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Photoluminescence of charged magneto-excitons in InAs single quantum dots
Applied Surface Science 190 (2002) 205±211
Photoluminescence of charged magneto-excitons in InAs single quantum dots Akiko Natori, Shin Ohnuma, Nguyen Hong Quang Department of Electro-Communications, University of Electro-Communications, Chofu, Tokyo 182-8585, Japan
Abstract We calculated the photoluminescence spectra of charged magneto-excitons in single two-dimensional parabolic quantum dots, using an unrestricted Hartree±Fock method. The calculated luminescence spectra explain well the observed red shifts of transition energies of InAs/GaAs single quantum dot by additional electron capture in a dot. The magnetic-®eld-induced transition of the ground state con®guration of trapped electrons causes drastic change in the photoluminescence spectra. The dependence of photoluminescence intensities of charged excitons on the excess energies of photogenerated carriers above the bulk GaAs energy gap is studied phenomenologically, by calculating the steady state electron population probability in a dot. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Charged exciton; Quantum dot; Photoluminescence; Unrestricted Hartree±Fock method
1. Introduction The study of quantum dots has been actively performed in recent years. In self-assembled quantum dots prepared by the Stranski±Krastanov growth mode, it is shown that the lateral potentials for the quasi-two-dimensional electrons can be treated approximately parabolic [1±4]. One of the main points which attracts much attention is the effect of electron± electron interaction on the optical properties of quantum dots. It is well known that the intraband excitations are insensitive to electron±electron interaction in quantum dots with parabolic con®ning potentials as a consequence of the generalized Kohn's theorem [5,6]. However, concerning the interband excitations, the situation is quite different. The many-electron effects in charged excitons have been studied in transmission experiments on a dot ensemble of InAs * Corresponding author. Tel./fax: 81-424-43-5145. E-mail address: [email protected] (A. Natori).
self-assembled quantum dots [7]. The authors have observed substantial redshift on the interband transition energies up to 20 meV, caused by occupation of dots until six electrons on average. Recently, the manyelectron effects in charged excitons have been studied by Holtz and coworkers [8,9] in photoluminescence (PL) experiments on a single InAs self-assembled quantum dot by means of micro-PL setup. The observed PL spectra at low excitation power consist of sharp peaks corresponding to a neutral exciton X and charged exciton states with additional electrons, and the authors found substantial red shifts of 3.1 and 7.8 meV in the lowest transition energies of X and X2 for excess one and two electrons, respectively. Furthermore, the PL intensities of X and X2 showed the dramatic out-of phase oscillations by tuning the pump-photon energy at a ®xed low excitation power. In our previous papers [10,11], we studied the charged excitons and the optical absorption spectra in parabolic quantum dots subjected to an external magnetic ®eld, using an unrestricted Hartree±Fock
0169-4332/02/$ ± see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 9 - 4 3 3 2 ( 0 1 ) 0 0 8 5 7 - 1
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A. Natori et al. / Applied Surface Science 190 (2002) 205±211
(UHF) method. The aim of the present work is to study both the observed charging effects of the luminescence spectra and the PL intensity dependence on the excess energy of photogenerated carriers [8,9]. 2. Formulation
Ci x1 ; . . . ; xN1 ; xo jc1 x1 ; . . . ; cN1 xN1 jco xo ;
2.1. Excitonic state The electronic structure of single electron in lensshaped self-assembled quantum dots is well approximated as parabolic lateral con®ning potential for adiabatic lowest sub-band wave function of the narrow quantum well of a wetting layer [3]. The Coulomb interaction between particles is weakened compared to the two-dimensional wave function, but the effect on the many-electron ground state is not so signi®cant [12]. Thus, we adopt two-dimensional approximation in the following. In the effective-mass approximation, the Hamiltonian of the system of N electrons and an exciton in a two-dimensional parabolic quantum dot in the presence of the perpendicular magnetic ®eld ~ Bkz can be written as H
N 1 X
h ~ ri h0 ~ ro
i1
N 1 2 X e E r i
N 1 X e2 ; Er i1 s io
(1)
where ~ ri and ~ ro are the two-dimensional position vectors of the ith electron and the hole, and h2 r2i me 1 2 2 h 2 ^zi ; h ~ ri oe oce ri oce L 4 2 2me 2 h0 ~ ro
h2 r2o mh 1 2 2 o o r2 h 4 ch o 2mh 2
(2) h ^zo : och L 2 (3)
me;h
oe;h
In the framework of the unrestricted Hartree±Fock approximation, the wave function of the excitonic excited state of N 1 electrons and one hole can be found in the form of a direct product of the wave function of the hole and the Slater determinant of N 1 identical electrons [10]. We have
Here and are the effective mass and the characteristic frequency of the con®nement potential of the electron or the hole, respectively, oce;ch eB=me;h c is the cyclotron frequency for the electron ^zi;zo is the z-component of orbital angular or the hole, L momentum operators of the ith electron or the hole, and Es is the static dielectric constant. In the single particle Hamiltonians given by Eqs. (2) and (3), the spin Zeeman term was neglected because of its smallness [13±15].
