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Non-delta-function electronic spectral densities in individual quantum dots
ARTICLE IN PRESS
Microelectronics Journal 39 (2008) 375–377 www.elsevier.com/locate/mejo
Non-delta-function electronic spectral densities in individual quantum dots Karel Kra´l Institute of Physics of the ASCR, v.v.i., Na Slovance 2, 18221 Praha 8, Czech Republic Available online 16 August 2007
Abstract Using a simplified model approach we estimate the optical line shape of the transition lines observable in photoluminescence experiments on quantum dots. We use the theory based on the interaction of electrons with the longitudinal optical phonons only. This theory gives, in the self-consistent Born approximation, the lowest-energy excited state line shape in the form of a very narrow peak with a shoulder on the low-energy side. We turn the attention to a comparison with experiments which appear to support this theoretical conclusion. This agreement emphasizes the role of the electronic multiple scattering on optical phonons in quantum dots. It is demonstrated that the optical line shape can give an information about the quantum dot system. r 2007 Elsevier Ltd. All rights reserved. PACS: 73.63.Kv; 73.21.La; 78.67.Hc Keywords: Quantum dots; Electron–phonon coupling; Luminescence
The quantum dots (QD) [1] are important for applications of nanostructures in optoelectronics. We could simply expect the optical lines of the luminescent transitions to be very narrow Lorentzians [2,3]. The experimental observations made with a sufficient spectral resolution tell us, however, that the actual line profiles are not always narrow and symmetrical [4–8]. Similar conclusions are found in the recent numerical experiments [9]. We shall pay attention to this question. The longitudinal optical (LO) phonons are important for the electronic relaxation in semiconductor QDs [10–12]. Using numerical calculations we show how the electron–LO (e–LO) phonon coupling influences the optical line shape. We shall confine our attention to the case of a single electron–hole pair in a single QD. The mutual annihilation of the particle pair gives the photon emitted in the photoluminescence. We shall neglect the effect of the motion of the hole particle regarding it as a static charge only. We shall limit the theory to two electronic bound states inside a single dot. We shall confine ourselves to the electronic intra-dot coupling to the dispersionless LO phonons only. E-mail address: [email protected] 0026-2692/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.mejo.2007.07.015
The electronic states in a QD will be approximated by electronic eigenstates in a cubic shaped QD, with infinitely deep potential. The electron ground state is the state with index n ¼ 0. The excited state (n ¼ 1) is one of the lowestenergy triple-degenerate exited states in this QD potential. The bath of LO phonons will be approximated by the bulk modes of dispersionless LO phonons of the full sample. Neglecting the electronic spin the Hamiltonian of the whole system is: H ¼ H e þ H ph þ H 1 , in which H e is the free electron Hamiltonian, H ph is the free phonon Hamiltonian and H 1 is the e–LO coupling operator. The details of the Hamiltonian are to be found e.g. in Ref. [12]. This Hamiltonian describes at the first sight a system with only a discrete structure of the density of states. Taking into account a multiple scattering of electrons on phonons, leading to the creation of virtual multiphonon states with a continuum of energy [13], we can expect the appearance of continua in the electronic spectral density function of energy parameter E. Upon using the Hamiltonian H and the multiple phonon scattering assumption it was earlier possible to provide an explanation of the fast electron energy relaxation in QDs [10,11] and similar effects (see Refs. [12,14]).
