Solid-State Electronics Vol. 37, Nos 4--6,pp. 1109-1112, 1994
Copyright ~ 1994ElsevierScienceLtd Printed in Great Britain.All rights reserved 0038-1101/94 $6.00+ 0.00
RELAXATION AND RADIATIVE DECAY OF EXCITONS IN GaAs QUANTUM DOTS U. BOCKELMANN,K. BRUNNER and G. ABSTREITER Walter Schottky Institut, Technische Universit/it Miinchen, Am Coulombwall, D-85748 Garching, Germany Abstract--Radiative and non-radiative exciton states of quantum dots with parabolic lateral potentials are calculated in an effective-mass approximation. Energy spectrum, radiative recombination rates and LA phonon scattering rates of the different excitons are discussed in their dependence on the lateral confinement potential. For sufficientlystrong lateral potentials the relaxation of excitons by LA phonon emission becomes so weak that several excitonic levels should be observable in a photoluminescence experiment even at low excitation intensity. The theoretical results are compared to microscopic photoluminescence data collected from a series of single GaAs quantum dots fabricated by laser induced local interdiffusion.
In quantum dots or quasi-zero-dimenional (0D) systems the charge carriers are confined in all three spatial directions and therefore the energy spectrum consists of atomic-like, discrete levels. In principle, artificial OD structures based on direct gap semiconductors (e.g. the GaAs/GaA1As system) could be studied in great detail by interband optical spectroscopy, i.e. by photoluminescence (PL) and photoluminescence excitation (PLE) spectroscopy. The interband spectra should reflect the details of the confining potentials as well as exciton and manybody effects. Information on the exciton relaxation dynamics could be deduced from the intensity ratios of the different PL peaks and also from a comparison of PL and PLE spectral1]. In practice, samples prepared for optical studies often contain a large number of quantum dots supposed to be equal. Differences in composition and size between the individual structures however lead to important inhomogeneous line broadening which usually covers the details of the interband spectra. This problem can be circumvented by local spectroscopy of a single quantum dot. This way, distinct narrow lines have been observed in microscopic PL and PLE spectroscopy on a series of single quantum dots fabricated by focused laser beam induced interdiffusion of GaAs/GaAIAs quantum wells. In this contribution, we present theoretical results on energy states, relaxation and radiative decay of excitons in quantum dots based on GaAs/GaAlAs quantum wells. The theoretical description is well suited for a comparison with the quantum dot samples fabricated by laser induced interdiffusion. We present PL spectra measured at low excitation power from a series of dots of different lateral size and compare them to the theoretical results. For quantum dots fabricated from quantum wells the confinement induced by the lateral potential is
usually weak compared to the confinement along the quantum well growth axis z. We therefore build the near band gap exciton states of the quantum dot on the electron and heavy-hole ground subbands of the underlying quantum well [envelope functions Z,(z) and Zh(Z), respectively], neglecting heavy-hole lighthole mixing and excited subbands. The envelope function of the quantum dot excitons are of the form:
~,x (r, ,r h ) = ~b (x~ ,y, ,Xh ,Yh )X* (Zc)X. (Zh).
The functions ¢#depending on the lateral electron and hole coordinates are the eigensolutions of the Hamiltonian: n = H 0 + H,h _h21
Ho= L - - 1 ~ + ~ ) ,=~h 2my \ a x , . Or;
+sm,,e°8,,( x ' "' ;+Y:)'
e2 neh = -- SSdzod2h IZe(ze)12lZh(zh)l2rh-----~" e, Ir~ We describe the lateral confinement by rotationalsymmetric, parabolic potentials with different angular frequency for electrons and holes (toe and coh). The Hamiltonian H is diagonalized numerically by expanding it on the set of analytical eigenfunctions of H0. H0 describes harmonic oscillators independent for the electrons and holes. The solutions, originally derived by Fock, are parametrized by the radial quantum numbers n,, n h (0,1,2 . . . . ) and the angular momentum quantum numbers le, lh (0,_+ 1, + 2. . . . ). The total angular momentum j = le + lh is a good quantum number of the Hamiltonian H because of the rotational symmetry of the lateral potential. The expansion on a basis of independent electron hole states has been used in earlier theoretical studies of radiative excitons in quantum dots[4-6]. In an alternative approach, H is written as a function of the center-off-mass (CM) coordinate R = (mere + mhr h)/