(4)
where xi f~ ri ; zi ; si g and xo f~ ro ; zo ; so g are the abbreviation for spatial ~ r, z and spin s variables of the ith electron and the hole. In the envelope function formalism, the single-particle wave functions ci,o(x) are factorized into a Bloch function and an envelope function ci x u ~ r; zfai ~ ra s; ci x u ~ r; zfbi ~ rb s;
or
co x v ~ r; zfao ~ ra s; co x v ~ r; zfbo ~ rb s;
or
(5) (6)
where u ~ r; z and v ~ r; z are the Bloch functions at the conduction and valence band edge, respectively. a(s) and b(s) are the spin functions of up- and down-spin states. In the Hartree±Fock±Roothaan formulation, the self-consistent envelope functions fai;o ~ r and fbi;o ~ r are expressed in expanded form in the basis functions. We choose the eigenfunctions wene me ~ r of the singleh electron Hamiltonian h ~ r and wnh mh ~ r of the hole Hamiltonian h0 ~ r as basis functions. The interband emission spectrum can be calculated according to the Fermi golden rule X g 2 s o jMif j d Ef Ei ho; (7) i;g
Mifg
where is the interband transition matrix element between the initial excitonic state with N excess electrons and the ®nal state of N electrons. The index g indicates that the transition electron is in the g-spin state. In the calculation of Mifg , we retain the exact value for electron±hole overlap integral Sij, unlike most previous works which always assumed electron±hole symmetric simpli®cation Sij hne me jnh mh i dne ;nh dme ;mh . 2.2. Rate equation The dependence of PL intensity of charged exciton states on the excess energies of photogenerated carriers above the bulk GaAs energy gap is studied phenom-
A. Natori et al. / Applied Surface Science 190 (2002) 205±211
207
enologically, by calculating the steady state population probabilities PN with N electrons in a quantum dot. Only three PL peaks X, X , and X2 were observed at a pump-photon energy larger than GaAs energy gap for low excitation power [8,9], corresponding to the ground state con®guration in a dot. Thus, we assume that the maximum number of trapped electrons in a dot is three for low excitation power. The electron diffusivity is much larger than hole diffusivity in GaAs due to difference of their effective masses, and hence electron capture rate is much higher than hole capture rate. Charged quantum dot states with N 1, 2 and 3 electrons are formed by electron capture process into a dot with N 1 electrons and they change to states with N 1 electrons by recombination process induced by capture of a hole. The rate equation on the occupation probability PN with N electrons in a dot can be written as d P0 C0 P0 R1 P1 ; dt d P1 C0 P0 C1 P1 R1 P1 R2 P2 ; dt d P2 C1 P1 C2 P2 R2 P2 R3 P3 ; dt d P3 C2 P2 R3 P3 : dt
(8)
Here, CN and RN are the electron capture rate and the recombination rate of a dot with N-trapped electrons. P3 Using the normalization condition P 1, the i i1 steady state occupation probability can be obtained easily as 1 ; 1 C0 =R1 C0 C1 =R1 R2 C0 C1 C2 =R1 R2 R3 C0 =R1 P1 ; 1 C0 =R1 C0 C1 =R1 R2 C0 C1 C2 =R1 R2 R3 C0 C1 =R1 R2 P2 ; 1 C0 =R1 C0 C1 =R1 R2 C0 C1 C2 =R1 R2 R3 C0 C1 C2 =R1 R2 R3 P3 : 1 C0 =R1 C0 C1 =R1 R2 C0 C1 C2 =R1 R2 R3 (9) P0
3. Numerical results For numerical calculations, we adopted the following parameters for InAs quantum dots [8]: me 0:067mo ,
Fig. 1. Photoluminescence spectra as a function of photon energy of a charged exciton with DN excess electrons, without magnetic ®eld (a) and with magnetic ®eld of 4 T (b). Here, the diagonal transition is shown by a thick solid line and the off-diagonal transition is shown by a thin solid line.
