(AlAs)n superlattices

4 February 2002

Physics Letters A 293 (2002) 272–276 www.elsevier.com/locate/pla

Phonon modes of short-period (GaAs)n/(AlAs)n superlattices T.R. Yang a,∗ , M.M. Dvoynenko a , A.V. Goncharenko b a Department of Physics, National Taiwan Normal University, 88 Sec. 4 Ting-Chou Rd., Taipei 117, Taiwan, ROC b Institute of Semiconductor Physics, National Academy of Sciences of Ukraine, 45 Prospect Nauki, 03028 Kyiv, Ukraine

Received 24 November 2001; accepted 15 December 2001 Communicated by L.J. Sham

Abstract In the work finding optical phonon energies from photoluminescence spectra and interpretation of their dependence on quantum well width are carried out. Remarkable decreasing both GaAs and AlAs phonon energies has been observed. It is shown that this dependence differs noticeably from that obtained from Raman spectra. The possible origin of this phenomenon is considered.  2002 Published by Elsevier Science B.V. PACS: 78.66.-w; 78.55.Cr; 63.22.+m Keywords: Short-period superlattice; Photoluminescence; Longitudinal optical phonon; Confined mode; Γ –X transfer

1. Introduction

Phonon parameters determine such important phenomena as carrier scattering, phonon-enhanced recombination losses (in particular, nonradiative recombination), spontaneous and stimulated luminescence, coupling parameters in superconductivity, properties of nonequilibrium plasma (expansion, some of nonlinear properties), etc. There are many works where the phonon characteristics in superlattices (SL) were investigated by methods of far-infrared and Raman spectroscopies (see, e.g., [1–7]). The method of the photoluminescence (PL) spectroscopy was less used than the above ones. In particular, in a set of previous

* Corresponding author.

E-mail address: [email protected] (T.R. Yang).

works a substantial decreasing energy for the surface and quantum well (QW) systems and a subsequent increasing the electron–phonon interaction probability were clearly demonstrated [8–11]. On the other hand, there are many works (see, e.g., [12–15]) where the importance of the phonon-assisted Γ –X transfer in GaAs/AlAs SLs has been revealed. Some physical phenomena where a consideration of this transfer is necessary are noted in the work [15]. It is reasonable to predict substantial peculiarities for the phonon parameters in different types of quantum-size structures, especially in short-period superlattices, where we have a continuous transition from direct-gap SLs (so-called SL-I) to quasi-directgap SLs (SL-II*) and, further, to strictly indirect-gap ones (SL-II). The main goal of the work is finding optical phonon energies E ph from PL spectra and interpretation of their dependence on n. As we shall see later, this de-

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pendence differs noticeably from that obtained from Raman spectra.

2. Experiment (GaAs)n /(AlAs)n SLs were grown by the molecular beam epitaxy method. The widths of QW and barriers were in the range n = 1–8, 10, 13–15 monolayers (ML) (1 ML = 0.283 nm). For comparison we will use some literature data (in particular, obtained in [16]). The main experimental technique was low-temperature photoluminescence. The samples were excited with 514.5 nm line of Ar+ laser. PL spectra were recorder at 4.2 K using a monochromator and a cooling photomultiplayer. PL measurements enable one to provide with a good accuracy data on the transition energies of free and bound excitons, phonon frequencies, whereas the intensities of exciton line I0 and its phonon replicas Ir depend on the degree of electron–phonon interaction (which means the probability of phonon capture during radiative recombination process), as well as on the changing of k-vector on the interface states and microcorrugation. The typical PL spectra of (GaAs)n /(AlAs)n SLs with different QW and barrier widths are presented, in particular, in [11–14]. According to these data, energy of exciton transition monotonically increases with decrease of well width. For SLs with n > 3 the most intensive line (which is labeled A in Ref. [14]) is due to zero-phonon recombination of quasi-direct localized excitons consisting of Γ heavy holes of GaAs and Xz electrons of AlAs. Besides two well-resolved weak lines (labeled B and C) on the low-energy side of spectrum are connected with GaAs and AlAs LO phonons, respectively. Under increasing of the QW width we have the transition from indirect-gap structure II (at n < 3) to quasi-direct (II*) (3 < n < 12) and to directgap structure I at n > 12 (see Fig. 1). PL spectra remarkably differ for the SLs with n = 1, 2. Zero-phonon line A becomes even weaker compared with its phonon satellites B and C. It means that exciton transitions in these SLs are indirect and the lowest electron state is Xx,y state of AlAs [17]. In this case we are able to detect even second phonon replica despite the strong background defect PL line. Additionally, the broad background PL line appears.

