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Exciton−longitudinal-optical phonon coupling in quantum wires and quantum dots
Physica B 322 (2002) 201–204
Excitonlongitudinal-optical phonon coupling in quantum wires and quantum dots W.Z. Shen Laboratory of Condensed Matter Spectroscopy and Opto-Electronic Physics, Department of Physics, Shanghai Jiao Tong University, 1954 Hua Shan Road, Shanghai 200030, People’s Republic of China Received 7 February 2002; accepted 8 March 2002
Abstract In this paper, by using the recently proposed theoretical approach for the study of excitonlongitudinal-optical (LO) phonon coupling in semiconductor quantum-well (QW) structures and bulk materials, we have deduced, for the first time, the excitonLO phonon coupling strength in quantum wire (QWR) and quantum dot (QD) structures, and explained quantitatively the typical experimental results of GaAs/AlAs QWR and CdSSe semiconductor microcrystallites. The theoretical approach has been shown clearly of being applied to other low dimensional semiconductor materials. r 2002 Elsevier Science B.V. All rights reserved. PACS: 78.66.Fd; 71.35.+z Keywords: ExcitonLO phonon coupling; Quantum wires; Quantum dots
1. Introduction The appearance of sharp, well-resolved exciton resonance at room temperature (at which most devices should be used) is one of the most important properties in quantum well structures (QWs). Several device applications have been made utilizing the large optical nonlinearities of excitons in QWs [1]. In attaining these large optical nonlinearities, the spectral width of excitons plays an important role [2]. As a result, the excitonlongitudinal-optical (LO) phonon coupling is a key parameter for the use of QWs as optoelectronic devices, and is still a subject of considerable experimental and theoretical interest E-mail address: [email protected] (W.Z. Shen).
[1–6]. Quite recently, the study of excitonLO phonon coupling has been summarized [3] as four distinct features: a large difference between (i) III– V and II–VI QWs, (ii) multiple QWs (MQWs) and single QWs (SQWs), as well as (iii) QWs and their corresponding bulk materials, and its linear dependence on well width in QWs. A quantitatively theoretical approach has been presented [3], which can well explain all the experimental observations of the large difference in exciton– LO phonon coupling and clarify the controversy in the literature. It is expected that additional confinement of carriers, such as in quantum wire structures (QWRs) and quantum dot structures (QDs), can lead to significant improvement in device performance [7]. Therefore, much attention has also been
0921-4526/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 2 ) 0 1 1 8 6 - 9
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paid to determine the exciton–phonon coupling in these structures. Typically, Gopal et al. [8] have performed a comparison study of the exciton–LO phonon coupling strength (Gc ) in GaAs/AlAs QWR structure with that of a reference GaAs/ AlAs MQW structure. They have found that the exciton–LO phonon scattering is enhanced to Gc of 16.571.0 meV for the one-dimensional (1D) excitons in the QWR, as compared with Gc of 12.071.0 meV for the two-dimensional (2D) excitons in the reference MQW. However, they did not give any quantitative analysis for the enhancement, simply by ascribing as the difference in the wave-functions between quasi-1D and 2D excitons [8]. On the other hand, Nomura et al. [9] have carried out a detailed experimental and theoretical investigation of the exciton–LO phonon coupling in CdS0.12Se0.88 semiconductor micro-crystallites. However, they still could not explain explicitly the experimentally obtained coupling strength Gc of 29 meV for the 1.75 nm micro-crystallites, and 30 meV for the 5.05 nm micro-crystallites, as well as the strong coupling Gc of 106 meV for bulk CdSe [9,10]. In this paper, by employing our theoretical approach [3] to these QWR and QD cases with additional confinement of the carriers, we analyze quantitatively the exciton–LO coupling strength of the above typically experimental results in the literature [8,9].