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Fig. 2. (a) Occupation probability Pi (i 1, 2, 3, 4) with i electrons in a dot as a function of C0/R1, and (b) C0/R1 as a function of excess energy above the GaAs energy gap.
A. Natori et al. / Applied Surface Science 190 (2002) 205±211
mh 0:25mo , oe 20:2 meV, oh 10:1 meV, Es 12:53. The corresponding units of length and energy are; effective Bohr radius aB 9:9 nm and twice the effective Rydberg constant 2Ry 11:61 meV. The oscillator lengths for electrons and holes in the absence of magnetic ®elds Le;h q h=me;h oe;h are 7.5 and 5.5 nm, respectively. These values are much smaller than the effective excitonic Bohr radius of 13 nm. This means that electrons and holes in small InAs dots are strongly con®ned. In Fig. 1, the calculated photoluminescence spectra from a charged exciton with DN excess electrons in a dot are shown, with and without the magnetic ®eld. We assumed the ground state con®guration for the initial electron con®guration in a dot, while excited states are also allowed in addition to the ground state for a hole. Here, 0±0 denotes the transition between the lowest states of an electron and a hole, 0±2 denotes that between the lowest state of an electron and the second excited state of a hole, and so on. The energies of 0±0 transition and 1±1 transition show red
209
shifts as DN increases, due to both the stronger attractive electron±hole interaction than the repulsive electron±electron interaction [16] and the exchange interaction in the ®nal state. The splitting of 0±0 transition observed at DN 2, 3 without magnetic ®eld is caused by two possible transitions of up or down spin electrons, and the splitting is attributed to difference of the exchange energies in the ®nal DN 1-electron states. The redshift of transition energies of X and X2 from that of X become 3.37 and 7.52 meV (lower energy peak), compared to the experimental values of 3.1 and 7.8 meV, respectively [8]. The higher energy PL peak of X2 almost overlaps with that of X , and they were not resolved experimentally. The PL peak corresponding to DN 3 was not observed experimentally and it suggests the maximum number of trapped electrons in the observed dot is three. It should be noticed that there exist the off-diagonal 0±2 transition peaks close to 1±1 transition peak. Experimentally, several ®ne PL peaks are observed around 1±1 transition peak which
Fig. 3. Occupation probability Pi (i 1, 2, 3, 4) with i electrons in a dot as a function of excess energy above the GaAs energy gap. Observed dependence of photoluminescence peak intensities [8] of X and X2 is also plotted by a solid circle and an open circle, respectively.
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A. Natori et al. / Applied Surface Science 190 (2002) 205±211
become signi®cant at higher excitation power. The splitting of 0±0 transition at DN 3 disappears at B 4 T, caused by the magnetic-®eld-induced transition of the ground state of four electrons [11]. Furthermore, the magnetic ®eld causes shifts of transition energies due to both the orbital Zeeman splitting (see 1±1 transition peak) and the diamagnetic shift (see 0±0 transition peak). The 0±0 transition intensity of charged exciton with DN excess electrons is approximately proportional to the occupation probability PDN1 with DN 1 electrons in a dot. Eq. (9) includes unknown three quantities Ci =Ri1 (i 0, 1, 2). We assumed simple relations about Ri of R2 R3 2R1 , since one of two occupied electrons in the lowest single particle state can recombine with a hole for R2 and R3 (see Fig. 1(a)). Further, we assume C2 C1 C0 for simplicity. Fig. 2(a) shows Pi (i 0, 1, 2, 3) as a function of C0/R1. We determined the excess energy dependence of C0/R1 as to reproduce the observed PL intensity ratio [8] of a neutral exciton X with DN 0 to a charged exciton X2 with DN 2. The obtained excess energy dependence of C0/R1 is shown in Fig. 2(b). The oscillating behavior of C0/R1 with the excess energy of about 40 meV is attributed to the energy dependence of electron capture rate C0 due to the optical phonon emission of electrons [8,9], and the monotonic decrease of the peak values with excess energy is attributed to monotonic increase of the recombination rate R1. R1 is proportional to the capture rate of a hole and the capture rate increases monotonically with excess energy, since increasing excess energy causes increases of the kinetic energy and hence the diffusivity of a hole. In the observed energy range, the optical phonon emission of a hole is not allowed. Fig. 3 represents the excess energy dependence of the occupation probabilities Pi. The out-of-phase oscillation of P1 and P3 by tuning excess energy is clearly seen except higher energy than 120 meV. The origin of this out-of-phase oscillation is that P3 is increasing function of C0/R1 but P1 is decreasing function of C0/R1 in the range of C0 =R1 > 1, as seen in Fig. 2(a). The excess energy dependence of P3 reproduces the observed dependence of PL intensity of X2 (open circles in Fig. 3) [8]. The calculated behavior of P1 reproduces the observed PL intensity dependence of X (solid circles in Fig. 3), but the agreement is poor in excess energy higher than 120 meV.
