Journal of Luminescence 87}89 (2000) 535}537
The valence-band coupling e!ect on Fano pro"les of magnetoexcitons in semiconductor quantum wells Ken-ichi Hino* Institute of Materials Science, University of Tsukuba, Tsukuba, Ibaraki 305-8573, Japan
Abstract The Fano pro"le of a magnetoexciton in a quantum well is investigated, based on the diabatic three-channel model. Both the Coulomb coupling and the valence-band coupling have conspicuous e!ects on the line-pro"le. The former coupling is responsible for the prominent asymmetric pro"le, having both a peak and a dip characteristic of the Fano resonance, whereas the latter one blurs the asymmetry and eventually smears the peak- and-dip structure. Such an e!ect of the valence-band coupling is analyzed in terms of the multi-channel resonant scattering theory. 2000 Elsevier Science B.V. All rights reserved. Keywords: Fano resonance; Valence-band coupling; Magnetoexcitons
The Fano resonance (FR)  is a key concept appearing in diverse "elds of physics, such as nuclear, atomic and molecular physics. It results from interference of a bound state with a continuum in which this state is embedded, and the associated spectral pro"le exhibits conspicuous asymmetry consisting of both a peak and a dip. The degree of asymmetry is, in general, determined by the magnitude of coupling. Recently, the existence of the FRs of quasi-two-dimensional (2D) excitons in semiconductor quantum wells and superlattices has been con"rmed [2}4]. In the present paper, we examine the FR phenomena of quasi-2D magnetoexcitons in the type-I semiconductor quantum well such as GaAs/AlV Ga\V As. The excitons are con"ned in the layer plane (the xy-plane) by the application of a magnetic "eld perpendicular to it as well as in the lamination direction of crystal (the z-axis). This e!ect of quantum con"nement makes all excitonic energy levels discrete. Both a mode of motion in the xy-plane and that in the z-direction are mutually coupled via the Coulomb interaction and the valence-band (VB) coupling. The FR under consideration results from the interaction between a hydrogenic bound level and
* Corresponding author. Fax: #81-298-53-4994 . E-mail address: [email protected]
a bunch of the Landau levels with energy-level intervals small enough to be regarded as quasi-continuum. It is understood that the Coulomb coupling between excitonic states with an in-plane angular momentum of the s-symmetry gives rise to a prominent asymmetric Fano pro"le, since only a state with the s-symmetry is optically active . As is well known, on the other hand, the VB coupling, which mixes excitonic states with di!erent in-plane angular momenta, causes an increase of binding energies, and in addition it leads to a signi"cant increase in an oscillator strength of an exciton with the p-symmetry for an optically forbidden transition (see for instance Ref. ). However, it seems that details of this e!ect on the Fano pro"le remains unnoticed, aside from Ref.  making a brief mention of its importance for accounting for details in experimental spectra, and Ref. , which showed an appearance of double structures in absorption spectra due to Fano-related interference caused by the VB coupling. The purpose of the present article is to show the VB e!ect on the Fano pro"le of quasi-2D magnetoexcitons in qualitative manners based on the diabatic three-channel model. The atomic units (a.u.) are used throughout unless otherwise stated. The method of the adiabatic expansion is introduced to problems of the quasi-2D exciton in Ref.  by choosing its adiabatic parameter as o, that is the distance
0022-2313/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 2 3 1 3 ( 9 9 ) 0 0 2 8 2 - 3
K.-i. Hino / Journal of Luminescence 87}89 (2000) 535}537
between an electron and a hole projected onto the xyplane. The validity and its e!ectiveness in several aspects are also discussed in it. Based on the diabatic expansion, which is mathematically equivalent to the adiabatic one, the kth spinor component WI of the envelope function satisfying the excitonic Luttinger}Hamiltonian associated with the present problem is given by [8,9] WI(o, X)"(o\ UI(o; X) t (o), (1) ? ? ? where X represents all excitonic coordinates but o and UI is termed as the ath (diabatic) channel function. In ? short, the ath channel corresponds to the ath sub-level of a mangetoexciton resulting from con"nement by the square well in the z-direction. The energy of this sub-level is denoted as eX. The present problem is approximated ? by the three-channel coupled equations for +t (o), (a"1}3) in the following : ? (E!h )t "< t #< t , (E!h )t "< t #< t , (E!h )t "< t #< t , (2) with E an energy of magnetoexciton. For simplicity, h is ? assumed to be a hydrogenic radial-Hamiltonian under a magnetic "eld with an excitonic mass k . The potential ? energy v incorporated in it is of the form ? 1 o m m!1/4 v (o)" ? ! # # ? #eX, ? ? 2k o eo 8k c 2k c ? ? ? (3) where m is an in-plane angular momentum quantum ? number, e is a static dielectric constant and c is the magnetic length de"ned as c"h/(2peB) with e, h and B the elementary electric charge, the Planck constant and the magnetic #ux density, respectively. It is assumed that m , m and m are set equal to 1, 0 and 0, corresponding to p-, s- and s-symmetries, respectively, and that the channel 1 has a heavy-hole character and the channels 2 and 3 have a light-hole character, with eX(eX(eX. The Coulomb coupling < couples channels 2 and 3 with the same in-plane angular momentum, and the VB mixing < couples channels 1 and @ b with di!erent in-plane angular-momenta with b equal to 2 or 3. At least three channels are needed to incorporate both the Coulomb coupling and the VB mixing simultaneously into theory. Fig. 1 shows a schematic diagram of v (o) versus o for the present model. ? The Landau}Zener (LZ) approximation is "rst applied to a problem of sequential curve-crossings at o and V o. According to this, the VB couplings, < and < , V are considered constant in the vicinity of the respective crossing points with < "< , and further the LZ @ @ wave functions are given in analytic and tractable forms.