ARTICLE IN PRESS K. Kra´l / Microelectronics Journal 39 (2008) 375–377
376
The above given Hamiltonian of our zero-dimensional structure has a very singular density of states already in the non-interacting case. From this reason, we cannot achieve a single-pole approximation for the Green’s function and a Lorentzian spectral density curve in the case of an interacting electron, as it is illustrated below. In the so-called diagonal approximation with respect to the electronic orbital state index n the electronic spectral density of the nth state is obtained from the retarded Green’s function in the standard way. We shall use the selfconsistent Born (SCB) approximation for the self-energy [15]. The optical spectra will be identified in this paper with the electronic spectral density An ðEÞ. This spectral density depends on the distribution of the electronic occupation N n , N 0 þ N 1 ¼ N tot among the two orbital states n ¼ 0; 1. Our present approximation, with the instant collision approximation and with the Kadanoff–Baym ansatz, have been recently shown to display a property of an overheating of the LO phonon bath. The details of this effect can be found in Ref. [16] and will not be repeated. This property of the kinetic equation for the LO phonon generation can be linked to the influence of the transverse terms (the interaction terms corresponding to the processes of emission or absorption of phonon without changing the orbital state of the electron) in the kinetic equations. The effect can be to a large extent suppressed in a simple way by eliminating the transverse terms from the Hamiltonian, based on the Lang–Firsov canonical transformation [16]. In the clearly specified cases we shall use the Hamiltonian with the transverse terms eliminated and speak about the Lang–Firsov (LF) approximation. Unless otherwise stated the numerical damping (an artificial increase of the imaginary part of self-energy) is chosen to be 0.01 MeV. In Fig. 1, we show the electronic
spectral density of the state n ¼ 0 in the CdSe QD when the single electron completely occupies the state n ¼ 0 and also in cases when the electron partially occupies the state n ¼ 1. The curve for the for N 1 ¼ 0 corresponds quite well to the experimental observations presented in Refs. [4–7]. This curve has the shape of a narrow line with a shoulder at the low-energy side. Let us note that the line profile is rather sensitive to the deviation from the case of N 1 ¼ 0. Fig. 2 shows a similar sensitivity of the spectral density to the change of the lattice temperature. We also see in Fig. 2 that in the LF approximation, in which the artificial effect of the LO phonon system overheating is partly suppressed, the spectral density curve is not so much sensitive to the temperature changes. In Fig. 3 the change of the position of the shoulder of the line profile, from the low-energy to the high-energy side, is shown when the electron approaches the full occupation of the upper-energy level. To the best knowledge of the author the line profile with a high-energy side shoulder has not been measured so far. In Fig. 4 the spectral density of the low- and high-energy states are shown for the material of CdSe. The numerical damping is chosen here to be 3 MeV for the purpose of easier comparison with the experiment [4]. The polaron constant of CdSe is large enough (about 0.47) for the phonon satellites of the lower-energy state to be observable both in theory and experiment. In conclusion, we have presented several calculated properties of the line profiles of the optical emission electronic transitions in QDs. The agreement of the line profile having the shoulder at the low-energy side with experiments provides a support for the opinion that the
Spectral density (1/meV)
Spectral density (1/meV)
3
2
3
2
1
1 0 0 0 15
16
17
18
19
20
21
Energy (meV) Fig. 1. The zero-phonon line of the electronic spectral density of the state n ¼ 0 calculated for the cubic quantum dot of CdSe with the lateral size of 5 nm at the lattice temperature of 10 K, non-LF approximation. The total electronic occupation of the dot N tot ¼ 1. Occupation of the state N 1 is 0 (full), 0.01 (thin) and 0.001 (dashed).
5
10 Energy (meV)
15
20
Fig. 2. The zero-phonon line of the spectral density of CdSe 5 nm large QD for the state n ¼ 0 with N tot ¼ 1 and N 1 ¼ 0. The full and the dotted curves are calculated with the standard model, while the dashed curve is calculated in the LF approximation (with the overheating partially eliminated, see text). Lattice temperatures: 10 (full), 50 K (dotted). The dashed curve is calculated in LF approximation for the lattice temperature of 50 K. The latter curve differs negligibly from the case of LF approximation at 10 K.
ARTICLE IN PRESS K. Kra´l / Microelectronics Journal 39 (2008) 375–377
multiple electronic scattering in QDs plays an important role.
Spectral density (1/meV)
6
Acknowledgement The authors acknowledge the support from the project OC 090 of MSˇMT and AVOZ10100520. Discussions with I. Kratochvı´ lova´ are acknowledged.
4
References
2
[1] [2] [3] [4]
0 -2
-1
0
1
2 [5]
Energy (meV) Fig. 3. The zero-phonon line of the spectral density A0 ðEÞ in GaAs QD with the lateral size of 29 nm at 10 K, N tot ¼ 1, non-LF approximation. The occupation of the low-energy level (n ¼ 0) is the following: 1 (full line), 0.75 (dashed), 0.5 (dash-dotted) and 0 (dotted).