4. Discussion and conclusion Our assumption about the ratio Ci =Ri1 is too simpli®ed, since it does not consider any charging effect of a dot on the electron capture rate Ci and the recombination rate Ri induced by hole capture. Consideration of the charging effect on the capture rates of electrons and holes will improve excess energy dependence of PL intensity of charged excitons. In conclusion, we calculated the photoluminescence spectra of charged magneto-excitons in single quantum dots, and the calculated results explain well the observed red shifts of transition energies of InAs single quantum dots. The dependence of PL intensities on the excess energy of photogenerated carriers is also studied phenomenologically, and it can reproduce the out-of-phase oscillation of PL intensities of X and X2 with excess energy.
Acknowledgements This work was supported in part by ``the New Frontier Program Grant-in-Aid for Scienti®c Research'', from the Ministry of Education, Science, Sports and Culture of Japan.
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A. Natori et al. / Applied Surface Science 190 (2002) 205±211 [10] N.H. Quang, S. Ohnuma, A. Natori, Phys. Rev. B 62 (2000) 12955. [11] A. Natori, S. Ohnuma, N.H. Quang, Jpn. J. Appl. Phys. 40 (2001) 1951. [12] M. Fujito, A. Natori, H. Yasunaga, Phys. Rev. B 53 (1996) 9952. [13] A. Natori, Y. Sugimoto, M. Fujito, Jpn. J. Appl. Phys. 36 (1997) 3960.
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[14] R. Rinaldi, P.V. Giugno, R. Cingolani, H. Lipsanen, M. Sopanen, J. Tulkki, J. Ahepelto, Phys. Rev. Lett. 77 (1996) 342. [15] M. BraskeÁn, M. Lindberg, J. Tulkki, Phys. Rev. B 55 (1997) 9275. [16] R.J. Warburton, B.T. Miller, C.S. Durr, C. Bodefeld, K. Karrai, J.P. Kotthaus, G. Medeiros-Ribeiro, P.M. Petroff, S. Huant, Phys. Rev. B 58 (1998) 16221.
Photoluminescence of charged magneto-excitons in InAs single quantum dots Akiko Natori, Shin Ohnuma, Nguyen Hong Quang Department of Electro-Communications, University of Electro-Communications, Chofu, Tokyo 182-8585, Japan
Abstract We calculated the photoluminescence spectra of charged magneto-excitons in single two-dimensional parabolic quantum dots, using an unrestricted Hartree±Fock method. The calculated luminescence spectra explain well the observed red shifts of transition energies of InAs/GaAs single quantum dot by additional electron capture in a dot. The magnetic-®eld-induced transition of the ground state con®guration of trapped electrons causes drastic change in the photoluminescence spectra. The dependence of photoluminescence intensities of charged excitons on the excess energies of photogenerated carriers above the bulk GaAs energy gap is studied phenomenologically, by calculating the steady state electron population probability in a dot. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Charged exciton; Quantum dot; Photoluminescence; Unrestricted Hartree±Fock method
1. Introduction The study of quantum dots has been actively performed in recent years. In self-assembled quantum dots prepared by the Stranski±Krastanov growth mode, it is shown that the lateral potentials for the quasi-two-dimensional electrons can be treated approximately parabolic [1±4]. One of the main points which attracts much attention is the effect of electron± electron interaction on the optical properties of quantum dots. It is well known that the intraband excitations are insensitive to electron±electron interaction in quantum dots with parabolic con®ning potentials as a consequence of the generalized Kohn's theorem [5,6]. However, concerning the interband excitations, the situation is quite different. The many-electron effects in charged excitons have been studied in transmission experiments on a dot ensemble of InAs * Corresponding author. Tel./fax: 81-424-43-5145. E-mail address: [email protected] (A. Natori).