U. BOCKELMANNet al.
(m~+ mh) and the relative coordinate r = r , - r h of the exciton. The expansion on a set of functions depending separately on R and r exhibits a faster convergency for weak lateral confinement but has the important disadvantage that the r-dependent basis functions are not analytical[5,6J. Excitons with a non-zero value o f j are non-radiative. From the wavefunction of the j = 0 excitons we determine the radiative lifetimes ~0 in the dipol approximation: o lOC I~dr~, (r,r)l 2.
Energies and radiative lifetimes of q u a n t u m dot excitons are presented in Fig. 1. We consider dots fabricated from a 30/~ wide GaAs/Ga07A10.3As q u a n t u m well with a constant ratio o&/~o~ to allow for a detailed comparison with our experimental results. Only radiative excitons are included in the figure where the length of the lines is chosen to be proportional to the rate of radiative recombination. At small ho3~, the spectrum is dominated by a series of equidistant lines. In this limit, the CM coordinate R and the relative coordinate r of the exciton are only weakly coupled. The equidistant lines originate from the quantization of the CM coordinate by the parabolic potential. The additional, weakly radiative excitons appearing at higher energies involve excited states of the relative motion. With increasing ht~o, the energy levels shift to higher values and the CM and the relative motion become increasingly coupled. The radiative lifetime of excitons originat-
Po~ o -o
L z =30
~.= % (~/mh) ~/2 -10
Fig. 1. Energy and inverse radiative lifetime r o' of the radiative excitons as a function of the confinement parameter hto~.. The value of r o Lis given by the length of the lines, in comparison to the double-arrow on the right. Zero-energy corresponds to the bandgap of the underlying quantum well, namely the sum of the bandgap of bulk GaAs and the confinement energies associated with Zc(z) and Zh(z).
1"-2 1"-3 ,'-4
t I LI I,l li, ll,llkJi,LI
3 I I I I, IL lh lill,(,(J 211 I II litllltuldl,ldl 1 I I I Ill tllill,lll 3/41 I I l,llI, Lli 1/2 I I IIILilLl It,h,ll,t,' I
ENERGY (meV) Fig. 2. Exciton energy spectra for a ratio to~i~oh varying from 0.5 to 5 but fixed sum ~o~+ ~oh of the electron and hole lateral confinement potentials. The length of the lines indicate the quantum number j of the total angular momentum.
ing from the ground state (excited states) of the relative coordinate increase (decrease) with increasing lateral confinement. The Coulomb attraction strongly influences the energy spectrum and radiative lifetimes of the quantum dot. The binding energy of the ground state exciton grows from 7.9 to 15.4meV with ho~e increasing from 1 to 10meV which is in no case small compared to the lateral confinement energies. When the electron hole Coulomb interaction is neglected, the radiative lifetime of the ground state of the h ~ = 6 meV dot is 0.3 ns, three times longer than the excitonic lifetime given in Fig. 1. The relation between the electron and hole confinement parameters used in Fig. I, is suggested by a simple model calculation of the process of laser induced thermal interdiffusion. Figure 2 demonstrates that the lowest exciton energies change only slightly when the ratio 60e/oA is varied. All exciton states (radiative and non-radiative ones) are plotted for fixed ~o~+ ~h. The total angular m o m e n t u m j of the exciton is indicated by the length of the plotted line. The exciton spectrum becomes increasingly dense with increasing energy separation from the ground-state. Let us now discuss the relaxation properties of the exciton. At low temperature and carrier density, emission of longitudinal acoustic (LA) phonons represent the dominant relaxation mechanism of excited states of energy below the optical phonon branches. It has been shown theoretically, that in q u a n t u m dots the electron LA phonon scattering rates decrease by several orders of magnitude when the separation of the discrete energy levels increases above a threshold value. How do the corresponding results look like for excitons? The rate of a transition from an initial exciton state i to a final
Relaxation and radiative decay of excitons in GaAs quantum dots i
statefaccompanied by emission of a phonon at zero temperature is given by the Fermi golden rule: _, _ 2 n I I -E.+ t,_j--~-Y,l<'l.'o, lWl'e~o,>l2 ~(E,, '
The sum extends over the wavevector q of the phonon of energy We consider bulk GaAs LA phonons neglecting confinement effects on the phonon spectrum. An isotropic dispersion relation COq= is assumed, with q = 3700m/s. To derive the exciton-phonon interaction potential Wwe first determine the strain tensor associated with the phonon. The interaction potential is then given by the sum of the strain induced shift of the electron and hole part of the exciton. The latter is obtained from the diagonal part of the Fs strain Hamiltonian, derived by Pikus and Bir. For heavy holes W takes the form:
Dot B a r ~ ,0~194.