2. Results and discussion We start with the obtained exciton–LO phonon coupling strength Gc in semiconductors [3]: Gc p 2pe2 ð1=eN 1=e0 Þ ½ðme þ mh Þ_oLO =ð2_2 Þ1=2 =ðq2 OÞ;
ð1Þ
where e is the free electron charge, eN and e0 are the high frequency and static dielectric constants, respectively, me and mh are the effective electron and hole masses, respectively, _oLO is the LO phonon energy, q is the phonon wave vector, and O is the volume unit. In low dimensional systems, . exciton–LO phonon interaction (Frohlich interaction) is affected not only by changes in the particle wave function due to the confining potential but
also by changes in the LO phonon modes caused by phonon confinement. In QWR structures with the confinement in the y and z axes, the phonon wave vector q ¼ ðqx ; mp=Ly ; np=Lz Þ with m and n integers, and O ¼ Lx Ly Lz is the dimension in the x; y; and z directions. As in the QW case where the contribution of the interface phonons to the exciton–LO phonon coupling is negligible [3], we concentrate only on the dominant m ¼ 1 and n ¼ 1 confined phonon modes, and have: Gc ¼ RLy =f½ðp=Ly Þ2 þ ðp=Lz Þ2 Ly Lz g;
ð2Þ
where R is the ratio constant, same as in Ref. [3]. Eq. (2) can be further simplified by assuming the same confinement size in the y and z axes (Ly ¼ Lz ), and Gc is found to have a linear dependence on the confinement width Lz (or Ly ): Gc ðQWRÞ ¼ RLz =ð2p2 Þ
ð3Þ
while in MQW structures, the exciton–LO phonon coupling strength has been deduced [3] by assuming that the three components (x; y; and z directions) of the phonon wave vector have the same contribution, and is given by Gc ðMQWÞ ¼ RLz =ð3p2 Þ:
ð4Þ
In comparison Eq. (3) with Eq. (4), it is clear that the exciton–LO phonon coupling will be enhanced by 50% for the 1D excitons in QWRs, as compared with that for 2D excitons in corresponding MQWs with the same material and same confinement size. The theoretical results can well explain the experimental results of the enhancement [8]: Gc ¼ 16:571:0 meV for 1D excitons in GaAs/AlAs QWR, while Gc ¼ 12:071:0 meV for 2D excitons in the reference GaAs/AlAs MQW with the same thickness of 10 nm. Furthermore, the agreement between the theory and experiment would be better if taking into account the small contribution of phonon mode in the x direction and the lateral barrier thickness. The above argument clearly reveals that the enhancement is mainly due to the changes in the LO phonon modes caused by phonon confinement, rather than the difference in the wave-functions between quasi-1D and 2D excitons, suggested in Ref. [8]. This can also explain the experimental fact [8] that the enhanced exciton–phonon scattering in 1D is
W.Z. Shen / Physica B 322 (2002) 201–204
unimportant for the temperature dependence of the exciton peak energy. In the QD case, the carriers are confined in the three directions x; y; and z: The density of states is known to be transformed from the continuum states of bulk semiconductors to discrete states by the confinement, and QDs exhibit the so-called blueshift of the transition energy. Following the same way, the exciton–LO phonon coupling strength can be easily obtained by considering only the dominant confined phonon modes (the lowest ones) and simply assuming the same confinement size in all three directions (Lx ¼ Ly ¼ Lz ): Gc ðQDÞ ¼ RLz =ð3p2 Þ
ð5Þ
which shows that the coupling strength in QDs is the same as that in MQWs. In order to verify the argument and explain the experimentally obtained Gc of 30 meV for the 5.05 nm CdS0.12Se0.88 semiconductor micro-crystallites [9], we compare with the experimental result of Gc ¼ 16 meV for a same II–VI Cd0.10Zn0.90Te/ZnTe MQW with a similar well width of 4.0 nm [5]. The above difference in exciton–LO phonon coupling is mainly due to the difference of the parameters of eN ; e0 ; me ; mh ; and _oLO ; and can be well explained by Eq. (1). The above parameters for the ternary wells (CdZnTe and CdSSe) are approximated as being temperature independent and can be estimated from these binary ones [5,9] by using a linear interpolation scheme. By employing the parameters of eN ¼ 7:26; e0 ¼ 10:11; me ¼ 0:119m0 ; mh ¼ 0:213m0 ; _oLO ¼ 25:6 meV for Cd0.10Zn0.90Te wells, and eN ¼ 6:23; e0 ¼ 9:64; me ¼ 0:13m0 ; mh ¼ 0:43m0 ; _oLO ¼ 28:0 meV for CdS0.12Se0.88 with m0 being the free electron mass, we get: Gc ðCdS0:12 Se0:88 Þ=Gc ðCd0:10 Zn0:90 TeÞ ¼ 1:98
203
[9] that Gc increases with the micro-crystallite radius, i.e., 29 meV for 1.75 nm and 30 meV for 5.05 nm micro-crystallites, is also in agreement with the conclusion of Eq. (5). Furthermore, we have confirmed that there is much stronger (B3.9 times in Gc ) exciton–LO phonon coupling in bulk materials than that of MQW structures [3]. Based on the above conclusion, the bulk materials will also have B3.9 times stronger in Gc than that of QD structures. The experimentally large value Gc of 106 meV for bulk CdSe [9,10] (comparing with 29–30 meV in microcrystallites [9]) provides additional evidence for our argument. The reduction in the coupling is mainly due to the confinement effect as pointed out by Schmitt-Rink et al. [2].
