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Light-absorption effect on Bragg interference in multilayer semiconductor heterostructures Alexey V. Kavokina) and Mikhail A. Kaliteevski A. F. Ioffe Physico-Technical Institute, Russian Academy of Sciences, 26, Polytechnicheskaya, 194021 St-Petersburg, Russia, CIS ~Received 22 August 1995; accepted for publication 9 October 1995! The transfer matrix method has been employed to study the effect of light absorption on optical spectra of GaAs/AlAs Bragg reflectors. Substantial distortion of the Bragg plateau due to the absorption in GaAs layers is found. The saturation value of the reflection coefficient of the infinite Bragg reflector is shown to change nonmonotonically with the absorption coefficient showing a minimum. The particular case of low-temperature light reflection from the Bragg mirror in the vicinity of the exciton resonance frequency in GaAs layer is considered. The dramatic enhancement ~by a factor of 50! of the exciton resonance amplitude in comparison with the reflectivity spectrum of GaAs is found in case of GaAs/AlAs Bragg structure covered by l/2-thick GaAs caplayer. © 1996 American Institute of Physics. @S0021-8979~96!01302-X# INTRODUCTION Bragg reflectors ~BRs! are widely used in various types of optical devices such as vertical cavity surface-emitting lasers and light-emitting diodes, waveguides, reflectivity modulators, and solar cells.1–5 Usually the transparent materials are used to construct BRs in order to exclude any effects connected with light absorption in Bragg mirrors. Evidently, the condition of the absence of absorption in the mirror substantially limits the useful spectral range of the Bragg reflector. In this article we present the theoretical study of the effect of light absorption on optical properties of Bragg reflectors constructed from multilayer semiconductor heterostructures such as GaAs/AlAs. Both cases of intersubband and excitonic absorption are considered. Let us consider a periodic structure containing N double layers consisted of layers of semiconductor materials A and B ~see Fig. 1!. In general, the refractive index is complex in both the materials A and B, n B5ñ B1ik B , ~1! where ñ A,B and k A,B are real and imaginary parts of the corresponding refractive index. In the following we will use the transfer-matrix method.6,7 The natural boundary conditions for the light wave at each heteroboundary can be written as a condition of continuity of the vector S D t 115cos~ q Aa ! cos~ q B b ! 2 t 2252 nA sin~ q Aa ! sin~ q B b ! , nB nB sin~ q Aa ! sin~ q B b ! 1cos~ q Aa ! cos~ q B b ! , nA nA cos~ q Aa ! sin~ q B b ! , t 125cos~ q B b ! sin~ q Aa ! 1 nB t 2152 BASIC EQUATIONS n A5ñ A1ik A , l is the wavelength of light in vacuum. The factor q 21 A is introduced to provide the same dimensionality of the components of the vector F. The transfer of the vector F from the interface between B and A layers by the period d is a result of the action of the transfer matrix t̂, with components ~3! nB cos~ q Aa ! sin~ q B b ! 2cos~ q B b ! sin~ q Aa ! , nA where a,b are the thickness of layers A, B, respectively. The eigenvalues of the matrix t̂ are F ~6! t 111t 22 6 5 2 FS D G t 111t 22 2 21 2 1/2 [exp~ 6iQd ! , ~4! where Q plays the role of an effective wave vector of light in a multilayer structure. The corresponding eigenvectors can be written as Ex F5 1 qA ]Ex , ]z ~2! where E x is the x component of the electric field ~for definiteness we consider a linearly polarized light!, the z direction is normal to layer planes, and q A~B!5n A~B!~2p/l!, where a! Electronic mail: [email protected] J. Appl. Phys. 79 (2), 15 January 1996 FIG. 1. The schema of the structure considered: N double layers consisted of A and B layers each one. r and t are the amplitude reflection and transmission coefficients. 0021-8979/96/79(2)/595/4/$6.00 © 1996 American Institute of Physics 595 SD SD 1 , x 1 , y ~5! F G F G 11r i ~ 12r ! with x5~F (6) 2t 11!/t 12, y5~F (2) 2t 11!/t 12 . To find out the amplitude reflection and transmission coefficients r and t of the structure under consideration one has to solve a system of two vector equations given by boundary conditions at left- and right-hand-side boundaries of the periodic structure, SD SD 1 1 n 0 5A 1B , x y nA SD ~6! SD t 1 1 n 1 5A exp~ iQNd ! 1B exp~ 2iQNd ! , x y i t n1 ~68! with n 0 and n 1 being refractive indices in the cap layer and the substrate, respectively ~see in inset in Fig. 1!. The solution of Eqs. ~6! and ~68! yields r5 ~ in 0 2n Ax !