exciton Bohr radius and the lateral extension of islands. We consider the macroscopic fluctuations when there is no effective localization of excitons. Such a situation is
O.L. LazarenkoÕa, A.N. Pikhtinr Applied Surface Science 166 (2000) 273–277
realized in heterostructures that grow according to island mechanisms. In the case of macroscopic fluctuations, it is necessary to convolute the initial spectrum with a Gaussian which width is proportional to the value of fluctuations and the absolute value of the first derivative of the exciton energy with respect to the fluctuating parameter: GndmL
Ž F. s
EEnexm Ž F . EL
k d L d L.
is defined only by electron and hole energy spectrum transformation: EEnemx
EEnexm EL
(
EEne EL
q
EEmh EL
,
s
ED Eg
EEne
q
ED Eg
EEmh ED Eg
,
which have been considered in Ref. w2x. In the second case, it is necessary to take into account the heterogeneity of the QW material energy gap. Therefore, EEnexm ED Eg
The parameter kd L depends on the fluctuation lateral extension and is about 1. Here
275
s
EEne ED Eg
q
EEmh ED Eg
y 1.
Note again that the difference in signs complicates the field dependence of the total broadening of excitonic peak. The field-induced changes in the reflection spectra of QWs with different fluctuation parameters of the same relative values are compared in Fig. 2. For
that is, the field dependence of the inhomogeneous broadening due to well width fluctuation is defined by the transformation of electron spectrum. The corresponding families of dimensionless field dependences have been presented in Ref. w2x. Note that the derivatives for electron and hole quantum levels may have different signs at some field values, that is, the exciton peak inhomogeneous broadening may be less than for the electron Žhole. one.
3. The effect of QW depth fluctuation We shall now examine, in a similar manner, the effect of QW depth fluctuation due to the fluctuation of alloy composition. There are two different situations: Ž1. the solid solution is the material of the barriers, and Ž2. the solid solution is the material of the QW. Both of them are equivalent to band offset fluctuation. In the first case, the energy gap of the QW material is constant and inhomogeneous broadening of the excitonic resonances
GndmD E g Ž F . s
EEnexm
Ž F.
ED Eg
kd D E gd D Eg
Fig. 2. The field-induced changes in the reflection spectra of QW with the following parameters: E g s 0.985 eV, Ve s120 meV, Vh s80 meV, m e s 0.06Pm 0 , m h h s 0.5Pm 0 , m l h s 0.07Pm 0 , L s 20 nm, for different fluctuated parameters of the same relative ˚ .; Žb. dD Eg r D Eg s 5% Ž10 meV.. values: Ža. d Lr Ls 5% Ž10 A
O.L. LazarenkoÕa, A.N. Pikhtinr Applied Surface Science 166 (2000) 273–277
276
our calculations, we used the typical parameters of the real ŽIn,Ga.ŽAs,P.rInP SQW: Eg s 0.935 eV, ˚ k s 12.5, Vc s 0.12 eV, VÕ s 0.08 eV, L s 200 A, m e s 0.06 P m 0 , m h h s 0.5 P m 0 , m l h s 0.07 P m 0 , and G T s 2.79 meV. At the zero electric field, this QW contains three electrons, three light holes, and seven heavy holes confined states. Thus, the structure of the reflection spectra is very complicated. One can see that the QW depth fluctuations determine the total broadening, if the solid solution is the material of the QW. One can see that it is impossible to consider that the exciton peaks are monotonously broadened in electric field, if there are any fluctuations of the QW parameters. When the electric field increases, the changes of the exciton resonances width may be very complicated. If you carry out a series of measurements at different values of electric field, you may obtain the main cause of the inhomogeneous broadening using presented in w2x dimensionless dependences.