[6] [7] [8]
[9] [10] [11] [12] [13]
0.06 Spectral density (1/meV)
377
0.04 [14] [15] [16]
0.02
0.00 0
200
400
Energy (meV) Fig. 4. Spectral density A0 ðEÞ in CdSe quantum dot with the lateral size of 5 nm at 10 K, N tot ¼ N 0 ¼ 1. The numerical damping is chosen to be 3 meV. The full and dashed lines denote the spectral density of the state n ¼ 0 and n ¼ 1, respectively.
A.J. Nozik, Ann. Rev. Phys. Chem. 52 (2001) 193. U. Bockelman, G. Bastard, Phys. Rev. B 42 (1990) 8947. T. Inoshita, H. Sakaki, Phys. Rev. B 46 (1992) 7260. S.A. Empedocles, D.J. Norris, M.G. Bawendi, Phys. Rev. Lett. 77 (1996) 3873. E. Pelucchi, M. Baier, Y. Ducommun, S. Watanabe, E. Kapon, Phys. Status Solidi B 238 (2003) 233. M. Vachon, Private communication. I. Sychugov, R. Juhasz, J. Valenta, J. Linnros, Phys. Rev. Lett. 94 (2005) 087405. F. Rol, B. Gayral, S. Founta, B. Daudin, J. Eymery, J.-M. Ge´rard, H. Mariette, Le Si Dang, D. Peyrade, Phys. Statics. Solids (b (2006) 1652. M.I. Vasilevskiy, E.V. Anda, S.S. Makler, Phys. Rev. B 70 (2004) 035318. K. Kra´l, Z. Kha´s, Phys. Status. Solidi B 208 (1998) R5. H. Tsuchiya, T. Miyoshi, J. Appl. Phys. 83 (1998) 2574. K. Kra´l, P. Zdeneˇk, Z. Kha´s, Surf. Sci. 566–568 (2004) 321. N. Terzi, in: B. DiBartolo (Ed.), Collective Excitations in Solids, NATO ASI Series, Series B, vol. 88, Plenum Press, New York, 1983, pp. 149-182. E.U. Rafailov, Appl. Phys. Lett. 88 (2006) 041101. K. Kra´l, Z. Kha´s, Phys. Rev. B 57 (1998) R2061. K. Kra´l, Czechoslovak J. Phys. 56 (2006) 33.
Microelectronics Journal 39 (2008) 375–377 www.elsevier.com/locate/mejo
Non-delta-function electronic spectral densities in individual quantum dots Karel Kra´l Institute of Physics of the ASCR, v.v.i., Na Slovance 2, 18221 Praha 8, Czech Republic Available online 16 August 2007
Abstract Using a simplified model approach we estimate the optical line shape of the transition lines observable in photoluminescence experiments on quantum dots. We use the theory based on the interaction of electrons with the longitudinal optical phonons only. This theory gives, in the self-consistent Born approximation, the lowest-energy excited state line shape in the form of a very narrow peak with a shoulder on the low-energy side. We turn the attention to a comparison with experiments which appear to support this theoretical conclusion. This agreement emphasizes the role of the electronic multiple scattering on optical phonons in quantum dots. It is demonstrated that the optical line shape can give an information about the quantum dot system. r 2007 Elsevier Ltd. All rights reserved. PACS: 73.63.Kv; 73.21.La; 78.67.Hc Keywords: Quantum dots; Electron–phonon coupling; Luminescence
The quantum dots (QD) [1] are important for applications of nanostructures in optoelectronics. We could simply expect the optical lines of the luminescent transitions to be very narrow Lorentzians [2,3]. The experimental observations made with a sufficient spectral resolution tell us, however, that the actual line profiles are not always narrow and symmetrical [4–8]. Similar conclusions are found in the recent numerical experiments [9]. We shall pay attention to this question. The longitudinal optical (LO) phonons are important for the electronic relaxation in semiconductor QDs [10–12]. Using numerical calculations we show how the electron–LO (e–LO) phonon coupling influences the optical line shape. We shall confine our attention to the case of a single electron–hole pair in a single QD. The mutual annihilation of the particle pair gives the photon emitted in the photoluminescence. We shall neglect the effect of the motion of the hole particle regarding it as a static charge only. We shall limit the theory to two electronic bound states inside a single dot. We shall confine ourselves to the electronic intra-dot coupling to the dispersionless LO phonons only. E-mail address: [email protected] 0026-2692/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.mejo.2007.07.015
The electronic states in a QD will be approximated by electronic eigenstates in a cubic shaped QD, with infinitely deep potential. The electron ground state is the state with index n ¼ 0. The excited state (n ¼ 1) is one of the lowestenergy triple-degenerate exited states in this QD potential. The bath of LO phonons will be approximated by the bulk modes of dispersionless LO phonons of the full sample. Neglecting the electronic spin the Hamiltonian of the whole system is: H ¼ H e þ H ph þ H 1 , in which H e is the free electron Hamiltonian, H ph is the free phonon Hamiltonian and H 1 is the e–LO coupling operator. The details of the Hamiltonian are to be found e.g. in Ref. [12]. This Hamiltonian describes at the first sight a system with only a discrete structure of the density of states. Taking into account a multiple scattering of electrons on phonons, leading to the creation of virtual multiphonon states with a continuum of energy [13], we can expect the appearance of continua in the electronic spectral density function of energy parameter E. Upon using the Hamiltonian H and the multiple phonon scattering assumption it was earlier possible to provide an explanation of the fast electron energy relaxation in QDs [10,11] and similar effects (see Refs. [12,14]).