3. Conclusions In summary, we have extended our theoretical approach for exciton–LO phonon coupling in semiconductor QWs to additional confinement cases of QWRs and QDs, by studying two typically experimental results in these two cases. The theoretical approach can well explain the enhancement of exciton–LO phonon scattering in GaAs/AlAs QWRs, as well as the coupling strength in semiconductor CdSSe micro-crystallites and their corresponding bulk materials. Based on the successful application in several QW materials [3], we think that the simply theoretical approach can be used in other QWR and QD materials. Better understanding of the coupling behavior is helpful for the application of these low dimensional semiconductor devices.
ð6Þ
in excellent agreement with the experimental results of 30 meV for CdS0.12Se0.88 micro-crystallites versus 16 meV for Cd0.10Zn0.90Te/ZnTe MQW, which clearly supports the above argument. It should be noted that the 1.05 nm difference in confinement size (4.0 to 5.05 nm) will increase Gc only 0.35 meV for Cd0.10Zn0.90Te/ ZnTe MQW [3]. The experimental observation
Acknowledgements This work is supported in part by the Natural Science Foundation of China under contract Nos. 10125416 and 60006005, Shanghai QMX and TRAPOYT of Minister Of Education, People’s Republic of China.
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References [1] D.S. Chemla, D.A.B. Miller, P.W. Smith, in: R. Dingle (Ed.), Semiconductors and Semimetals, Vol. 24, Academic, San Diego, 1987, p. 279. [2] S. Schmitt-Rink, D.A.B. Miller, D.S. Chemla, Phys. Rev. B 35 (1987) 8113. [3] W.Z. Shen, Appl. Phys. Lett. 79 (2001) 1285. [4] W.Z. Shen, W.G. Tang, S.C. Shen, S.M. Wang, T.G. Andersson, Appl. Phys. Lett. 65 (1994) 2728; W.Z. Shen, S.C. Shen, W.G. Tang, S.M. Wang, T.G. Andersson, J. Appl. Phys. 78 (1995) 1178.
[5] D. Lee, A.M. Johnson, J.E. Zucker, R.D. Feldman, R.F. Austin, J. Appl. Phys. 69 (1991) 6722. . [6] J. Feldmann, G. Peter, E.O. Gobel, P. Dawson, K. Morre, C. Foxon, R.J. Elliott, Phys. Rev. Lett. 59 (1987) 2337. [7] E. Kapon, D.M. Hwang, R. Bhat, Phys. Rev. Lett. 63 (1989) 430. [8] A.V. Gopal, R. Kumar, A.S. Vengurlekar, T. Melin, F. Laruelle, B. Etienne, Appl. Phys. Lett. 74 (1999) 2474. [9] S. Nomura, T. Kobayashi, Phys. Rev. B 45 (1992) 1305. [10] J. Voigt, F. Spielgelberg, M. Senoner, Phys. Stat. Sol. B 91 (1979) 189.