~ / I
1000 A Si3N. l ~ o A C~As 200A AIGOAs 30A GOA$ 2 0 0 A AIGQA|
W°cql/2(De'°+[ l +rn(q~q2
~teK~/,Jr, ecl Re.or, a25 hen / 0.6 $
Dot Size w
i .J 0
(5) where D is the conduction band deformation potential and/, m are valence band deformation potentials. Figure 3 shows the relaxation rate ~ ~-: of the first excited radiative exciton in comparison to its radiative decay rate z o '- The quantity z/- ~is defined as the sum over the LA phonon scattering rates to all exciton states below the initial state at zero temperature. We assume that the final states are unoccupied. Scattering between any pair ofexciton states is possible. This also means that non-radiative excitons are important for the relaxation dynamics of radiative states. The contribution of a given transition depends strongly on the energy separation of the involved levels which is equal to the phonon energy [energy conservation expressed by the delta function in eqn (4)]. For small q the exciton phonon matrix element increases with q due to the prefactor q ].,2ofeqn (5). For q above about 2 the matrix element decreases strongly with increasing q
80 FIRST EXCITED
2O RADIATIVE DECAY
6 lh(~, (meV)
Fig. 3. Relaxation rate ~ ~-' (solid line) and radiative recombination rate To ~(dashed line) of the first excited radiative exciton as a function of lateral confinement. SSF 37 4"-(~-K.K
ENERGY (meV) Fig. 4. PL spectra of single dot structures for three lateral sizes w. Laser excitation was at 1.96eV with a power of 1 gW (125 nW for w = 450 nm) focused to a spot of 1.5 gm diameter. The dot structure fabricated by focused laser beam interdiffusion of a single quatum well is presented schematically in the inset. (increasing energy). Roughly speaking, when the phonon wavelength becomes small compared to the smallest dimension of the quantum dot (usually the well width L..) the exciton-phonon coupling becomes weak, in a similar way as in the case of the electron-phonon interaction. The relaxation rate of the first excited j = 0 state, plotted in Fig. 3, is dominated by scattering into the j = + I ground state. Typically, PL spectra ofone, two and three dimensional semiconductor structures show only one peak corresponding to the exciton ground state. This is because the relaxation of excited states is more efficient than their radiative decay. From Fig. 3 it appears clearly, that in 0D systems the relaxation rate of excited excitons can be comparable or even smaller than the rate of radiative recombination. From a detailed rate equation analysis based on calculated rates of exciton relaxation and radiative recombination follows that it should be possible to observe strong PL signals from excited quantum dot states even for very low excitation intensity. For the experimental studies a series of single quantum dots of different lateral size has been fabricated by laser-induced thermal interdiffusion of a 30 ,~, wide GaAs/Ga0 65A1035As single quantum well. A schematic cross section of the dot structures is given in the inset of Fig. 4. The lateral potential is defined by drawing a square frame of size w with a