Fig. 1. Schematic illustration of types of (GaAs)n /(AlAs)n structures. (LH) Light hole, (HH) heavy hole.

Fig. 2. LO phonon energies for (GaAs)n /(AlAs)n SLs as a function of n obtained with photoluminescence spectra according to [19] (squares) and [16] (triangles). For comparison, presented here are also the energies of confined phonon modes with n = 1 (open circles), n = 3 (cross-marked circles) and n = 5 (full circle) obtained from Raman spectra [2,5,22].

According to [18], it is connected with defects. From PL spectra one can calculate the phonon energies. The results are presented in Fig. 2. For comparison, here the energies of confined LO phonons are also shown which are obtained from Raman spectra. Three peculiarities in Fig. 2 have engaged our attention, namely: (i) the energies of LO phonons obtained from PL spectra are distinctly lower than these obtained from Raman spectra; (ii) an energy jump in the dependence Eph (n) of AlAs takes place at n = 3; (iii) an energy jump in the dependence Eph (n) of GaAs takes place at n = 12.

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Below we will attempt to account for these phenomena from some rather general reasoning.

3. Interpretation It has been known (see, e.g., [20]) that matrix element of the phonon-assisted electron transition from the state with the wave function ψ1
to that with the wave function ψ2 is proportional to ψ2∗ ϕψ1 d r, where ϕ is the potential of the electron–phonon interaction. In the case of electron transitions in SLs from the X point to Γ point of the Brillouin zone
appropriate matrix element may be specified as ψΓ∗ ϕψX d r. For nonexciton states corresponding wave functions are defined as [15]    z)FΓ (z) exp i kρ , ψΓ ∝ UΓ (ρ, (1)   π ψXx ∝ UXx (ρ, (2)  z)FXx (z) exp i kρ + i x , a   π ψXz ∝ UXz (ρ, (3)  z)FXz (z) exp i kρ + i z , a where UK (ρ,  z) is the Bloch function, FK (z) is the electron envelope function, k is the electron wave vector in the SL layers, i.e., in the (x, y) plane, ρ is the radius-vector in the (x, y) plane, and a is the lattice constant. For indirect-gap SLs (n = 1, 2) the electron transitions occur from the points Xx and Xy to the point Γ . For bulk material the potential of electron–phonon interaction is the same as the electron function [21] ϕ ∝ V (r ) exp(i q r),

(4)

where V (r ) is a periodic function with the period a, r is the radius-vector, q is the phonon wave vector. For SLs the potential ϕ is varied in the same manner as the electron wave function, namely the envelope function arises which describes phonons localization. It should be noted that the harmonic factor and translation symmetry of the Bloch function in the (x, y) plane in the expressions for ψΓ , ψXx and ϕ lead to the pulse conservation law. Therefore, in going from the point Xx to point Γ , electrons interact with phonons which propagate along the axis Ox and have the wave vector π/a. By this is meant that such the phonon frequency may be determined from the bulk dispersion dependence of GaAs or AlAs (see Fig. 3). For GaAs it equals

Fig. 3. Dispersion curves of LO phonons in bulk GaAs and AlAs, and LA phonons in bulk GaAs.

29.5 meV approximately [2,22]. For AlAs the LO phonon dispersion dependencies are not such uniquely determined as for GaAs (see, e.g., [2,5,22,23]). We use the dispersion dependencies obtained from Raman spectroscopy data in SLs (GaAs)n /(AlAs)n [5,23]. According to these data, the LO phonon energy at the X point of the Brillouin zone for bulk AlAs equals approximately 45.9 meV. These data agree well with our results (see Fig. 2). Consequently, for indirectgap Xx,y → Γ transfers which take place in SLs (GaAs)n /(AlAs)n with n = 1, 2, the phonons are generated that propagate along the axis Ox (Oy) and have the wave vector π/a. For quasi-direct-gap SLs (n = 3–11) the phonon generation occurs in Xz → Γ transfer. This means that the phonons generate which propagate along the axis Oz (normally to the QW surface). These are the confined phonons. Experimental values of their energies taken from the works [2,5,22] are given in Fig. 2. These values are extracted from Raman spectra, i.e., the LO phonons with zeroth wave vector are presented (similar results can be also obtained from infrared transmission spectra using polarized light and oblique incidence; see, e.g., [6]). It should be noted that in Raman spectra the confined LO phonon modes of lowest order having highest energies are best pronounced. Confined modes of higher order have lower energies. So for SL (GaAs)5 /(AlAs)5 the confined modes localized in the GaAs layers have