d R dn dm1 s
ER n m Ed1
d d1sd d1
4. The top layer effect The reflected light from the QW interferes with the light reflected from the sample surface. The interference depends on the numbers of wavelength consisting the top layer. The form of the absorption spectrum periodically changes. It may look like a single resonance, antiresonance or antisymmetrical structure. In real samples, the surface may be very rough. In such samples, the top layer width has an uncertainty d d1. Thus, the symmetry of the reflection spectrum near the excitonic resonance distorts: R dn dm1 s R n m q d R dn dm1 . This distortion periodically depends on the top layer width and for the normal incidence may be calculated as
( ž (k y 1 / k "G ž (k q 1 /
16 k 0
0
1
0
3
0
`
= Ý Ž 2 i y 1. is1
y3
Ž Ei syn m q D E0 n m y "v . cos Ž 2 k 1 d1 . y Ž "Gn m q "G0 n m . sin Ž 2 k 1 d1 . 2 2 Ž Ei syn m q D E0 n m y "v . q Ž "Gn m q "G0 n m .
Here, k 0 denotes the effective dielectric constant of the structure, k 1 s v k 0 rc, G 0 n m is the radiative broadening of excitonic resonance, Ei s y n m is the energy of excitonic resonance, D E0 n m is the renormalization of the exciton resonance energy, and d1 denotes the width of the top layer. Such distortion of the absorption spectrum symmetry may be the cause of misunderstandings in the interpretation of experimental data for samples with an appreciably rough surface.
(
.
In order to illustrate our approach to simulate optical spectra in an electric field, the calculated and experimental PR ŽFig. 3. and PT spectra of the same laser structure ŽIn,Ga.ŽAs,P.rInP have been compared. For our calculations, we used the same parameters of SQW as in Fig. 2.
5. Characterization of low dimensional structures by photomodulation spectroscopy The results obtained for the influence of electric field on absorption and reflection spectra of QW structure makes it possible to simulate PR and PT spectra of real low dimensional systems.
Fig. 3. Theoretical Žsolid line. and experimental Žsquares. PR spectra of laser structure based on SQW.
O.L. LazarenkoÕa, A.N. Pikhtinr Applied Surface Science 166 (2000) 273–277
The best fitting of calculated and experimental data is achieved at the mean value of built-in electric field in the area of the QW equals to 16 kVrcm, the reduction of the electric field due to the laser pump equals to 1 kVrcm, and the inhomogeneous broadening is 20 meV. It is necessary to make experiments with different F to determine the cause of the broadening. Note that the evaluated values of these parameters are the same as for PR and PT spectra.
6. Conclusion Thus, we have demonstrated that the electric field modifies not only excitonic spectrum of QW but also its transformation due to different tapes of inhomo-
277
geneity of real structures. The effect of QW width and depth fluctuations, the top layer effect and the effect of gradient of electric field in MQW on optical absorption, reflection, and PR spectra have been calculated for real semiconductor heterostructures used in lasers and optical modulators.
References w1x O.L. Lazarenkova, A.N. Pikhtin, Phys. Status Solidi A 175 Ž1999. 51. w2x O.L. Lazarenkova, A.N. Pikhtin, Semiconductors 32 Ž1998. 992. w3x D.A.B. Miller, D.S. Chemla, T.C. Damen, A.C. Gossard, W. Wiegmann, T.H. Wood, C.A. Burrus, Phys. Rev. B 32 Ž1985. 1043.
