- Email: [email protected]
GaAs quantum dots
ARTICLE IN PRESS
Physica E 26 (2005) 267–270 www.elsevier.com/locate/physe
Microscopic approach to many-exciton complexes in self-assembled InGaAs/GaAs quantum dots Weidong Sheng, Marek Korkusinski, Pawel Hawrylak Institute for Microstructural Sciences, National Research Council of Canada, 1200 Montreal Road, Ottawa, Ontario, Canada, K1A 0R6 Available online 23 November 2004
Abstract We present a numerical calculation of many-exciton complexes in self-assembled InAs/GaAs quantum dots. We apply continuum elasticity theory and atomistic valence-force-field method to calculate strain distribution, and make use of various methods, ranging from a quasi-atomistic tight-binding approach to the single-band effective-mass approximation, to obtain single-particle energy levels. The effect of strain is incorporated by the deformation potential theory. We expand multiexciton states in the basis of Slater determinants and solve the many-body problem by the configuration-interaction method. The dynamics of multiexcitons is studied by solving the rate equations, from which the excitation–power dependence of emission spectrum is obtained. The emission spectra calculated by the microscopic tight-binding approach are found to be in good agreement with those obtained by the simple effective-mass method. r 2004 Elsevier B.V. All rights reserved. PACS: 71.35.Cc; 73.21.La; 78.67.Hc Keywords: Excitons Quantum dots; Configuration-interaction method
1. Introduction There has been a great interest in semiconductor self-assemble quantum dots for many years because of their potential applications in nanoelectronics and quantum computing [1]. These semiconductor nanostructures form due to the strain that arises from the lattice mismatch Corresponding author. Tel.: +1-613-993-4428; fax: +1-
613-990-0202. E-mail address: [email protected] (W. Sheng).
between two different kinds of materials. Depending on the types of the island material and the host matrix as well as growth conditions, the size, shape, and chemical composition profile of the dots may vary in a large range [2]. Due to many uncertainties in the growth process, an accurate simulation to predict structural properties of quantum dots would be very difficult. Hence, for a numerical study, one usually begins with a model system whose structural parameters are assumed from experimental information. In this study, we choose our model structure as a disklike
1386-9477/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2004.08.071
ARTICLE IN PRESS W. Sheng et al. / Physica E 26 (2005) 267–270
268
intermixing In0.5Ga0.5As dot whose base diameter is 25.4 nm and height is 2.3 nm [3]. For the electronic structure calculation, we will apply several methods, ranging from the quasiatomistic tight-binding-like effective bond-orbital method [4], eight-band k p [5], to the single-band effective-mass [6]. After obtaining single-particle states, we calculate Coulomb matrix elements and solve multiexciton states by using the configuration-interaction method [7]. Finally, we will compare the multiexciton emission spectra obtained by using different methods.
2. Formalism To calculate the strain distribution for a dot of given size and shape, we can apply either the continuum elasticity (CE) theory or the atomistic valence-force field method [8]. The strain energy ECE is given by V C 11 2xx þ 2yy þ 2zz E CE ¼ 2 V þ C 44 2xy þ 2yz þ 2zx 2 þ VC 12 xx yy þ yy zz þ zz xx : In both methods, strain tensor is calculated by minimizing the strain energy by moving each unit cell or atom according to the applied force. In this work, we use the CE theory to calculate the strain. For the electronic calculation, we start with a simple single-band effective-mass approximation. The Hamiltonians, including the strain effect, are given by 2 _2 q q2 _2 q2 He ¼ þ 2mexy qx2 qy2 2mez qz2 þ aC H S þ V ebo ; 2 _2 q q2 _2 q2 Hh ¼ þ 2mhxy qx2 qy2 2mhz qz2
the band dispersion (effective mass). For simplicity, only the hydrostatic (HS) and biaxial (BS) strain components are taken into account. aC, aV, and bV are the deformation potential parameters [8]. Vbo’s are the band offsets between the island and matrix material, which are chosen as 630 meV and 42 meV between conduction and valence bands, respectively. In the eight-band k p formalism, the mixing between the conduction and valence bands, and between valence bands are taken into account. Spin-orbital interaction is also included. For smaller quantum dots, a method that can describe the structure at atomistic level is required. In this work, we adopt the improved effective bondorbital method (EBOM) [9]. EBOM is a empirical sp3 tight-binding method based on the effective FCC lattice. The improved version includes the interactions up to the second-nearest neighbor and therefore has more fitting parameters to fit the band edges at both the G and X points. When the Coulomb matrix elements are calculated from the single-particle states, the Hamiltonian for many-exciton complex can be defined by X X X þ þ H^ ¼ E ei cþ E hi hþ V he i hi i ci ijkl hi cj hk cl i