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energies 36.03, 35.5, 34.5, 32.9 and 30.6 meV as the mode index changes from 1 to 5, respectively. When analyzing experimental data, the dispersion of confined modes has to be also taken into account. Xz → Γ transfer generates the phonons which propagate along the axis Oz with the wave vector π/b where b is the SL period in a direction of the axis Oz. By this is meant that the energies of the confined LO phonons interacting with electrons have to be lower than these derived from Raman spectra. For the case of quasidirect-gap Xz → Γ transfers it is hard to make a certain conclusion on the energy of phonons interacting with electrons. Mathematically this is due to the fact that in the expression for the potential of the electron– phonon interaction ϕ, Eq. (4), the harmonic factor and translation symmetry with the period a in the z direction disappear. At the same time, the availability of the harmonic function exp(iπz/a) in the integrand leads to the filtration of another integrand factor at the space frequency π/a. Since several terms are presented in the integrand which contain harmonic signals with the space frequency π/a, the situation becomes more complicated. As a result, the problem on the quantity of the electron interaction with a certain confined phonon mode in the process of Xz → Γ transfer remains unsolved especially for short-period SLs. It should be noted that at low temperatures exciton transitions are most probably to be. Since in the process of light absorption indirect-gap excitons can be generated with the participation of phonons only [24] our preceding reasoning on the phonon energies and wave vectors have to be also true for excitons. Different behavior of the dependence of the LO phonon energy on n for GaAs and AlAs has engaged our attention (particularly striking is this difference at small n). One can assume that the band in the photoluminescence spectra identifying as LO phonon of GaAs is caused by LA phonons too. Indeed, the AlAs LA phonon energy at q = π/a can reach 27 meV [13, 14] (see Fig. 3). We believe that this LA phonon is capable to change the position of phonon replicas in the experimental PL spectra. It would appear reasonable that for confined LO phonon modes localized in the AlAs layers the modes of the lowest order are the most strongly interacted with electrons. Because of this, a jump takes place in passing from the indirect-gap SLs to quasi-direct-gap ones at n = 3. A subsequent rise of the phonon en-

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ergy with n can be caused by the rise of the energy of the confined LO modes with n [14]. The direct-gap transitions are inherent in SL-I (n
12). Similar transitions generate the phonons with q = 0. Most likely these are LO phonons, which occur in the process of Raman scattering in the bulk GaAs (with the energy 36.3 meV) and AlAs (with the energy 49.7 meV). Consequently, according to our assumptions, for the case of the quasi-direct-gap transitions the interaction of electrons with confined LO phonon modes localized in the AlAs layers is strongest for the modes of lowest order. For LO modes localized in the GaAs layers the situation gets complicated by the possible electron interaction with LA phonons localized in the layers AlAs having the wave vector π/a. This leads to the energy jump in passing from the indirect-gap SLs to quasi-direct-gap ones (n = 3) that is observed in the case of LO phonons localized in the AlAs layers. In going from the quasi-direct-gap SLs to direct-gap ones (n = 12) an energy jump is also possible. However, it is observed only for LO phonons localized in the GaAs layers. We believe that in AlAs layers this does not take place because in these layers electrons interact mainly with first (lowest) confined mode. At the same time, at q = 0 this mode frequency coincides in fact with that of the bulk LO phonon.

4. Conclusion In this Letter we present the data of systematical investigations of LO phonons for different types of symmetrical short-period GaAs/AlAs SLs. The emphasis is on the LO phonon frequencies derived from low-temperature photoluminescence spectra. In particular, we show that the peculiar behavior of the dependence of the LO phonon energy on the QW (barrier) widths is related to Γ –X transfer taking place in the indirect-gap and quasi-direct-gap SLs. One question left unanswered may be formulated as follows: What is the value of the electron–phonon interaction when the phonons are the confined ones in short-period SLs? Here we have considered the Eph (n) dependence only. One can outline also some close problems which are in sight. The first problem is to investigate the Ir (n) dependence, i.e., to consider the intensity of phonon replicas as a function of n. Earlier [25] we observed a noticeable increase of the reflectance value at maxi-

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mum in the AlAs vibration range as n increased from 1 to 4. We related this phenomenon to interface broadening whose role becomes more important as the SL period decreases. On the other hand, the intensities (probabilities) of the phonon-assisted Γ –X transfer are considered in [15,26]. It is shown that that interface broadening lowers the Γ –X transfer probability. One can expect that such the dependence can give an additional information on processes accompanying Γ –X transfer. The second problem is to investigate the width of phonon replicas in PL spectra as a function of n. Some physical mechanisms of exciton linewidh broadening in QWs and experimental data are reported, in particular, in the works [10,27]. However, successive measurements are necessary to investigate the problem as a whole.

Acknowledgements We are grateful to V. Kochelap and B. Glavin for helpful discussions, as well as to V. Litovchenko kindly providing with experimental data.

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