ARTICLE IN PRESS K. Kra´l / Microelectronics Journal 39 (2008) 375–377
376
The above given Hamiltonian of our zero-dimensional structure has a very singular density of states already in the non-interacting case. From this reason, we cannot achieve a single-pole approximation for the Green’s function and a Lorentzian spectral density curve in the case of an interacting electron, as it is illustrated below. In the so-called diagonal approximation with respect to the electronic orbital state index n the electronic spectral density of the nth state is obtained from the retarded Green’s function in the standard way. We shall use the selfconsistent Born (SCB) approximation for the self-energy [15]. The optical spectra will be identified in this paper with the electronic spectral density An ðEÞ. This spectral density depends on the distribution of the electronic occupation N n , N 0 þ N 1 ¼ N tot among the two orbital states n ¼ 0; 1. Our present approximation, with the instant collision approximation and with the Kadanoff–Baym ansatz, have been recently shown to display a property of an overheating of the LO phonon bath. The details of this effect can be found in Ref. [16] and will not be repeated. This property of the kinetic equation for the LO phonon generation can be linked to the influence of the transverse terms (the interaction terms corresponding to the processes of emission or absorption of phonon without changing the orbital state of the electron) in the kinetic equations. The effect can be to a large extent suppressed in a simple way by eliminating the transverse terms from the Hamiltonian, based on the Lang–Firsov canonical transformation [16]. In the clearly specified cases we shall use the Hamiltonian with the transverse terms eliminated and speak about the Lang–Firsov (LF) approximation. Unless otherwise stated the numerical damping (an artificial increase of the imaginary part of self-energy) is chosen to be 0.01 MeV. In Fig. 1, we show the electronic
spectral density of the state n ¼ 0 in the CdSe QD when the single electron completely occupies the state n ¼ 0 and also in cases when the electron partially occupies the state n ¼ 1. The curve for the for N 1 ¼ 0 corresponds quite well to the experimental observations presented in Refs. [4–7]. This curve has the shape of a narrow line with a shoulder at the low-energy side. Let us note that the line profile is rather sensitive to the deviation from the case of N 1 ¼ 0. Fig. 2 shows a similar sensitivity of the spectral density to the change of the lattice temperature. We also see in Fig. 2 that in the LF approximation, in which the artificial effect of the LO phonon system overheating is partly suppressed, the spectral density curve is not so much sensitive to the temperature changes. In Fig. 3 the change of the position of the shoulder of the line profile, from the low-energy to the high-energy side, is shown when the electron approaches the full occupation of the upper-energy level. To the best knowledge of the author the line profile with a high-energy side shoulder has not been measured so far. In Fig. 4 the spectral density of the low- and high-energy states are shown for the material of CdSe. The numerical damping is chosen here to be 3 MeV for the purpose of easier comparison with the experiment [4]. The polaron constant of CdSe is large enough (about 0.47) for the phonon satellites of the lower-energy state to be observable both in theory and experiment. In conclusion, we have presented several calculated properties of the line profiles of the optical emission electronic transitions in QDs. The agreement of the line profile having the shoulder at the low-energy side with experiments provides a support for the opinion that the
Spectral density (1/meV)
Spectral density (1/meV)
3
2
3
2
1
1 0 0 0 15
16
17
18
19
20
21
Energy (meV) Fig. 1. The zero-phonon line of the electronic spectral density of the state n ¼ 0 calculated for the cubic quantum dot of CdSe with the lateral size of 5 nm at the lattice temperature of 10 K, non-LF approximation. The total electronic occupation of the dot N tot ¼ 1. Occupation of the state N 1 is 0 (full), 0.01 (thin) and 0.001 (dashed).