Excitonlongitudinal-optical phonon coupling in quantum wires and quantum dots W.Z. Shen Laboratory of Condensed Matter Spectroscopy and Opto-Electronic Physics, Department of Physics, Shanghai Jiao Tong University, 1954 Hua Shan Road, Shanghai 200030, People’s Republic of China Received 7 February 2002; accepted 8 March 2002
Abstract In this paper, by using the recently proposed theoretical approach for the study of excitonlongitudinal-optical (LO) phonon coupling in semiconductor quantum-well (QW) structures and bulk materials, we have deduced, for the first time, the excitonLO phonon coupling strength in quantum wire (QWR) and quantum dot (QD) structures, and explained quantitatively the typical experimental results of GaAs/AlAs QWR and CdSSe semiconductor microcrystallites. The theoretical approach has been shown clearly of being applied to other low dimensional semiconductor materials. r 2002 Elsevier Science B.V. All rights reserved. PACS: 78.66.Fd; 71.35.+z Keywords: ExcitonLO phonon coupling; Quantum wires; Quantum dots
1. Introduction The appearance of sharp, well-resolved exciton resonance at room temperature (at which most devices should be used) is one of the most important properties in quantum well structures (QWs). Several device applications have been made utilizing the large optical nonlinearities of excitons in QWs [1]. In attaining these large optical nonlinearities, the spectral width of excitons plays an important role [2]. As a result, the excitonlongitudinal-optical (LO) phonon coupling is a key parameter for the use of QWs as optoelectronic devices, and is still a subject of considerable experimental and theoretical interest E-mail address: [email protected] (W.Z. Shen).
[1–6]. Quite recently, the study of excitonLO phonon coupling has been summarized [3] as four distinct features: a large difference between (i) III– V and II–VI QWs, (ii) multiple QWs (MQWs) and single QWs (SQWs), as well as (iii) QWs and their corresponding bulk materials, and its linear dependence on well width in QWs. A quantitatively theoretical approach has been presented [3], which can well explain all the experimental observations of the large difference in exciton– LO phonon coupling and clarify the controversy in the literature. It is expected that additional confinement of carriers, such as in quantum wire structures (QWRs) and quantum dot structures (QDs), can lead to significant improvement in device performance [7]. Therefore, much attention has also been
0921-4526/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 2 ) 0 1 1 8 6 - 9
W.Z. Shen / Physica B 322 (2002) 201–204
202
paid to determine the exciton–phonon coupling in these structures. Typically, Gopal et al. [8] have performed a comparison study of the exciton–LO phonon coupling strength (Gc ) in GaAs/AlAs QWR structure with that of a reference GaAs/ AlAs MQW structure. They have found that the exciton–LO phonon scattering is enhanced to Gc of 16.571.0 meV for the one-dimensional (1D) excitons in the QWR, as compared with Gc of 12.071.0 meV for the two-dimensional (2D) excitons in the reference MQW. However, they did not give any quantitative analysis for the enhancement, simply by ascribing as the difference in the wave-functions between quasi-1D and 2D excitons [8]. On the other hand, Nomura et al. [9] have carried out a detailed experimental and theoretical investigation of the exciton–LO phonon coupling in CdS0.12Se0.88 semiconductor micro-crystallites. However, they still could not explain explicitly the experimentally obtained coupling strength Gc of 29 meV for the 1.75 nm micro-crystallites, and 30 meV for the 5.05 nm micro-crystallites, as well as the strong coupling Gc of 106 meV for bulk CdSe [9,10]. In this paper, by employing our theoretical approach [3] to these QWR and QD cases with additional confinement of the carriers, we analyze quantitatively the exciton–LO coupling strength of the above typically experimental results in the literature [8,9].
