- Email: [email protected]
Electronic spectral densities in quantum dots
Materials Science and Engineering B 147 (2008) 267–270
Electronic spectral densities in quantum dots Karel Kr´al ∗ Institute of Physics of ASCR, v.v.i., Na Slovance 2, 18221 Prague 8, Czech Republic Received 12 June 2007; accepted 20 July 2007
Abstract The spectral line profiles of the luminescence transitions in individual quantum dots have been studied both theoretically and experimentally in the recent years. In these works the spectral line profiles have been shown to be clearly different from the simple form of a broadened delta function of frequency variable. In the present work we turn attention to these earlier measurements and calculations and compare them with the electronic spectral line profiles in individual quantum dots calculated under the assumption of quantum dot electrons multiple scattering on the longitudinal optical phonons. The calculated luminescence spectral line profiles of the optical transition will be expressed with help of the electronic spectral densities dependent on the electronic statistical distribution, as it develops in the course of the relaxation processes or other kinetic processes in quantum dots. © 2007 Elsevier B.V. All rights reserved. Keywords: Quantum dots; Electron–phonon interaction; Luminescence; Relaxation phenomena; Quantum structures
1. Introduction The kinetics of the charge carriers in the bulk semiconductors of the III–V types were earlier found to be strongly influenced by the interaction of electrons and holes with the longitudinal optical (LO) phonons [1]. In the zero-dimensional nanostructures like the compound semiconductor quantum dots (QD) it is thus quite likely that the electron kinetics, together with the electron dynamics, will be at least strongly influenced by the same mechanism. In accordance with this expectation the electron-optical-phonon (e-LO) coupling was shown to provide an efficient mechanism of fast electron energy relaxation [2,3]. This kinetic theory was based on the nonequilibrium Green’s functions with the applied assumption of the instant collisions and Kadanoff–Baym ansatz [3,2]. The same theoretical approach gave us the electronic upconversion effect [4], which provides the explanation of why the quantum dot lasers may emit light from higher excited states [5]. Already these two pieces of agreement between theory and experiment support the view that the electron–phonon kinetics in quantum dots, based on the multiple scattering of electrons on LO phonons, at least contribute to other possible mechanisms of electronic energy relaxation [6].
∗
Corresponding author. E-mail address: [email protected].
0921-5107/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.mseb.2007.07.069
This theory of the electron-LO phonon motion in quantum dots displays also an unpleasant property of an optical phonon system overheating [7]. This effect means that at any state of the distribution functions the optical phonons are generated and do not thus play the role of a passive bath. This effect was ascribed in Ref. [7] to a shortcomming of the phonon kinetic equation, namely to the absence of such terms in it, which would correspond to a buildup of a lattice deformation after the electronic subsystem changes the occupation of the electronic states. An analysis based on the well-known Lang–Firsov canonical transformation leads to finding a simple tool with which the effect of the overheating is suppressed in magnitude by a factor of about five. This tool is simply the neglection of the so called transverse terms in the electron-LO phonon interaction operator [8]. The theory with the overheating suppressed will thus be called Lang-Firsov approximation, while the original version of the electron–phonon (LF) theory in quantum dot, with the overheating uncompensated, is called a non-LF theory. In the present work we turn the attention to the remarkable property of the luminescence optical transition peaks in individual quantum dots. We shall first remind the significant experimental data on this topic and then we present numerical results of calculations based on an approximative calculation of the electronic spectral density based on the interaction of electrons with LO phonons in quantum dots. The numerical results will show the dependence of the lineshape on the statistical state of the electronic system, on the size of the quantum dot, and also
268
K. Kr´al / Materials Science and Engineering B 147 (2008) 267–270