i
ijkl
1 X ee þ þ 1 X hh þ þ þ V ijkl ci cj ck cl þ V h h hk hl : 2 ijkl 2 ijkl ijkl i j To calculate multiexciton states, we use the configuration-interaction method [10] to solve the many-body Hamiltonian. The emission spectrum is calculated by I sN ð_oÞ
X eE iN =kT ¼ P E j =kT N i je + 2 * X X f pnm hn cm C iN C N1 nm f dðE iN E fN1 _oÞ
aV H S bV BS þ V ebo : Here the effective mass along the growth direction (z) and in-plane (xy) are treated as fitting parameters because the strain would not only modify the band edges but also the curvature of
3. Results Fig. 1 shows the calculated energy levels of the intermixing disklike dot. The result by the
ARTICLE IN PRESS W. Sheng et al. / Physica E 26 (2005) 267–270
269
the calculated spectra cover emissions over different shells, several thousands of multiexciton eigenstates need to be calculated. Fig. 2 shows the calculated emission spectra for up to six excitons by both the microscopic tightbinding approach and the effective-mass approximations. It is seen that except for the emission
1.48 1.46 1.44 1.42 1.40 1.38 Energy (eV)
EMA 6→5 5→4
0.10
4→3 3→2
0.09
2→1 1→0 1.28
0.08
1.29
1.3
1.31
1.32
1.33
K.P 6→5
0.07 EMA
K.P
EBOM
single-band effective-mass method is fitted to that by the eight-band k p method. An overall shift is seen between the results by the eight-band k p method and EBOM. Compared with that obtained by the eight-band k p method, the effective band gap is smaller by about 25 meV by the EBOM calculation. The energy separation in conduction bands is also found smaller. Based on these single-particle states, we construct multiexciton configurations by using the Slater determinants and solve the many-body problem by using the configuration-interaction approach. As the total number of configurations grows exponentially with the number of excitons, we need to apply a truncation to limit the total number of configuration under 100,000. In order to calculate the emission spectrum, one needs to obtain both the energy levels and wave functions of multiexciton states. To make sure that
Intensity
5→4
Fig. 1. Energy levels of the intermixing disklike dot calculated by different methods.
4→3 3→2 2→1 1→0
1.28
1.29
1.3
1.31
1.32
1.33
EBOM 6→5 5→4 4→3 3→2 2→1 1→0 1.26
1.27
1.28
1.29 Energy (eV)
1.3
1.31
Fig. 2. Multiexciton emission spectra calculated for up to six excitons by different methods.
ARTICLE IN PRESS W. Sheng et al. / Physica E 26 (2005) 267–270
270
energies, all the three methods show qualitatively similar structures in the emission spectra. For a specific major emission peak, e.g., the biexciton peak, different methods are found to give the different relative intensity. This is due to the quantum interference effect that arises from the coherent summation of transition probability between each configuration in the initial multiexciton state and that in the final state. A detail assignment for major emission peaks is found in good agreement with the analysis made by a simple model [11]. In experiments, one usually observes a spectrum that combines those from different many-exciton complexes and also strongly depends on the external excitation power. In order to obtain the dependence of emission spectrum on the excitation power, we first calculate the radiative lifetimes for each many-exciton complex. Then we compute the corresponding probabilities by solving the rate equations [12]. The combined emission spectra are shown in Fig. 3. It is seen, at low excitation power, only emission from a single exciton and a little from biexciton. As excitation intensity increases, the biexciton emission becomes stronger, at the same time, P = 8.0
Intensity
P = 4.0
P = 1.0
1.26
1.27
1.28 Energy (eV)
1.29
1.3
Fig. 3. Emission spectra calculated at various excitation power by the EBOM.
emission from three-exciton complex appears in the second shell. When excitation power increases further, stronger emission peaks can be found in the second shell while there appear more small peaks in the first shell. In conclusion, we have presented a microscopic study of many-exciton complexes in self-assembled quantum dots. We have found that the quasiatomistic tight-binding approach gives qualitatively similar emission spectra as those obtained by the simple effective-mass calculation.