2. Results and discussion We start with the obtained exciton–LO phonon coupling strength Gc in semiconductors [3]: Gc p 2pe2 ð1=eN 1=e0 Þ ½ðme þ mh Þ_oLO =ð2_2 Þ1=2 =ðq2 OÞ;
ð1Þ
where e is the free electron charge, eN and e0 are the high frequency and static dielectric constants, respectively, me and mh are the effective electron and hole masses, respectively, _oLO is the LO phonon energy, q is the phonon wave vector, and O is the volume unit. In low dimensional systems, . exciton–LO phonon interaction (Frohlich interaction) is affected not only by changes in the particle wave function due to the confining potential but
also by changes in the LO phonon modes caused by phonon confinement. In QWR structures with the confinement in the y and z axes, the phonon wave vector q ¼ ðqx ; mp=Ly ; np=Lz Þ with m and n integers, and O ¼ Lx Ly Lz is the dimension in the x; y; and z directions. As in the QW case where the contribution of the interface phonons to the exciton–LO phonon coupling is negligible [3], we concentrate only on the dominant m ¼ 1 and n ¼ 1 confined phonon modes, and have: Gc ¼ RLy =f½ðp=Ly Þ2 þ ðp=Lz Þ2 Ly Lz g;
ð2Þ
where R is the ratio constant, same as in Ref. [3]. Eq. (2) can be further simplified by assuming the same confinement size in the y and z axes (Ly ¼ Lz ), and Gc is found to have a linear dependence on the confinement width Lz (or Ly ): Gc ðQWRÞ ¼ RLz =ð2p2 Þ
ð3Þ
while in MQW structures, the exciton–LO phonon coupling strength has been deduced [3] by assuming that the three components (x; y; and z directions) of the phonon wave vector have the same contribution, and is given by Gc ðMQWÞ ¼ RLz =ð3p2 Þ:
ð4Þ
In comparison Eq. (3) with Eq. (4), it is clear that the exciton–LO phonon coupling will be enhanced by 50% for the 1D excitons in QWRs, as compared with that for 2D excitons in corresponding MQWs with the same material and same confinement size. The theoretical results can well explain the experimental results of the enhancement [8]: Gc ¼ 16:571:0 meV for 1D excitons in GaAs/AlAs QWR, while Gc ¼ 12:071:0 meV for 2D excitons in the reference GaAs/AlAs MQW with the same thickness of 10 nm. Furthermore, the agreement between the theory and experiment would be better if taking into account the small contribution of phonon mode in the x direction and the lateral barrier thickness. The above argument clearly reveals that the enhancement is mainly due to the changes in the LO phonon modes caused by phonon confinement, rather than the difference in the wave-functions between quasi-1D and 2D excitons, suggested in Ref. [8]. This can also explain the experimental fact [8] that the enhanced exciton–phonon scattering in 1D is
W.Z. Shen / Physica B 322 (2002) 201–204
unimportant for the temperature dependence of the exciton peak energy. In the QD case, the carriers are confined in the three directions x; y; and z: The density of states is known to be transformed from the continuum states of bulk semiconductors to discrete states by the confinement, and QDs exhibit the so-called blueshift of the transition energy. Following the same way, the exciton–LO phonon coupling strength can be easily obtained by considering only the dominant confined phonon modes (the lowest ones) and simply assuming the same confinement size in all three directions (Lx ¼ Ly ¼ Lz ): Gc ðQDÞ ¼ RLz =ð3p2 Þ
ð5Þ
which shows that the coupling strength in QDs is the same as that in MQWs. In order to verify the argument and explain the experimentally obtained Gc of 30 meV for the 5.05 nm CdS0.12Se0.88 semiconductor micro-crystallites [9], we compare with the experimental result of Gc ¼ 16 meV for a same II–VI Cd0.10Zn0.90Te/ZnTe MQW with a similar well width of 4.0 nm [5]. The above difference in exciton–LO phonon coupling is mainly due to the difference of the parameters of eN ; e0 ; me ; mh ; and _oLO ; and can be well explained by Eq. (1). The above parameters for the ternary wells (CdZnTe and CdSSe) are approximated as being temperature independent and can be estimated from these binary ones [5,9] by using a linear interpolation scheme. By employing the parameters of eN ¼ 7:26; e0 ¼ 10:11; me ¼ 0:119m0 ; mh ¼ 0:213m0 ; _oLO ¼ 25:6 meV for Cd0.10Zn0.90Te wells, and eN ¼ 6:23; e0 ¼ 9:64; me ¼ 0:13m0 ; mh ¼ 0:43m0 ; _oLO ¼ 28:0 meV for CdS0.12Se0.88 with m0 being the free electron mass, we get: Gc ðCdS0:12 Se0:88 Þ=Gc ðCd0:10 Zn0:90 TeÞ ¼ 1:98
203
[9] that Gc increases with the micro-crystallite radius, i.e., 29 meV for 1.75 nm and 30 meV for 5.05 nm micro-crystallites, is also in agreement with the conclusion of Eq. (5). Furthermore, we have confirmed that there is much stronger (B3.9 times in Gc ) exciton–LO phonon coupling in bulk materials than that of MQW structures [3]. Based on the above conclusion, the bulk materials will also have B3.9 times stronger in Gc than that of QD structures. The experimentally large value Gc of 106 meV for bulk CdSe [9,10] (comparing with 29–30 meV in microcrystallites [9]) provides additional evidence for our argument. The reduction in the coupling is mainly due to the confinement effect as pointed out by Schmitt-Rink et al. [2].