self-assembled quantum dots [7]. The authors have observed substantial redshift on the interband transition energies up to 20 meV, caused by occupation of dots until six electrons on average. Recently, the manyelectron effects in charged excitons have been studied by Holtz and coworkers [8,9] in photoluminescence (PL) experiments on a single InAs self-assembled quantum dot by means of micro-PL setup. The observed PL spectra at low excitation power consist of sharp peaks corresponding to a neutral exciton X and charged exciton states with additional electrons, and the authors found substantial red shifts of 3.1 and 7.8 meV in the lowest transition energies of X and X2 for excess one and two electrons, respectively. Furthermore, the PL intensities of X and X2 showed the dramatic out-of phase oscillations by tuning the pump-photon energy at a ®xed low excitation power. In our previous papers [10,11], we studied the charged excitons and the optical absorption spectra in parabolic quantum dots subjected to an external magnetic ®eld, using an unrestricted Hartree±Fock
0169-4332/02/$ ± see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 9 - 4 3 3 2 ( 0 1 ) 0 0 8 5 7 - 1
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A. Natori et al. / Applied Surface Science 190 (2002) 205±211
(UHF) method. The aim of the present work is to study both the observed charging effects of the luminescence spectra and the PL intensity dependence on the excess energy of photogenerated carriers [8,9]. 2. Formulation
Ci x1 ; . . . ; xN1 ; xo jc1 x1 ; . . . ; cN1 xN1 jco xo ;
2.1. Excitonic state The electronic structure of single electron in lensshaped self-assembled quantum dots is well approximated as parabolic lateral con®ning potential for adiabatic lowest sub-band wave function of the narrow quantum well of a wetting layer [3]. The Coulomb interaction between particles is weakened compared to the two-dimensional wave function, but the effect on the many-electron ground state is not so signi®cant [12]. Thus, we adopt two-dimensional approximation in the following. In the effective-mass approximation, the Hamiltonian of the system of N electrons and an exciton in a two-dimensional parabolic quantum dot in the presence of the perpendicular magnetic ®eld ~ Bkz can be written as H
N 1 X
h ~ ri h0 ~ ro
i1
N 1 2 X e E r i
N 1 X e2 ; Er i1 s io
(1)
where ~ ri and ~ ro are the two-dimensional position vectors of the ith electron and the hole, and h2 r2i me 1 2 2 h 2 ^zi ; h ~ ri oe oce ri oce L 4 2 2me 2 h0 ~ ro
h2 r2o mh 1 2 2 o o r2 h 4 ch o 2mh 2
(2) h ^zo : och L 2 (3)
me;h
oe;h
In the framework of the unrestricted Hartree±Fock approximation, the wave function of the excitonic excited state of N 1 electrons and one hole can be found in the form of a direct product of the wave function of the hole and the Slater determinant of N 1 identical electrons [10]. We have
Here and are the effective mass and the characteristic frequency of the con®nement potential of the electron or the hole, respectively, oce;ch eB=me;h c is the cyclotron frequency for the electron ^zi;zo is the z-component of orbital angular or the hole, L momentum operators of the ith electron or the hole, and Es is the static dielectric constant. In the single particle Hamiltonians given by Eqs. (2) and (3), the spin Zeeman term was neglected because of its smallness [13±15].