Acknowledgments This work was supported by NRC High Performance Computing project and Canadian Institute for Advanced Research (CIAR). References [1] L. Jacak, P. Hawrylak, A. Wojs, Quantum Dots, Springer, Berlin, 1998. [2] D. Bimberg, M. Grundmann, N.N. Ledentsov, Quantum Dot Heterostructures, Wiley, Chichester, 1998. [3] S. Raymond, et al., Phys. Rev. Lett. 92 (2004) 187402. [4] Y.-C. Chang, Phys. Rev. B 37 (1988) 8215; S.V. Nair, L.M. Ramaniah, K.C. Rustagi, Phys. Rev. B 45 (1992) 5969; S.J. Sun, Y.-C. Chang, Phys. Rev. B 62 (2000) 13631. [5] T.B. Bahder, Phys. Rev. B 41 (1990) 11992; H. Jiang, J. Singh, Phys. Rev. B 56 (1997) 4696; C. Pryor, Phys. Rev. B 57 (1998) 7190; O. Stier, M. Grundmann, D. Bimberg, Phys. Rev. B 59 (1999) 5688; W. Sheng, J.-P. Leburton, Appl. Phys. Lett. 80 (2002) 2755. [6] P. Hawrylak, M. Korkusinski, Electronic properties of self-assembled quantum dots, in: P. Michler (Ed.), Single Quantum Dots, Springer, Berlin, 2003. [7] A. Hartmann, Y. Ducommun, E. Kapon, U. Hohenester, E. Molinari, Phys. Rev. Lett. 84 (2000) 5648; P. Hawrylak, G.A. Narvaez, M. Bayer, A. Forchel, Phys. Rev. Lett. 85 (2000) 389. [8] C. Pryor, J. Kim, L.W. Wang, A.J. Williamson, A. Zunger, J. Appl. Phys. 83 (1998) 2548. [9] J.P. Loehr, Phys. Rev. B 50 (1994) 5429. [10] A. Szabo, N.S. Ostlund, Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory, McGraw-Hill, New York, 1989. [11] P. Hawrylak, Phys. Rev. B 60 (1999) 5597. [12] E. Dekel, D. Gershoni, E. Ehrenfreund, J.M. Garcia, P.M. Petroff, Phys. Rev. B 61 (2000) 11009.
Physica E 26 (2005) 267–270 www.elsevier.com/locate/physe
Microscopic approach to many-exciton complexes in self-assembled InGaAs/GaAs quantum dots Weidong Sheng, Marek Korkusinski, Pawel Hawrylak Institute for Microstructural Sciences, National Research Council of Canada, 1200 Montreal Road, Ottawa, Ontario, Canada, K1A 0R6 Available online 23 November 2004
Abstract We present a numerical calculation of many-exciton complexes in self-assembled InAs/GaAs quantum dots. We apply continuum elasticity theory and atomistic valence-force-field method to calculate strain distribution, and make use of various methods, ranging from a quasi-atomistic tight-binding approach to the single-band effective-mass approximation, to obtain single-particle energy levels. The effect of strain is incorporated by the deformation potential theory. We expand multiexciton states in the basis of Slater determinants and solve the many-body problem by the configuration-interaction method. The dynamics of multiexcitons is studied by solving the rate equations, from which the excitation–power dependence of emission spectrum is obtained. The emission spectra calculated by the microscopic tight-binding approach are found to be in good agreement with those obtained by the simple effective-mass method. r 2004 Elsevier B.V. All rights reserved. PACS: 71.35.Cc; 73.21.La; 78.67.Hc Keywords: Excitons Quantum dots; Configuration-interaction method
1. Introduction There has been a great interest in semiconductor self-assemble quantum dots for many years because of their potential applications in nanoelectronics and quantum computing [1]. These semiconductor nanostructures form due to the strain that arises from the lattice mismatch Corresponding author. Tel.: +1-613-993-4428; fax: +1-
613-990-0202. E-mail address: [email protected] (W. Sheng).
between two different kinds of materials. Depending on the types of the island material and the host matrix as well as growth conditions, the size, shape, and chemical composition profile of the dots may vary in a large range [2]. Due to many uncertainties in the growth process, an accurate simulation to predict structural properties of quantum dots would be very difficult. Hence, for a numerical study, one usually begins with a model system whose structural parameters are assumed from experimental information. In this study, we choose our model structure as a disklike
1386-9477/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2004.08.071
ARTICLE IN PRESS W. Sheng et al. / Physica E 26 (2005) 267–270
268
intermixing In0.5Ga0.5As dot whose base diameter is 25.4 nm and height is 2.3 nm [3]. For the electronic structure calculation, we will apply several methods, ranging from the quasiatomistic tight-binding-like effective bond-orbital method [4], eight-band k p [5], to the single-band effective-mass [6]. After obtaining single-particle states, we calculate Coulomb matrix elements and solve multiexciton states by using the configuration-interaction method [7]. Finally, we will compare the multiexciton emission spectra obtained by using different methods.