U. BOCKELMANNet al.
focused Ar + laser beam on the sample surface. Afterwards an area of 6 x 6/~m around the dot is interdiffused by scanning the beam continuously. This procedure gives rise to a lateral modulation of the band-gap as shown schematically in the lower part of the inset. The strongly nonlinear temperature dependence of the interdiffusion allows to realize steep potential barriers for electrons and holes. PL and PLE spectroscopy have been performed with a spatial resolution of about 1.5/~m at liquid helium temperature. Only one quantum dot is measured at a time and consequently there should be no inhomogeneous broadening of the experimental spectra. Details of sample preparation and experimental setup are given in Ref.. In Fig. 4 we present PL spectra of three quantum dots of different lateral size w. Excitation was performed with a HeNe laser of low power density. The PL linewidths have been found to increase with intensity. The energy shift of the lowest PL peak with increasing structure size w is a combined effect of lateral confinement and alloying of the dot center. The effective size of the lateral confinement is given by the interdiffusion profile and not just by the geometrical size w. Decreasing w from large values, the laser induced barriers move closer together which leads to an increasing lateral quantization. Maximum confinement is obtained when the barriers are about to meet at the dot center. We expect this to be realized in the w = 450 nm dot where a maximum line splitting of about 10meV is observed. Further decreasing w, the increasing AI content near to the center of the dot leads to a strong blue shift of the PL ground state but the lateral barrier height decreases. For w --, 0, we approach the case of a homogeneously alloyed 2D layer without lateral confinement. Near the dot center, a simple model of the local quantum well interdiffusion gives nearly isotropic and parabolic lateral potentials with a similar curvature for electrons and heavy holes. This means m~tn~ = mhco ~ or equivalently tOh = °~e(me/mh)l 2 the relation used above in calculating Figs l and 3. The PL spectra of the 400 and 500 nm dots are dominated by the lowest energy peak but display clear structures of almost equidistant spacing at higher photon energies. The position of the experimental lines compare quite well with Fig. l if we consider hta~ ~ 3 meV (2 meV) for w = 400 nm (500nm). For the 450nm dot the interdiffusion model gives hto~=6meV. The 10meV separation between the lowest two peaks agrees with Fig. I but the threefold splitting of the excited state is not born out by the calculation. Figure 2 shows that the fine structure cannot be explained either by varying the ratio ogh/e0~. However, the first excited radiative (j = 0) state is almost degenerate with the j = +2 ground states (Fig. 2). A deviation of the lateral potential from the rotational symmetry could mix the
three states, i.e. the non-radiative excitons would gain oscillator strength. Such a deviation can be due to charged defects introduced by the laser processing in the surrounding area or simply by not perfectly constant interdiffusion conditions during writing the square frame which defines the dot. PLE spectra of the 450rim dot as shown for example in Ref. display for detection at the lowest luminescence peak small peaks with roughly an equidistant energy separation of about 2 meV between the luminescence ground state and first excited state. These weak structures cannot be understood from the calculated exciton energies, even if we assume a much weaker confinement for the holes than for the electrons (see Fig. 2). The main features of the experimental PL lines (Fig. 4), especially the fact that the excited states are most prominant in the dot with the largest level splitting are in agreement with a strong decrease of the exciton relaxation rate with increasing lateral confinement. In conclusion, relaxation and radiative recombination of excitons in quantum dots have been studied theoretically. The calculations show that the radiative lifetimes of one part of the exciton states increase with increasing lateral confinement (in particular the lowest states) but there exist also exciton states that exhibit the opposite dependence. In quantum dots of sufficiently strong lateral confinement the relaxation of excited excitons by emission of LA phonons can be less efficient than their radiative decay. This suggests that excited excitons can be observed in PL, which is in agreement with recent microscopic PL experiments on single quantum dots. The variation of position and intensity of the PL peaks with lateral structure size qualitatively confirms the theoretical results. Acknowledgements--This work has been supported in part by the "'Deutsche Forschunggemeinschaft(SFB 348)" the "Bundesministerium fiir Forschung und Technologic (DFE Verbundprogramm TK363/2)'" and by a "PROCOPE" contract.
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