(4)
where xi f~ ri ; zi ; si g and xo f~ ro ; zo ; so g are the abbreviation for spatial ~ r, z and spin s variables of the ith electron and the hole. In the envelope function formalism, the single-particle wave functions ci,o(x) are factorized into a Bloch function and an envelope function ci x u ~ r; zfai ~ ra s; ci x u ~ r; zfbi ~ rb s;
or
co x v ~ r; zfao ~ ra s; co x v ~ r; zfbo ~ rb s;
or
(5) (6)
where u ~ r; z and v ~ r; z are the Bloch functions at the conduction and valence band edge, respectively. a(s) and b(s) are the spin functions of up- and down-spin states. In the Hartree±Fock±Roothaan formulation, the self-consistent envelope functions fai;o ~ r and fbi;o ~ r are expressed in expanded form in the basis functions. We choose the eigenfunctions wene me ~ r of the singleh electron Hamiltonian h ~ r and wnh mh ~ r of the hole Hamiltonian h0 ~ r as basis functions. The interband emission spectrum can be calculated according to the Fermi golden rule X g 2 s o jMif j d Ef Ei ho; (7) i;g
Mifg
where is the interband transition matrix element between the initial excitonic state with N excess electrons and the ®nal state of N electrons. The index g indicates that the transition electron is in the g-spin state. In the calculation of Mifg , we retain the exact value for electron±hole overlap integral Sij, unlike most previous works which always assumed electron±hole symmetric simpli®cation Sij hne me jnh mh i dne ;nh dme ;mh . 2.2. Rate equation The dependence of PL intensity of charged exciton states on the excess energies of photogenerated carriers above the bulk GaAs energy gap is studied phenom-
A. Natori et al. / Applied Surface Science 190 (2002) 205±211
207
enologically, by calculating the steady state population probabilities PN with N electrons in a quantum dot. Only three PL peaks X, X , and X2 were observed at a pump-photon energy larger than GaAs energy gap for low excitation power [8,9], corresponding to the ground state con®guration in a dot. Thus, we assume that the maximum number of trapped electrons in a dot is three for low excitation power. The electron diffusivity is much larger than hole diffusivity in GaAs due to difference of their effective masses, and hence electron capture rate is much higher than hole capture rate. Charged quantum dot states with N 1, 2 and 3 electrons are formed by electron capture process into a dot with N 1 electrons and they change to states with N 1 electrons by recombination process induced by capture of a hole. The rate equation on the occupation probability PN with N electrons in a dot can be written as d P0 C0 P0 R1 P1 ; dt d P1 C0 P0 C1 P1 R1 P1 R2 P2 ; dt d P2 C1 P1 C2 P2 R2 P2 R3 P3 ; dt d P3 C2 P2 R3 P3 : dt
(8)
Here, CN and RN are the electron capture rate and the recombination rate of a dot with N-trapped electrons. P3 Using the normalization condition P 1, the i i1 steady state occupation probability can be obtained easily as 1 ; 1 C0 =R1 C0 C1 =R1 R2 C0 C1 C2 =R1 R2 R3 C0 =R1 P1 ; 1 C0 =R1 C0 C1 =R1 R2 C0 C1 C2 =R1 R2 R3 C0 C1 =R1 R2 P2 ; 1 C0 =R1 C0 C1 =R1 R2 C0 C1 C2 =R1 R2 R3 C0 C1 C2 =R1 R2 R3 P3 : 1 C0 =R1 C0 C1 =R1 R2 C0 C1 C2 =R1 R2 R3 (9) P0
3. Numerical results For numerical calculations, we adopted the following parameters for InAs quantum dots [8]: me 0:067mo ,
Fig. 1. Photoluminescence spectra as a function of photon energy of a charged exciton with DN excess electrons, without magnetic ®eld (a) and with magnetic ®eld of 4 T (b). Here, the diagonal transition is shown by a thick solid line and the off-diagonal transition is shown by a thin solid line.
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A. Natori et al. / Applied Surface Science 190 (2002) 205±211
Fig. 2. (a) Occupation probability Pi (i 1, 2, 3, 4) with i electrons in a dot as a function of C0/R1, and (b) C0/R1 as a function of excess energy above the GaAs energy gap.