Fig. 1. The schematic diagram of diabatic potentials v (o) (a"1}3) versus o for the three-channel model. o and ? ? o are classical turning points. See the text for meanings of other notations appearing herein.
By employing them as basis functions, Eq. (2) is cast into the standard algebraic eigenvalue equation. Moreover, the additional approximation is made on it that a matrix element 1< 2 of < with respect to LZ wave func tions, is regarded as a constant parameter . Fig. 2 shows calculated absorption spectra I(E) at B"0.08 T for three sets of coupling parameters, +< , < , 1< 2,. The value of I(E) is proportional to the associated oscillator strength. The material parameters for GaAs are employed. A state of magnetoexciton is designated as ns+a,, meaning the nth s-state supported by v . ? In Fig. 2(a), where the VB couplings are "xed as relatively small, the Fano pro"le with a prominent peakand-dip structure is discernible around the energy position of 1s+3,, say, E . This is the very result from interference of 1s+3, with the degenerate quasi-continuum pertinent to the channel 2, generated by the Coulomb coupling. The spectral dip with no absorption is termed as a window, which is a unique character of the FR caused by the interaction between one bound state and one continuum state. On the contrary, as is shown in Fig. 2(b), where the VB couplings are increased and the Coulomb coupling is reduced from those of Fig. 2(a), the e!ect of < does not cause any asymmetry even if 1s+3, is embedded in the quasi-continuum relevant to channel 1. The e!ect of < is found weak due to a small con tinuum}continuum interaction between channels 1 and 2, though it is not shown here. In Fig. 2(c), the VB couplings become larger than those of Fig. 2(a) and the Coulomb coupling becomes stronger than that of Fig. 2(b). The e!ect of < blurs out the asymmetry around E which stands out in Fig. 2(a). On the other hand, the asymmetric pro"le, which is missing in Fig. 2(b), is somewhat retrieved due to the stronger Coulomb coupling. Next, let us "gure out the provided results that the VB coupling smears the asymmetry and the window in the
K.-i. Hino / Journal of Luminescence 87}89 (2000) 535}537
Fig. 2. The calculated absorption spectra I(E) of magnetoexcitons at B"0.08 (T) versus E. Each value of I(E) is connected by lines for a guide to eyes. Here, eX (a"1}3) are equal to 1.0, ? 2.0 and 3.0 (10\ a.u.), respectively. 1< 2 and < (a.u.) are set in respective "gures equal to (a) 1;10\ and 4;10\; (b) 2;10\ and 3;10\; (c) 1;10\ and 3;10\. In every "gure, < equals < . For more detail of notations, see the text.
Fano pro"le around E . It is speculated that this FR can be interpreted from the point of view of the multi-channel resonant scattering theory, which is followed by the corresponding FR at B"0 . In this case, there are two choices of an incoming boundary condition imposed on a solution for the FR, since the two channels 1 and 2 are open and 1s+3, is embedded in both continua at the same time. Contribution to I(E) from the open channel c and the associated dipole matrix element are denoted by IA(E) and dA(E), respectively. The gross amount of contribution to I(E) is the superposition of the two. Note that the wave function t does not a!ect I(E) since it is optically inactive and t (0)"0.
For c"2, both the incident-channel wave function t and that of the bound state t dominate d(E), and their interference results in the asymmetric pro"le in I(E). The spectral width depends mostly upon < . As to c"1, the incoming scattering wave function t con tributes to d(E), instead of the incident wave function t which has no in#uence on it. However, its contribu tion is proportional to the transition matrix element ¹\, corresponding to the inelastic scattering from the channel 1 to the channel 2, and the value of ¹\ is considered negligibly small because of the aforementioned weak e!ect of < . As a result, just the closed channel 3 a!ects d(E) and it causes little interference with the open channel 2 in I(E). Thus, the pro"le leads to the Breit}Wigner shape without asymmetry and window. Its spectral width is mostly due to < . According to this speculation, the spectral pro"le I(E) for the quasi-2D magnetoexciton is, in general, the superposition of the symmetric Breit}Wigner pro"le I(E) upon the asymmetric Fano pro"le I(E). In the case of a weak VB coupling, I(E) dominates over I(E), giving rise to the asymmetry and the window as seen in Fig. 2(a). On the other hand, in the case of a small Coulomb coupling, I(E) smears the asymmetric pro"le of I(E), as seen in Fig. 2(b). The case of Fig. 2(c) is intermediate between the two extremes. To summarize, the VB coupling as well as the Coulomb coupling has the decisive e!ect on the spectral pro"les of magnetoexciton. This suggests that an observed Lorentzian pro"le is thought due in part to this e!ect, not always due to inhomogeneous broadening. This statement would be con"rmed by close comparison of experiments with sophisticated theories with switching the VB coupling on and o!. This research is "nancially supported by Grant-in-Aid for Scienti"c Research (C) from the Ministry of Education, Science, Sports and Culture of Japan.
References         
U. Fano, Phys. Rev. 124 (1961) 1866. D.Y. Oberli et al., Phys. Rev. B 49 (1994) 5757. S. Bar-Ad et al., Phys. Rev. Lett. 78 (1997) 1363. C.P. Holfeld et al., Phys. Rev. Lett. 81 (1998) 874. U. Ekenberg, M. Altarelli, Phys. Rev. B 35 (1987) 7585, and references therein. R. Winkler, Phys. Rev. B 51 (1995) 14 395. G. Rau et al., Phys. Rev. B 58 (1998) 7210. K. Hino, J. Phys. Soc. Japan 67 (1998) 3159. K. Hino, J. Phys. Soc. Japan 69 (2000) in press.