on the version of the electron–phonon interaction theory used, whether it is that one with the overheating uncompensated or the one with the overheating partially suppressed. 2. The present state of experimental findings One of the first measurements of the luminescent optical line shape measured on the individual nanocrystals or quantum dots is the work of Empedocles et al. [9]. In this work the line shape was shown to deviate from a delta function. The line shape was observed to be non-symmetrical, with a rather well pronounced shoulder on the low-energy side of the peak. Later the measument of the individual dot line shape was performed on the individual quantum dots of the material Inx Ga1−x As/Al0.30 Ga0.70 As [10]. In this work the line shape was remarkably well determined displaying the very narrow peak, in the sense of the full width at the half maximum, accompanied by a well pronounced shoulder at the low-energy side. A very similar quality of the measured line shape was presented recently [11] on InAs/GaAs quantum dots. This work has not yet been published to the best knowledge of the author. The authors of the work [11] showed again a very narrow peak complemented by a well-developed shoulder at the low-energy side. At this moment it is suitable to remind the measurements of the luminescence line shape of the individual nanocrystals of Si [12]. This work shows again a relatively narrow peak with a shoulder. Both the peak and the shoulder come out from the experiment somewhat broader than the corresponding features in the binary material quantum dots mentioned above. Nevertheless, the qualitative features of the main peak of the lowest energy luminescent transition has similar properties to those reported above. We shall not introduce more experiments in this respect. Instead we turn the attention of the reader to a numerical experiment performed by Vasilevskiy et al. [13], who determined the optical line shape calculating the electronic spectral density from the definition, assuming the coupling of the electrons with optical and acoustic phonons. Although the numerical calculation was based on approximating the state of the phonon system by the Fock’s states with up to only several phonons, one can clearly see in the resulting data [13] that they remind strongly the form of a narrow maximum with a shoulder. We shall not make any detailed comparison with the paper [13] in the present work.
have only two non-degenerate bound states in the electronic QD. The QD will be assumed to be cubic shape with infinitly deep potential. We shall take into account only the electronic ground state (n = 0) and one of the triply degenerate excited states (n = 1). The electronic spin will be neglected. The total number of electrons Ntot , Ntot = N0 + N1 , in the dot will not be larger that one. Here Nn is the electronic occupation of the state n. The phonons will be approximated by the bulk modes of optical phonons of the whole sample. Restricting ourselves to the interaction of the electron with the LO phonons only we express the electronic spectral density in the self-consistent Born approximation to the electronic self-energy [2,8]. In the non-LF approximation mentioned above the electronic spectral density will becalculated with the full operator of the e-LO coupling, + H1 = q,m,n=0,1 Aq Φ(n, m, q)(bq − b−q )cn+ cm , with ci being annihilation operator of an electron in the state i and with bq being LO phonon annihilation operator in the state with phonon momentum q. Aq is Fr¨ohlich’s coupling constant and Φ is a form factor [2,8]. In the so called LF approximation, in which the phonon overheating is decreased by a factor of about 5, the transverse terms of H1 , which are those with n = m, are completely eliminated. The electronic spectral density is calculated from the retarded Green’s function with the electronic self-energy given by the self-consistent Born approximation (see e.g. [4]). 4. Numerical results The numerical calculations are performed for the material parameters of GaAs crystal. The lattice temperature is set to 10 K. In Fig. 1 we see the optical transition line shape which coincides with the shape observed in the experiments cited above. Let us note that the occupation of the electronic levels is integer. The shoulder is found at the low-energy side. In contrast to the latter Figure, the next Fig. 2 shows the shoulder at the highenergy side. To the best knowledge of the author this high-energy position of the shoulder has not yet been reported in experiment. Although the peaks are narrow in the non-LF approximation when the level occupation is an integer number, a small
3. Theory We assume the optical transition in quantum dots in which an electron in the conduction band states and a hole in the valence band states annihilate while photon is emitted. In this process we neglect the motion of holes and substitute their influence by a static charge added to the potential well of the electron [4]. Assuming that the spectral density of such a heavy hole can be expressed as a delta function, the spectral line shape of the luminescence transition can then be expressed in this approximation as the spectral density of the electron in the conduction band states of quantum dot. The electron will be assumed to
Fig. 1. The electronic spectral density main peak of the lower-energy state in QD with the lateral size of 17 nm. Calculated in the non-LF approximation (the overheating of the phonons is not suppressed). The electronic distribution is N0 = 1, N1 = 0 (integer occupation of the electronic states).