Simulation of excitonic spectra in electric field to characterize the quality of low dimensional structures O.L. Lazarenkova) , A.N. Pikhtin St. Petersburg Electrotechnical UniÕersity, Professor PopoÕ str. 5, St. Petersburg 197376, Russian Federation
Abstract In order to characterize the real low dimensional structures, the simulation of excitonic spectra has been done. Absorption spectrum of multiple quantum well ŽMQW. in graded electric field, the effect of macroscopic quantum well width and depth fluctuations, and the influence of surface roughness have been considered. The simulated spectra are in a good agreement with experimental data. q 2000 Published by Elsevier Science B.V. PACS: 73.40 and 78.20; 71.35; S7.12; S7.15 Keywords: Quantum well; Electric field; Inhomogeneity; Modulation optical spectroscopy
A lot of modern devices of nanoelectronics and optoelectronics are based on specific properties of low dimensional structures. Their characteristics are very dependent on the quality of ultra thin epitaxial layers. Therefore, it is necessary to develop the techniques for investigation of quantum-size structures after the formation as during the technology process. Photoreflectance ŽPR. and phototransmittance ŽPT. as contactless techniques of modulation spectroscopy are the major tools for quantum well ŽQW. characterization. Their essence is to record the change of a reflected ŽPR. or transmitted ŽPT. probe light beam under the periodic modulation of the sample charac-
) Corresponding author. Tel.: q7-812-234-31-64; fax: q7-812234-31-64. E-mail address: omk.meeeltech.ru ŽO.L. Lazarenkova..
teristics by the second light beam, which changes the built-in electric field. Thus, the relatively sharp features are observed in the spectra, even at room temperature. However, some difficulties are currently encountered in analyzing the observed spectra of QW structures. The excitonic effects have to be taken into account on principle. In this paper, we develop an approach previously proposed by ourselves w1x to simulate PR and PT excitonic spectra for QWs with different kinds of heterogeneity. To solve this problem, one should know the effect of electric field on reflection and absorption of QW structure. In Ref. w1x we have demonstrated that for simulation of optical spectra in electric field in the approximation of weakly interacting quantum states w2x one can use formulas that deal with optical transitions between quasi bound electron and hole states with finite broadening. The energy of the electron Žhole. state corresponds to the real part of the resolvent
0169-4332r00r$ - see front matter q 2000 Published by Elsevier Science B.V. PII: S 0 1 6 9 - 4 3 3 2 Ž 0 0 . 0 0 4 0 5 - 0
274
O.L. LazarenkoÕa, A.N. Pikhtinr Applied Surface Science 166 (2000) 273–277
poles of the one-electron Hamiltonian, and the fieldinduced homogeneous broadening of the states corresponds to the imaginary part of the resolvent poles. The effect of electric field on the energy and homogeneous broadening of electron Žhole. state has been discussed in detail in Ref. w2x. The probabilities of interband optical transitions may have a very intricate field dependence as a consequence of the electron and hole envelope wave function symmetry transformation in an electric field w1x.
1. Absorption spectrum of multiple quantum well (MQW) in graded electric field In real heterostructures, the electric field may be uniform or graded as in optical modulators based on MQW structures placed in the i-area of a p-i-n diode Žsee Fig. 1a.. In MQW structures, it is impossible to neglect the electric field gradient inside the structures as it is possible for an ultra thin layer of a single quantum well ŽSQW.. The interference of signals of QWs influenced by different values of electric field may be the cause of some changes of optical spectra. In Fig. 1 we compare the calculated spectra of GaAsr Ga 0.32 Al 0.68 As SQW Ždotted line. and MQW Žsolid line. structures for different values of the mean electric field F in the active region. The width of the ˚ well was the same in both structures: L s 95 A, ˚ barrier width is L B s 98 A, the quantity of QWs is N s 50. The electric field in the active region differs on about 25 kVrcm. We have not used any fitting parameters in our calculations. One can see that the effect of the electric field gradient varies for different mean values of electric fields. It is not equivalent to the similar broadening of each excitonic resonance because of complicated field-dependence of the parameters that define absorption. The comparison of calculated spectra with experimental data Žcircles in Fig. 1. reported in Ref. w3x has demonstrated a good agreement. One can follow the redistribution in electric field of the different excitonic transitions intensity both in experimental and calculated spectra. The fact that the experimental excitonic peaks are wider than the calculated ones
Fig. 1. The distribution of electric field in real optical modulator Ža. and the effect of gradient electric field on the absorption spectra of GaAsrGa 0.32 Al 0.68 As SQW Ždotted line. and MQW Žsolid line. structures for different mean electric field: Žb. F s10 kVrcm, Žc. F s 47 kVrcm, Žd. F s 73 kVrcm. The width of well is the same in both cases: Ls9.5 nm, barrier width is L B s9.8 nm, the number of QWs is N s 50. The increase of the electric field in the active region is about 25 kVrcm. The experimental data Žcircles. have been taken from Ref. w3x. The measurements and simulations refer to room temperature spectra.
convinces that one should take into account inhomogeneous broadening caused by well width and depth fluctuations.