3. Conclusions In summary, we have extended our theoretical approach for exciton–LO phonon coupling in semiconductor QWs to additional confinement cases of QWRs and QDs, by studying two typically experimental results in these two cases. The theoretical approach can well explain the enhancement of exciton–LO phonon scattering in GaAs/AlAs QWRs, as well as the coupling strength in semiconductor CdSSe micro-crystallites and their corresponding bulk materials. Based on the successful application in several QW materials [3], we think that the simply theoretical approach can be used in other QWR and QD materials. Better understanding of the coupling behavior is helpful for the application of these low dimensional semiconductor devices.
ð6Þ
in excellent agreement with the experimental results of 30 meV for CdS0.12Se0.88 micro-crystallites versus 16 meV for Cd0.10Zn0.90Te/ZnTe MQW, which clearly supports the above argument. It should be noted that the 1.05 nm difference in confinement size (4.0 to 5.05 nm) will increase Gc only 0.35 meV for Cd0.10Zn0.90Te/ ZnTe MQW [3]. The experimental observation
Acknowledgements This work is supported in part by the Natural Science Foundation of China under contract Nos. 10125416 and 60006005, Shanghai QMX and TRAPOYT of Minister Of Education, People’s Republic of China.
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W.Z. Shen / Physica B 322 (2002) 201–204
References [1] D.S. Chemla, D.A.B. Miller, P.W. Smith, in: R. Dingle (Ed.), Semiconductors and Semimetals, Vol. 24, Academic, San Diego, 1987, p. 279. [2] S. Schmitt-Rink, D.A.B. Miller, D.S. Chemla, Phys. Rev. B 35 (1987) 8113. [3] W.Z. Shen, Appl. Phys. Lett. 79 (2001) 1285. [4] W.Z. Shen, W.G. Tang, S.C. Shen, S.M. Wang, T.G. Andersson, Appl. Phys. Lett. 65 (1994) 2728; W.Z. Shen, S.C. Shen, W.G. Tang, S.M. Wang, T.G. Andersson, J. Appl. Phys. 78 (1995) 1178.
[5] D. Lee, A.M. Johnson, J.E. Zucker, R.D. Feldman, R.F. Austin, J. Appl. Phys. 69 (1991) 6722. . [6] J. Feldmann, G. Peter, E.O. Gobel, P. Dawson, K. Morre, C. Foxon, R.J. Elliott, Phys. Rev. Lett. 59 (1987) 2337. [7] E. Kapon, D.M. Hwang, R. Bhat, Phys. Rev. Lett. 63 (1989) 430. [8] A.V. Gopal, R. Kumar, A.S. Vengurlekar, T. Melin, F. Laruelle, B. Etienne, Appl. Phys. Lett. 74 (1999) 2474. [9] S. Nomura, T. Kobayashi, Phys. Rev. B 45 (1992) 1305. [10] J. Voigt, F. Spielgelberg, M. Senoner, Phys. Stat. Sol. B 91 (1979) 189.