K. Kr´al / Materials Science and Engineering B 147 (2008) 267–270
Fig. 2. The main peak of the higher-energy state (n = 1) in the quantum dot with the lateral size of 17 nm, in the non-LF approximation. Electronic level occupation is: N0 = 0 and N1 = 1.
deviation from this integer occupation causes a considerable broadening of the spectral density calculated. This is shown in the Fig. 3. This broad shape of the main peak, together with its obvious deviation from a shape of a narrow peak with a low-energy shoulder, remains to be compared with future experimental data obtained desirably with a simultaneous knowledge of the electronic distribution. The situation becomes different when the electronic line shape is calculated in the approximation in which the overheating of phonons is suppressed (LF approximation). This is shown in Fig. 4. In this figure the main peak remains very narrow at all values of the quantum dot lateral size. We can clearly see that the position of the shoulder can move to the high-energy side for a certain range of the quantum dot size. More experimental material is needed for to compare with the theory at a given size of the dot and at a given level occupation. The experiment also could help to decide which approach to the suppression of the overheating is suitable. The calculated spectral densities show that the mechanism of the multiple electronic scattering on the longitudinal optical phonons is able to give us the line shapes observable in the individual quantum dot experiments.
269
Fig. 4. The main peak of the lower-energy state spectral density in the LF approximation (with the phonon overheating effect reduced down to about 20%) as a function of the lateral size. Level occupations: N0 = 0.8, N1 = 0.2.
5. Conclusions The multiple electronic scattering on the dispersionless optical phonons is able to reproduce the main features of the optical line shapes observed on individual dots in experiments. The model used has many simplifications in comparison with the real situation in quantum dot samples. The influence of the over-barrier continua of states has not yet been included, and the influence of the acoustic phonons, although expected to be weak, remains neglected. The results presented suggest however that the theoretically assumed mechanism may play a significant role in both the electron phonon dynamics and in the electron and phonon kinetics in quantum dots. The multiple phonon scattering may in other words be spoken about as a mechanism beyond the adiabatic approximation to the motion of electrons and phonons. Besides this, the mechanism of the multiple scattering of electrons on LO phonons, giving the fast electronic energy relaxation and the electronic upconversion [4] is intrinsic and is effective even at a single electron in a single quantum dot in the quantum dot sample. Acknowledgements The author acknowledges the support form the projects ME ˇ 866 OS of MSMT and from the institutional project AVOZ 10100520. Discussions with I. Kratochv´ılov´a are acknowledged. References
Fig. 3. The lateral size dependence of the low-energy level main peak spectral density in the non-LF approximation. Level occupation is: N0 = 0.9, N1 = 0.1 (non-integer level occupation).
[1] B.R. Nag, Electron Transport in Compound Semiconductors, Springer Series in Solid-State Sciences 11, Springer-Verlag, Berlin, Heidelberg, New York, 1980. [2] K. Kr´al, Z. Kh´as, Phys. Status Solidus (B) 208 (1998) R5. [3] H. Tsuchiya, T. Miyoshi, J. Appl. Phys. 83 (1998) 2574. [4] K. Kr´al, P. Zdenˇek, Z. Kh´as, Surf. Sci. 566-568 (2004) 321. [5] E.U. Rafailov, A.D. McRobbie, M.A. Cataluna, L. O’Faolain, W. Sibbett, D.A. Livshits, Appl. Phys. Lett. 88 (2006) 041101. [6] V.I. Klimov, Los Alamos Sci. 28 (2003) 214. [7] K. Kr´al, Czech. J. Phys. 56 (2006) 33.
270
K. Kr´al / Materials Science and Engineering B 147 (2008) 267–270
[8] K. Kr´al, Z. Kh´as, Phys. Rev. B 57 (1998) R2061. [9] S.A. Empedocles, D.J. Norris, M.G. Bawendi, Phys. Rev. Lett. 77 (1996) 3873. [10] E. Pelucchi, M. Baier, Y. Ducommun, S. Watanabe, E. Kapon, Phys. Status Solidus (B) 238 (2003) 233.
[11] M. Vachon, S. Raymond, A. Babinski, J. Lapointe, Z. Wasilewski, M. Potemski, Presentation at ISCS, Vancouver, private communication (2006). [12] I. Sychugov, R. Juhasz, J. Valenta, J. Linnros, Phys. Rev. Lett. 94 (2005) 087405. [13] M.I. Vasilevskiy, E.V. Anda, S.S. Makler, Phys. Rev. B 70 (2004) 035318.