5
10 Energy (meV)
15
20
Fig. 2. The zero-phonon line of the spectral density of CdSe 5 nm large QD for the state n ¼ 0 with N tot ¼ 1 and N 1 ¼ 0. The full and the dotted curves are calculated with the standard model, while the dashed curve is calculated in the LF approximation (with the overheating partially eliminated, see text). Lattice temperatures: 10 (full), 50 K (dotted). The dashed curve is calculated in LF approximation for the lattice temperature of 50 K. The latter curve differs negligibly from the case of LF approximation at 10 K.
ARTICLE IN PRESS K. Kra´l / Microelectronics Journal 39 (2008) 375–377
multiple electronic scattering in QDs plays an important role.
Spectral density (1/meV)
6
Acknowledgement The authors acknowledge the support from the project OC 090 of MSˇMT and AVOZ10100520. Discussions with I. Kratochvı´ lova´ are acknowledged.
4
References
2
[1] [2] [3] [4]
0 -2
-1
0
1
2 [5]
Energy (meV) Fig. 3. The zero-phonon line of the spectral density A0 ðEÞ in GaAs QD with the lateral size of 29 nm at 10 K, N tot ¼ 1, non-LF approximation. The occupation of the low-energy level (n ¼ 0) is the following: 1 (full line), 0.75 (dashed), 0.5 (dash-dotted) and 0 (dotted).
[6] [7] [8]
[9] [10] [11] [12] [13]
0.06 Spectral density (1/meV)
377
0.04 [14] [15] [16]
0.02
0.00 0
200
400
Energy (meV) Fig. 4. Spectral density A0 ðEÞ in CdSe quantum dot with the lateral size of 5 nm at 10 K, N tot ¼ N 0 ¼ 1. The numerical damping is chosen to be 3 meV. The full and dashed lines denote the spectral density of the state n ¼ 0 and n ¼ 1, respectively.
A.J. Nozik, Ann. Rev. Phys. Chem. 52 (2001) 193. U. Bockelman, G. Bastard, Phys. Rev. B 42 (1990) 8947. T. Inoshita, H. Sakaki, Phys. Rev. B 46 (1992) 7260. S.A. Empedocles, D.J. Norris, M.G. Bawendi, Phys. Rev. Lett. 77 (1996) 3873. E. Pelucchi, M. Baier, Y. Ducommun, S. Watanabe, E. Kapon, Phys. Status Solidi B 238 (2003) 233. M. Vachon, Private communication. I. Sychugov, R. Juhasz, J. Valenta, J. Linnros, Phys. Rev. Lett. 94 (2005) 087405. F. Rol, B. Gayral, S. Founta, B. Daudin, J. Eymery, J.-M. Ge´rard, H. Mariette, Le Si Dang, D. Peyrade, Phys. Statics. Solids (b (2006) 1652. M.I. Vasilevskiy, E.V. Anda, S.S. Makler, Phys. Rev. B 70 (2004) 035318. K. Kra´l, Z. Kha´s, Phys. Status. Solidi B 208 (1998) R5. H. Tsuchiya, T. Miyoshi, J. Appl. Phys. 83 (1998) 2574. K. Kra´l, P. Zdeneˇk, Z. Kha´s, Surf. Sci. 566–568 (2004) 321. N. Terzi, in: B. DiBartolo (Ed.), Collective Excitations in Solids, NATO ASI Series, Series B, vol. 88, Plenum Press, New York, 1983, pp. 149-182. E.U. Rafailov, Appl. Phys. Lett. 88 (2006) 041101. K. Kra´l, Z. Kha´s, Phys. Rev. B 57 (1998) R2061. K. Kra´l, Czechoslovak J. Phys. 56 (2006) 33.