A. Natori et al. / Applied Surface Science 190 (2002) 205±211
mh 0:25mo , oe 20:2 meV, oh 10:1 meV, Es 12:53. The corresponding units of length and energy are; effective Bohr radius aB 9:9 nm and twice the effective Rydberg constant 2Ry 11:61 meV. The oscillator lengths for electrons and holes in the absence of magnetic ®elds Le;h q h=me;h oe;h are 7.5 and 5.5 nm, respectively. These values are much smaller than the effective excitonic Bohr radius of 13 nm. This means that electrons and holes in small InAs dots are strongly con®ned. In Fig. 1, the calculated photoluminescence spectra from a charged exciton with DN excess electrons in a dot are shown, with and without the magnetic ®eld. We assumed the ground state con®guration for the initial electron con®guration in a dot, while excited states are also allowed in addition to the ground state for a hole. Here, 0±0 denotes the transition between the lowest states of an electron and a hole, 0±2 denotes that between the lowest state of an electron and the second excited state of a hole, and so on. The energies of 0±0 transition and 1±1 transition show red
209
shifts as DN increases, due to both the stronger attractive electron±hole interaction than the repulsive electron±electron interaction [16] and the exchange interaction in the ®nal state. The splitting of 0±0 transition observed at DN 2, 3 without magnetic ®eld is caused by two possible transitions of up or down spin electrons, and the splitting is attributed to difference of the exchange energies in the ®nal DN 1-electron states. The redshift of transition energies of X and X2 from that of X become 3.37 and 7.52 meV (lower energy peak), compared to the experimental values of 3.1 and 7.8 meV, respectively [8]. The higher energy PL peak of X2 almost overlaps with that of X , and they were not resolved experimentally. The PL peak corresponding to DN 3 was not observed experimentally and it suggests the maximum number of trapped electrons in the observed dot is three. It should be noticed that there exist the off-diagonal 0±2 transition peaks close to 1±1 transition peak. Experimentally, several ®ne PL peaks are observed around 1±1 transition peak which
Fig. 3. Occupation probability Pi (i 1, 2, 3, 4) with i electrons in a dot as a function of excess energy above the GaAs energy gap. Observed dependence of photoluminescence peak intensities [8] of X and X2 is also plotted by a solid circle and an open circle, respectively.
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become signi®cant at higher excitation power. The splitting of 0±0 transition at DN 3 disappears at B 4 T, caused by the magnetic-®eld-induced transition of the ground state of four electrons [11]. Furthermore, the magnetic ®eld causes shifts of transition energies due to both the orbital Zeeman splitting (see 1±1 transition peak) and the diamagnetic shift (see 0±0 transition peak). The 0±0 transition intensity of charged exciton with DN excess electrons is approximately proportional to the occupation probability PDN1 with DN 1 electrons in a dot. Eq. (9) includes unknown three quantities Ci =Ri1 (i 0, 1, 2). We assumed simple relations about Ri of R2 R3 2R1 , since one of two occupied electrons in the lowest single particle state can recombine with a hole for R2 and R3 (see Fig. 1(a)). Further, we assume C2 C1 C0 for simplicity. Fig. 2(a) shows Pi (i 0, 1, 2, 3) as a function of C0/R1. We determined the excess energy dependence of C0/R1 as to reproduce the observed PL intensity ratio [8] of a neutral exciton X with DN 0 to a charged exciton X2 with DN 2. The obtained excess energy dependence of C0/R1 is shown in Fig. 2(b). The oscillating behavior of C0/R1 with the excess energy of about 40 meV is attributed to the energy dependence of electron capture rate C0 due to the optical phonon emission of electrons [8,9], and the monotonic decrease of the peak values with excess energy is attributed to monotonic increase of the recombination rate R1. R1 is proportional to the capture rate of a hole and the capture rate increases monotonically with excess energy, since increasing excess energy causes increases of the kinetic energy and hence the diffusivity of a hole. In the observed energy range, the optical phonon emission of a hole is not allowed. Fig. 3 represents the excess energy dependence of the occupation probabilities Pi. The out-of-phase oscillation of P1 and P3 by tuning excess energy is clearly seen except higher energy than 120 meV. The origin of this out-of-phase oscillation is that P3 is increasing function of C0/R1 but P1 is decreasing function of C0/R1 in the range of C0 =R1 > 1, as seen in Fig. 2(a). The excess energy dependence of P3 reproduces the observed dependence of PL intensity of X2 (open circles in Fig. 3) [8]. The calculated behavior of P1 reproduces the observed PL intensity dependence of X (solid circles in Fig. 3), but the agreement is poor in excess energy higher than 120 meV.
4. Discussion and conclusion Our assumption about the ratio Ci =Ri1 is too simpli®ed, since it does not consider any charging effect of a dot on the electron capture rate Ci and the recombination rate Ri induced by hole capture. Consideration of the charging effect on the capture rates of electrons and holes will improve excess energy dependence of PL intensity of charged excitons. In conclusion, we calculated the photoluminescence spectra of charged magneto-excitons in single quantum dots, and the calculated results explain well the observed red shifts of transition energies of InAs single quantum dots. The dependence of PL intensities on the excess energy of photogenerated carriers is also studied phenomenologically, and it can reproduce the out-of-phase oscillation of PL intensities of X and X2 with excess energy.
Acknowledgements This work was supported in part by ``the New Frontier Program Grant-in-Aid for Scienti®c Research'', from the Ministry of Education, Science, Sports and Culture of Japan.
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