Electronic spectral densities in quantum dots Karel Kr´al ∗ Institute of Physics of ASCR, v.v.i., Na Slovance 2, 18221 Prague 8, Czech Republic Received 12 June 2007; accepted 20 July 2007
Abstract The spectral line profiles of the luminescence transitions in individual quantum dots have been studied both theoretically and experimentally in the recent years. In these works the spectral line profiles have been shown to be clearly different from the simple form of a broadened delta function of frequency variable. In the present work we turn attention to these earlier measurements and calculations and compare them with the electronic spectral line profiles in individual quantum dots calculated under the assumption of quantum dot electrons multiple scattering on the longitudinal optical phonons. The calculated luminescence spectral line profiles of the optical transition will be expressed with help of the electronic spectral densities dependent on the electronic statistical distribution, as it develops in the course of the relaxation processes or other kinetic processes in quantum dots. © 2007 Elsevier B.V. All rights reserved. Keywords: Quantum dots; Electron–phonon interaction; Luminescence; Relaxation phenomena; Quantum structures
1. Introduction The kinetics of the charge carriers in the bulk semiconductors of the III–V types were earlier found to be strongly influenced by the interaction of electrons and holes with the longitudinal optical (LO) phonons [1]. In the zero-dimensional nanostructures like the compound semiconductor quantum dots (QD) it is thus quite likely that the electron kinetics, together with the electron dynamics, will be at least strongly influenced by the same mechanism. In accordance with this expectation the electron-optical-phonon (e-LO) coupling was shown to provide an efficient mechanism of fast electron energy relaxation [2,3]. This kinetic theory was based on the nonequilibrium Green’s functions with the applied assumption of the instant collisions and Kadanoff–Baym ansatz [3,2]. The same theoretical approach gave us the electronic upconversion effect [4], which provides the explanation of why the quantum dot lasers may emit light from higher excited states [5]. Already these two pieces of agreement between theory and experiment support the view that the electron–phonon kinetics in quantum dots, based on the multiple scattering of electrons on LO phonons, at least contribute to other possible mechanisms of electronic energy relaxation [6].
∗
Corresponding author. E-mail address: [email protected].
0921-5107/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.mseb.2007.07.069
This theory of the electron-LO phonon motion in quantum dots displays also an unpleasant property of an optical phonon system overheating [7]. This effect means that at any state of the distribution functions the optical phonons are generated and do not thus play the role of a passive bath. This effect was ascribed in Ref. [7] to a shortcomming of the phonon kinetic equation, namely to the absence of such terms in it, which would correspond to a buildup of a lattice deformation after the electronic subsystem changes the occupation of the electronic states. An analysis based on the well-known Lang–Firsov canonical transformation leads to finding a simple tool with which the effect of the overheating is suppressed in magnitude by a factor of about five. This tool is simply the neglection of the so called transverse terms in the electron-LO phonon interaction operator [8]. The theory with the overheating suppressed will thus be called Lang-Firsov approximation, while the original version of the electron–phonon (LF) theory in quantum dot, with the overheating uncompensated, is called a non-LF theory. In the present work we turn the attention to the remarkable property of the luminescence optical transition peaks in individual quantum dots. We shall first remind the significant experimental data on this topic and then we present numerical results of calculations based on an approximative calculation of the electronic spectral density based on the interaction of electrons with LO phonons in quantum dots. The numerical results will show the dependence of the lineshape on the statistical state of the electronic system, on the size of the quantum dot, and also
268
K. Kr´al / Materials Science and Engineering B 147 (2008) 267–270