2. The effect of QW width fluctuation In real structures the island-like fluctuation of the epilayer thickness may exist. It results in QW width fluctuation. The character of exciton-fluctuation interaction depends on the ratio of the exciton Bohr radius and the lateral extension of islands. We consider the macroscopic fluctuations when there is no effective localization of excitons. Such a situation is
O.L. LazarenkoÕa, A.N. Pikhtinr Applied Surface Science 166 (2000) 273–277
realized in heterostructures that grow according to island mechanisms. In the case of macroscopic fluctuations, it is necessary to convolute the initial spectrum with a Gaussian which width is proportional to the value of fluctuations and the absolute value of the first derivative of the exciton energy with respect to the fluctuating parameter: GndmL
Ž F. s
EEnexm Ž F . EL
k d L d L.
is defined only by electron and hole energy spectrum transformation: EEnemx
EEnexm EL
(
EEne EL
q
EEmh EL
,
s
ED Eg
EEne
q
ED Eg
EEmh ED Eg
,
which have been considered in Ref. w2x. In the second case, it is necessary to take into account the heterogeneity of the QW material energy gap. Therefore, EEnexm ED Eg
The parameter kd L depends on the fluctuation lateral extension and is about 1. Here
275
s
EEne ED Eg
q
EEmh ED Eg
y 1.
Note again that the difference in signs complicates the field dependence of the total broadening of excitonic peak. The field-induced changes in the reflection spectra of QWs with different fluctuation parameters of the same relative values are compared in Fig. 2. For
that is, the field dependence of the inhomogeneous broadening due to well width fluctuation is defined by the transformation of electron spectrum. The corresponding families of dimensionless field dependences have been presented in Ref. w2x. Note that the derivatives for electron and hole quantum levels may have different signs at some field values, that is, the exciton peak inhomogeneous broadening may be less than for the electron Žhole. one.
3. The effect of QW depth fluctuation We shall now examine, in a similar manner, the effect of QW depth fluctuation due to the fluctuation of alloy composition. There are two different situations: Ž1. the solid solution is the material of the barriers, and Ž2. the solid solution is the material of the QW. Both of them are equivalent to band offset fluctuation. In the first case, the energy gap of the QW material is constant and inhomogeneous broadening of the excitonic resonances
GndmD E g Ž F . s
EEnexm
Ž F.
ED Eg
kd D E gd D Eg
Fig. 2. The field-induced changes in the reflection spectra of QW with the following parameters: E g s 0.985 eV, Ve s120 meV, Vh s80 meV, m e s 0.06Pm 0 , m h h s 0.5Pm 0 , m l h s 0.07Pm 0 , L s 20 nm, for different fluctuated parameters of the same relative ˚ .; Žb. dD Eg r D Eg s 5% Ž10 meV.. values: Ža. d Lr Ls 5% Ž10 A
O.L. LazarenkoÕa, A.N. Pikhtinr Applied Surface Science 166 (2000) 273–277
276
our calculations, we used the typical parameters of the real ŽIn,Ga.ŽAs,P.rInP SQW: Eg s 0.935 eV, ˚ k s 12.5, Vc s 0.12 eV, VÕ s 0.08 eV, L s 200 A, m e s 0.06 P m 0 , m h h s 0.5 P m 0 , m l h s 0.07 P m 0 , and G T s 2.79 meV. At the zero electric field, this QW contains three electrons, three light holes, and seven heavy holes confined states. Thus, the structure of the reflection spectra is very complicated. One can see that the QW depth fluctuations determine the total broadening, if the solid solution is the material of the QW. One can see that it is impossible to consider that the exciton peaks are monotonously broadened in electric field, if there are any fluctuations of the QW parameters. When the electric field increases, the changes of the exciton resonances width may be very complicated. If you carry out a series of measurements at different values of electric field, you may obtain the main cause of the inhomogeneous broadening using presented in w2x dimensionless dependences.