~ n Ay2in 1 ! 1 ~ in 0 2n Ay !~ in 1 2n Ax ! exp~ 2iQNd ! , ~ in 0 1n Ax !~ n Ay2in 1 ! 1 ~ in 0 1n Ay !~ in 1 2n Ax ! exp~ 2iQNd ! ~7! t5 in 0 1n Ax in 0 2n Ax 2r . in 1 2n Ax in 1 2n Ax ~8! The general formulas ~7! and ~8! can be simplified in a few important particular cases. For N→`, r5 in 0 2n Ax , in 0 1n Ax y50. ~9! At the Bragg interference condition, 2p p 2p ñ a5 ñ b5 , l A l B 2 ~10! this picture is dramatically changed. The magnitude of the reflectivity at the plateau gets lower and the oscillations disappear so that all the spectrum is substantially smoothed and only one central maximum is kept. Correspondingly, f changes its sign only one. Figure 3 shows a comparison between the experimental spectrum of an absorbing Bragg reflector8 and our theoretical calculation. The studied structure contains 10 periods of GaAs/AlAs superlattice with layer thickness a546 nm, if k A5k B50, ñ A,ñ B , well-known expressions can be obtained, r5 12 ~ n A /n B! 2N , 11 ~ n A /n B! 2N ~11! t5 2 . 11 ~ n A /n B! 2N ~12! We note, however, that the formulas ~11! and ~12! are not valid in case of nonzero absorption, and one should use Eqs. ~7! and ~8!. RESULTS AND DISCUSSION To have an idea about the magnitude of the effect of the absorption on the spectra of Bragg mirrors we have considered a structure schematically shown in Fig. 1. This is a periodic multilayer GaAs/AlAs structure surrounded by semiinfinite layers of GaAs. All GaAs layers in the structure are supposed to be absorbing unlike AlAs layers. Figures 2~a! and 2~b! shows the spectra of the squared modulus and the phase of the reflection coefficient from such a structure for different levels of absorption. The used parameters are: thickness of layers in the superlattice a~GaAs!546 nm, b~AlAs!561 nm; the number of periods N510. At the wavelength l5715 nm the Bragg interference condition is satisfied. One can see for low absorption levels the plateau and oscillating tails in uru2 spectrum. At high absorption levels 596 J. Appl. Phys., Vol. 79, No. 2, 15 January 1996 FIG. 2. Calculated ~a! squared module and ~b! phase of the amplitude reflection coefficient for GaAs/AlAs periodic structure surrounded by semiinfinite media of GaAs, N510, a546 nm, b561 nm, for different levels of absorption in GaAs: ~1! k50; ~2! k50.01; ~3! k50.05; ~4! k50.1; ~5! k50.5. A. V. Kavokin and M. A. Kaliteevski FIG. 3. Experimentally measured Ref. 8 ~solid! and calculated ~dashed and dotted! spectra of GaAs/AlAs absorbing Bragg reflector covered by GaAs cap layer. Dashed and dotted curves represent calculations taking into account and ignoring absorption in GaAs layers, respectively. b561 nm grown on GaAs substrate and covered by 88-nmthick GaAs cap layer. All GaAs layers were doped with the donor ~Si! concentration n51018. There are no fitting parameters in the calculation, while the spectral dependence of k A is taken from Ref. 9. Note that this is a case of the intersubband absorption, and excitonic effects are removed due to the high screening. Peculiarities of the systems with lowtemperature excitonic absorption are discussed below. One can see that quite a good agreement is achieved between theory ~dashed line! and experiment ~solid line!. On the other hand, the spectrum calculated neglecting the absorption in GaAs layers ~dotted line! is strongly different from the experimental curve. Figure 4 shows the reflection coefficient R5 u r u 2, when the Bragg condition ~10! is satisfied, as a function of the number of periods of the superlattice N for different values FIG. 5. The saturative value of the reflection coefficient of the infinite GaAs/AlAs Bragg reflector as a function of the level of absorption in GaAs layers. Light is reflected to GaAs media. of k A . One can see that this dependence always has a saturating character while the saturation value of the reflection coefficient R ` drastically depends on absorption. It is worth noting from Fig. 4 that the Bragg mirror containing 10 periods of a superlattice has roughly the same peak value of the reflection coefficient as an infinite Bragg mirror. The saturation value of the reflection coefficient R ` as a function of k A is shown in Fig. 5. This dependence is nonmonotonic. The initial decrease of R ` with k A manifests the negative effect of light absorption on the quality of a Bragg mirror. The minimum of R ` ~k A! corresponds to k A}l /a. For higher k A the interference effects get negligible due to strong absorption in the first GaAs layer. This first layer acts as a metallic film, which results in the increase of the reflection coefficient. An important particular case of the absorbing Bragg mirror is realized if the condition ~10! is satisfied at the wavelength close to the wavelength of the exciton resonance in one of the materials of the mirror. ~The temperature is supposed to be low enough for appearance of excitonic effects.! The reflection coefficient of such structure in the vicinity of the exciton resonance can be found by using the same formalism as earlier, bearing in mind that the dielectric constant e5n 2 in an absorbing layer can be written in the form S e ~ v ! 