on the version of the electron–phonon interaction theory used, whether it is that one with the overheating uncompensated or the one with the overheating partially suppressed. 2. The present state of experimental findings One of the first measurements of the luminescent optical line shape measured on the individual nanocrystals or quantum dots is the work of Empedocles et al. [9]. In this work the line shape was shown to deviate from a delta function. The line shape was observed to be non-symmetrical, with a rather well pronounced shoulder on the low-energy side of the peak. Later the measument of the individual dot line shape was performed on the individual quantum dots of the material Inx Ga1−x As/Al0.30 Ga0.70 As [10]. In this work the line shape was remarkably well determined displaying the very narrow peak, in the sense of the full width at the half maximum, accompanied by a well pronounced shoulder at the low-energy side. A very similar quality of the measured line shape was presented recently [11] on InAs/GaAs quantum dots. This work has not yet been published to the best knowledge of the author. The authors of the work [11] showed again a very narrow peak complemented by a well-developed shoulder at the low-energy side. At this moment it is suitable to remind the measurements of the luminescence line shape of the individual nanocrystals of Si [12]. This work shows again a relatively narrow peak with a shoulder. Both the peak and the shoulder come out from the experiment somewhat broader than the corresponding features in the binary material quantum dots mentioned above. Nevertheless, the qualitative features of the main peak of the lowest energy luminescent transition has similar properties to those reported above. We shall not introduce more experiments in this respect. Instead we turn the attention of the reader to a numerical experiment performed by Vasilevskiy et al. [13], who determined the optical line shape calculating the electronic spectral density from the definition, assuming the coupling of the electrons with optical and acoustic phonons. Although the numerical calculation was based on approximating the state of the phonon system by the Fock’s states with up to only several phonons, one can clearly see in the resulting data [13] that they remind strongly the form of a narrow maximum with a shoulder. We shall not make any detailed comparison with the paper [13] in the present work.
have only two non-degenerate bound states in the electronic QD. The QD will be assumed to be cubic shape with infinitly deep potential. We shall take into account only the electronic ground state (n = 0) and one of the triply degenerate excited states (n = 1). The electronic spin will be neglected. The total number of electrons Ntot , Ntot = N0 + N1 , in the dot will not be larger that one. Here Nn is the electronic occupation of the state n. The phonons will be approximated by the bulk modes of optical phonons of the whole sample. Restricting ourselves to the interaction of the electron with the LO phonons only we express the electronic spectral density in the self-consistent Born approximation to the electronic self-energy [2,8]. In the non-LF approximation mentioned above the electronic spectral density will becalculated with the full operator of the e-LO coupling, + H1 = q,m,n=0,1 Aq Φ(n, m, q)(bq − b−q )cn+ cm , with ci being annihilation operator of an electron in the state i and with bq being LO phonon annihilation operator in the state with phonon momentum q. Aq is Fr¨ohlich’s coupling constant and Φ is a form factor [2,8]. In the so called LF approximation, in which the phonon overheating is decreased by a factor of about 5, the transverse terms of H1 , which are those with n = m, are completely eliminated. The electronic spectral density is calculated from the retarded Green’s function with the electronic self-energy given by the self-consistent Born approximation (see e.g. [4]). 4. Numerical results The numerical calculations are performed for the material parameters of GaAs crystal. The lattice temperature is set to 10 K. In Fig. 1 we see the optical transition line shape which coincides with the shape observed in the experiments cited above. Let us note that the occupation of the electronic levels is integer. The shoulder is found at the low-energy side. In contrast to the latter Figure, the next Fig. 2 shows the shoulder at the highenergy side. To the best knowledge of the author this high-energy position of the shoulder has not yet been reported in experiment. Although the peaks are narrow in the non-LF approximation when the level occupation is an integer number, a small
3. Theory We assume the optical transition in quantum dots in which an electron in the conduction band states and a hole in the valence band states annihilate while photon is emitted. In this process we neglect the motion of holes and substitute their influence by a static charge added to the potential well of the electron [4]. Assuming that the spectral density of such a heavy hole can be expressed as a delta function, the spectral line shape of the luminescence transition can then be expressed in this approximation as the spectral density of the electron in the conduction band states of quantum dot. The electron will be assumed to
Fig. 1. The electronic spectral density main peak of the lower-energy state in QD with the lateral size of 17 nm. Calculated in the non-LF approximation (the overheating of the phonons is not suppressed). The electronic distribution is N0 = 1, N1 = 0 (integer occupation of the electronic states).