2. Formalism To calculate the strain distribution for a dot of given size and shape, we can apply either the continuum elasticity (CE) theory or the atomistic valence-force field method [8]. The strain energy ECE is given by V C 11 2xx þ 2yy þ 2zz E CE ¼ 2 V þ C 44 2xy þ 2yz þ 2zx 2 þ VC 12 xx yy þ yy zz þ zz xx : In both methods, strain tensor is calculated by minimizing the strain energy by moving each unit cell or atom according to the applied force. In this work, we use the CE theory to calculate the strain. For the electronic calculation, we start with a simple single-band effective-mass approximation. The Hamiltonians, including the strain effect, are given by 2 _2 q q2 _2 q2 He ¼ þ 2mexy qx2 qy2 2mez qz2 þ aC H S þ V ebo ; 2 _2 q q2 _2 q2 Hh ¼ þ 2mhxy qx2 qy2 2mhz qz2
the band dispersion (effective mass). For simplicity, only the hydrostatic (HS) and biaxial (BS) strain components are taken into account. aC, aV, and bV are the deformation potential parameters [8]. Vbo’s are the band offsets between the island and matrix material, which are chosen as 630 meV and 42 meV between conduction and valence bands, respectively. In the eight-band k p formalism, the mixing between the conduction and valence bands, and between valence bands are taken into account. Spin-orbital interaction is also included. For smaller quantum dots, a method that can describe the structure at atomistic level is required. In this work, we adopt the improved effective bondorbital method (EBOM) [9]. EBOM is a empirical sp3 tight-binding method based on the effective FCC lattice. The improved version includes the interactions up to the second-nearest neighbor and therefore has more fitting parameters to fit the band edges at both the G and X points. When the Coulomb matrix elements are calculated from the single-particle states, the Hamiltonian for many-exciton complex can be defined by X X X þ þ H^ ¼ E ei cþ E hi hþ V he i hi i ci ijkl hi cj hk cl i
i
ijkl
1 X ee þ þ 1 X hh þ þ þ V ijkl ci cj ck cl þ V h h hk hl : 2 ijkl 2 ijkl ijkl i j To calculate multiexciton states, we use the configuration-interaction method [10] to solve the many-body Hamiltonian. The emission spectrum is calculated by I sN ð_oÞ
X eE iN =kT ¼ P E j =kT N i je + 2 * X X f pnm hn cm C iN C N1 nm f dðE iN E fN1 _oÞ
aV H S bV BS þ V ebo : Here the effective mass along the growth direction (z) and in-plane (xy) are treated as fitting parameters because the strain would not only modify the band edges but also the curvature of
3. Results Fig. 1 shows the calculated energy levels of the intermixing disklike dot. The result by the
ARTICLE IN PRESS W. Sheng et al. / Physica E 26 (2005) 267–270
269
the calculated spectra cover emissions over different shells, several thousands of multiexciton eigenstates need to be calculated. Fig. 2 shows the calculated emission spectra for up to six excitons by both the microscopic tightbinding approach and the effective-mass approximations. It is seen that except for the emission
1.48 1.46 1.44 1.42 1.40 1.38 Energy (eV)
EMA 6→5 5→4
0.10
4→3 3→2
0.09
2→1 1→0 1.28
0.08
1.29
1.3
1.31
1.32
1.33
K.P 6→5
0.07 EMA
K.P
EBOM
single-band effective-mass method is fitted to that by the eight-band k p method. An overall shift is seen between the results by the eight-band k p method and EBOM. Compared with that obtained by the eight-band k p method, the effective band gap is smaller by about 25 meV by the EBOM calculation. The energy separation in conduction bands is also found smaller. Based on these single-particle states, we construct multiexciton configurations by using the Slater determinants and solve the many-body problem by using the configuration-interaction approach. As the total number of configurations grows exponentially with the number of excitons, we need to apply a truncation to limit the total number of configuration under 100,000. In order to calculate the emission spectrum, one needs to obtain both the energy levels and wave functions of multiexciton states. To make sure that
Intensity
5→4
Fig. 1. Energy levels of the intermixing disklike dot calculated by different methods.