5 e B 11 D v LT , v 0 2 v 2iG ~13! where eB is the background dielectric constant, v0 and vLT are the exciton resonance frequency and longitudinal transverse splitting, respectively, and G is the exciton damping. Figure 6 shows the calculated reflectivity spectrum of the GaAs/AlAs BR ~N58! with layer thickness a,b such as 2p p 2p ñ Aa5 ñ B b5 , l0 l0 2 FIG. 4. Reflection coefficient of N-period-thick GaAs/AlAs Bragg reflector surrounded by the semi-infinite media of GaAs as a function of N, for different levels of absorption in GaAs: ~1! k50; ~2! k50.01; ~3! k50.05; ~4! k50.1; ~5! k50.5. J. Appl. Phys., Vol. 79, No. 2, 15 January 1996 with l05v 0c ~curve 2!. One can see that the amplitude of the exciton resonance in reflection is an order higher in Bragg structure in comparison with bulk GaAs ~curve 1!. In accordance with Ref. 10 we have taken \vLT50.1 meV, \G51 meV. The reflectivity modulation of almost 100% is A. V. Kavokin and M. A. Kaliteevski 597 Bragg plateau is only about 1 meV. On the other hand, the effect discussed in the present work takes place for all values of l0 belonging to the Bragg plateau, which is of width about 50 meV in the considered structures. CONCLUSION FIG. 6. Calculated reflectivity spectrum of ~1! bulk GaAs, ~2! Bragg reflector consisting of eight periods of GaAs/AlAs structure, and ~3! the same Bragg reflector covered by GaAs caplayer of the thickness equal to a halfwavelength of light at the exciton resonance frequency in GaAs. The Bragg interference condition is satisfied at the exciton resonance frequency in GaAs. achieved in case of a Bragg structure covered by GaAs cap layer of thickness l5l 0 /2ñ A ~curve 3!. This dramatic enhancement of the modulation is caused by destructive interference of light waves reflected by the surface and the interface cap layer/superlattice. High sharpness of the excitonic modulation of the reflection spectra of Bragg structures provides extremely high precision in detection of exciton parameters such as oscillator strength and radiative lifetime. Note that Bragg reflection from multiple-quantum-well structure near the exciton resonance frequency has been analyzed in a recent work.11 In that case, however, the enhancement of the excitonic modulation of the reflection spectrum is substantial only in the close vicinity of the wavelength corresponding to Bragg interference condition. The reason is that in a multiple-quantum-well structure the width of the 598 J. Appl. Phys., Vol. 79, No. 2, 15 January 1996 In conclusion, the light-absorption effect on optical properties of Bragg mirrors has been considered in the framework of transfer matrix method. Dramatic distortion of the reflectivity spectra of Bragg mirrors due to absorption is revealed. We have established that the peak value of the reflection coefficient from Bragg mirror depends nonmonotonically on the absorption with a minimum corresponding to the threshold between interference regime and metallic reflectance regime. The interference between exciton resonances in Bragg structures is shown to result in the giant increase of the reflectivity modulation in the vicinity of the exciton resonance, so that in particular cases the amplitude of the excitonic peak in reflection is almost 100%. 1 T. Saka, M. Hirotani, T. Kato, and H. Susava, J. Appl. Phys. 73, 380 ~1993!; H. Sugawara, K. Itaya, and G. Hatakoshi, ibid. 74, 3189 ~1993!. 2 A. Von Lehmen, T. Banwell, L. Carrion, N. Stoffel, L. 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Kochereshko, I. N. Uraltsev, D. R. Yakovlev, M. A. Ivanov, B. Ya. Meltser, and M. A. Kaliteevski, Fiz. Tech. Poluprovodn. 22, 784 ~1988! @Sov. Phys. Semicond. 22, 495 ~1988!#. 11 E. L. Ivchenko, S. Jorda, and A. I. Nesvizhskii, Fiz. Tverd. Tela 36, 2118 ~1994! @Phys. Solid State 36, 1156 ~1994!#. A. V. Kavokin and M. A. Kaliteevski