K. Kr´al / Materials Science and Engineering B 147 (2008) 267–270
Fig. 2. The main peak of the higher-energy state (n = 1) in the quantum dot with the lateral size of 17 nm, in the non-LF approximation. Electronic level occupation is: N0 = 0 and N1 = 1.
deviation from this integer occupation causes a considerable broadening of the spectral density calculated. This is shown in the Fig. 3. This broad shape of the main peak, together with its obvious deviation from a shape of a narrow peak with a low-energy shoulder, remains to be compared with future experimental data obtained desirably with a simultaneous knowledge of the electronic distribution. The situation becomes different when the electronic line shape is calculated in the approximation in which the overheating of phonons is suppressed (LF approximation). This is shown in Fig. 4. In this figure the main peak remains very narrow at all values of the quantum dot lateral size. We can clearly see that the position of the shoulder can move to the high-energy side for a certain range of the quantum dot size. More experimental material is needed for to compare with the theory at a given size of the dot and at a given level occupation. The experiment also could help to decide which approach to the suppression of the overheating is suitable. The calculated spectral densities show that the mechanism of the multiple electronic scattering on the longitudinal optical phonons is able to give us the line shapes observable in the individual quantum dot experiments.
269
Fig. 4. The main peak of the lower-energy state spectral density in the LF approximation (with the phonon overheating effect reduced down to about 20%) as a function of the lateral size. Level occupations: N0 = 0.8, N1 = 0.2.
5. Conclusions The multiple electronic scattering on the dispersionless optical phonons is able to reproduce the main features of the optical line shapes observed on individual dots in experiments. The model used has many simplifications in comparison with the real situation in quantum dot samples. The influence of the over-barrier continua of states has not yet been included, and the influence of the acoustic phonons, although expected to be weak, remains neglected. The results presented suggest however that the theoretically assumed mechanism may play a significant role in both the electron phonon dynamics and in the electron and phonon kinetics in quantum dots. The multiple phonon scattering may in other words be spoken about as a mechanism beyond the adiabatic approximation to the motion of electrons and phonons. Besides this, the mechanism of the multiple scattering of electrons on LO phonons, giving the fast electronic energy relaxation and the electronic upconversion [4] is intrinsic and is effective even at a single electron in a single quantum dot in the quantum dot sample. Acknowledgements The author acknowledges the support form the projects ME ˇ 866 OS of MSMT and from the institutional project AVOZ 10100520. Discussions with I. Kratochv´ılov´a are acknowledged. References
Fig. 3. The lateral size dependence of the low-energy level main peak spectral density in the non-LF approximation. Level occupation is: N0 = 0.9, N1 = 0.1 (non-integer level occupation).
[1] B.R. Nag, Electron Transport in Compound Semiconductors, Springer Series in Solid-State Sciences 11, Springer-Verlag, Berlin, Heidelberg, New York, 1980. [2] K. Kr´al, Z. Kh´as, Phys. Status Solidus (B) 208 (1998) R5. [3] H. Tsuchiya, T. Miyoshi, J. Appl. Phys. 83 (1998) 2574. [4] K. Kr´al, P. Zdenˇek, Z. Kh´as, Surf. Sci. 566-568 (2004) 321. [5] E.U. Rafailov, A.D. McRobbie, M.A. Cataluna, L. O’Faolain, W. Sibbett, D.A. Livshits, Appl. Phys. Lett. 88 (2006) 041101. [6] V.I. Klimov, Los Alamos Sci. 28 (2003) 214. [7] K. Kr´al, Czech. J. Phys. 56 (2006) 33.
270
K. Kr´al / Materials Science and Engineering B 147 (2008) 267–270
[8] K. Kr´al, Z. Kh´as, Phys. Rev. B 57 (1998) R2061. [9] S.A. Empedocles, D.J. Norris, M.G. Bawendi, Phys. Rev. Lett. 77 (1996) 3873. [10] E. Pelucchi, M. Baier, Y. Ducommun, S. Watanabe, E. Kapon, Phys. Status Solidus (B) 238 (2003) 233.
[11] M. Vachon, S. Raymond, A. Babinski, J. Lapointe, Z. Wasilewski, M. Potemski, Presentation at ISCS, Vancouver, private communication (2006). [12] I. Sychugov, R. Juhasz, J. Valenta, J. Linnros, Phys. Rev. Lett. 94 (2005) 087405. [13] M.I. Vasilevskiy, E.V. Anda, S.S. Makler, Phys. Rev. B 70 (2004) 035318.