4→3 3→2 2→1 1→0
1.28
1.29
1.3
1.31
1.32
1.33
EBOM 6→5 5→4 4→3 3→2 2→1 1→0 1.26
1.27
1.28
1.29 Energy (eV)
1.3
1.31
Fig. 2. Multiexciton emission spectra calculated for up to six excitons by different methods.
ARTICLE IN PRESS W. Sheng et al. / Physica E 26 (2005) 267–270
270
energies, all the three methods show qualitatively similar structures in the emission spectra. For a specific major emission peak, e.g., the biexciton peak, different methods are found to give the different relative intensity. This is due to the quantum interference effect that arises from the coherent summation of transition probability between each configuration in the initial multiexciton state and that in the final state. A detail assignment for major emission peaks is found in good agreement with the analysis made by a simple model [11]. In experiments, one usually observes a spectrum that combines those from different many-exciton complexes and also strongly depends on the external excitation power. In order to obtain the dependence of emission spectrum on the excitation power, we first calculate the radiative lifetimes for each many-exciton complex. Then we compute the corresponding probabilities by solving the rate equations [12]. The combined emission spectra are shown in Fig. 3. It is seen, at low excitation power, only emission from a single exciton and a little from biexciton. As excitation intensity increases, the biexciton emission becomes stronger, at the same time, P = 8.0
Intensity
P = 4.0
P = 1.0
1.26
1.27
1.28 Energy (eV)
1.29
1.3
Fig. 3. Emission spectra calculated at various excitation power by the EBOM.
emission from three-exciton complex appears in the second shell. When excitation power increases further, stronger emission peaks can be found in the second shell while there appear more small peaks in the first shell. In conclusion, we have presented a microscopic study of many-exciton complexes in self-assembled quantum dots. We have found that the quasiatomistic tight-binding approach gives qualitatively similar emission spectra as those obtained by the simple effective-mass calculation.
Acknowledgments This work was supported by NRC High Performance Computing project and Canadian Institute for Advanced Research (CIAR). References [1] L. Jacak, P. Hawrylak, A. Wojs, Quantum Dots, Springer, Berlin, 1998. [2] D. Bimberg, M. Grundmann, N.N. Ledentsov, Quantum Dot Heterostructures, Wiley, Chichester, 1998. [3] S. Raymond, et al., Phys. Rev. Lett. 92 (2004) 187402. [4] Y.-C. Chang, Phys. Rev. B 37 (1988) 8215; S.V. Nair, L.M. Ramaniah, K.C. Rustagi, Phys. Rev. B 45 (1992) 5969; S.J. Sun, Y.-C. Chang, Phys. Rev. B 62 (2000) 13631. [5] T.B. Bahder, Phys. Rev. B 41 (1990) 11992; H. Jiang, J. Singh, Phys. Rev. B 56 (1997) 4696; C. Pryor, Phys. Rev. B 57 (1998) 7190; O. Stier, M. Grundmann, D. Bimberg, Phys. Rev. B 59 (1999) 5688; W. Sheng, J.-P. Leburton, Appl. Phys. Lett. 80 (2002) 2755. [6] P. Hawrylak, M. Korkusinski, Electronic properties of self-assembled quantum dots, in: P. Michler (Ed.), Single Quantum Dots, Springer, Berlin, 2003. [7] A. Hartmann, Y. Ducommun, E. Kapon, U. Hohenester, E. Molinari, Phys. Rev. Lett. 84 (2000) 5648; P. Hawrylak, G.A. Narvaez, M. Bayer, A. Forchel, Phys. Rev. Lett. 85 (2000) 389. [8] C. Pryor, J. Kim, L.W. Wang, A.J. Williamson, A. Zunger, J. Appl. Phys. 83 (1998) 2548. [9] J.P. Loehr, Phys. Rev. B 50 (1994) 5429. [10] A. Szabo, N.S. Ostlund, Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory, McGraw-Hill, New York, 1989. [11] P. Hawrylak, Phys. Rev. B 60 (1999) 5597. [12] E. Dekel, D. Gershoni, E. Ehrenfreund, J.M. Garcia, P.M. Petroff, Phys. Rev. B 61 (2000) 11009.