d R dn dm1 s
ER n m Ed1
d d1sd d1
4. The top layer effect The reflected light from the QW interferes with the light reflected from the sample surface. The interference depends on the numbers of wavelength consisting the top layer. The form of the absorption spectrum periodically changes. It may look like a single resonance, antiresonance or antisymmetrical structure. In real samples, the surface may be very rough. In such samples, the top layer width has an uncertainty d d1. Thus, the symmetry of the reflection spectrum near the excitonic resonance distorts: R dn dm1 s R n m q d R dn dm1 . This distortion periodically depends on the top layer width and for the normal incidence may be calculated as
( ž (k y 1 / k "G ž (k q 1 /
16 k 0
0
1
0
3
0
`
= Ý Ž 2 i y 1. is1
y3
Ž Ei syn m q D E0 n m y "v . cos Ž 2 k 1 d1 . y Ž "Gn m q "G0 n m . sin Ž 2 k 1 d1 . 2 2 Ž Ei syn m q D E0 n m y "v . q Ž "Gn m q "G0 n m .
Here, k 0 denotes the effective dielectric constant of the structure, k 1 s v k 0 rc, G 0 n m is the radiative broadening of excitonic resonance, Ei s y n m is the energy of excitonic resonance, D E0 n m is the renormalization of the exciton resonance energy, and d1 denotes the width of the top layer. Such distortion of the absorption spectrum symmetry may be the cause of misunderstandings in the interpretation of experimental data for samples with an appreciably rough surface.
(
.
In order to illustrate our approach to simulate optical spectra in an electric field, the calculated and experimental PR ŽFig. 3. and PT spectra of the same laser structure ŽIn,Ga.ŽAs,P.rInP have been compared. For our calculations, we used the same parameters of SQW as in Fig. 2.
5. Characterization of low dimensional structures by photomodulation spectroscopy The results obtained for the influence of electric field on absorption and reflection spectra of QW structure makes it possible to simulate PR and PT spectra of real low dimensional systems.
Fig. 3. Theoretical Žsolid line. and experimental Žsquares. PR spectra of laser structure based on SQW.
O.L. LazarenkoÕa, A.N. Pikhtinr Applied Surface Science 166 (2000) 273–277
The best fitting of calculated and experimental data is achieved at the mean value of built-in electric field in the area of the QW equals to 16 kVrcm, the reduction of the electric field due to the laser pump equals to 1 kVrcm, and the inhomogeneous broadening is 20 meV. It is necessary to make experiments with different F to determine the cause of the broadening. Note that the evaluated values of these parameters are the same as for PR and PT spectra.
6. Conclusion Thus, we have demonstrated that the electric field modifies not only excitonic spectrum of QW but also its transformation due to different tapes of inhomo-
277
geneity of real structures. The effect of QW width and depth fluctuations, the top layer effect and the effect of gradient of electric field in MQW on optical absorption, reflection, and PR spectra have been calculated for real semiconductor heterostructures used in lasers and optical modulators.
References w1x O.L. Lazarenkova, A.N. Pikhtin, Phys. Status Solidi A 175 Ž1999. 51. w2x O.L. Lazarenkova, A.N. Pikhtin, Semiconductors 32 Ž1998. 992. w3x D.A.B. Miller, D.S. Chemla, T.C. Damen, A.C. Gossard, W. Wiegmann, T.H. Wood, C.A. Burrus, Phys. Rev. B 32 Ž1985. 1043.