Physica E 7 (2000) 250–254
Intersubband electro-absorption and retardation in coupled quantum wells: the role of interface scattering R. Kapona , N. Cohena , V. Thierry-Miegb , R. Planelb;1 , A. Sa’ara; ∗ a Department
of Applied Physics, The Fredi and Nadine Hermann School of Applied Science, The Hebrew University of Jerusalem, Jerusalem 91904, Israel b Laboratoire de Microstructures et Microelectronique-CNRS, 196 Avenue H. Ravera, BP107, 92225 Bagneux, France
Abstract In this work we present a systematic experimental study aimed at resolving the various contributions to electro-optical modulation in a multiple coupled quantum wells structure. Using a set of eight-cross=parallel polarizer–analyzer measurements we were able to resolve the spectral dependence of the DC electric- eld-induced absorption and phase-retardation due to intersubband transitions. The results of our experiment were tted to a model that allows all quantum properties of the structure to vary with the external DC electric eld and estimate the contribution of each term to the overall modulation. The experimental results suggest that, apart from the Stark shift of the energy levels, a major contribution to electro-optical modulation comes from line width modulation. We propose a model that correlates this eect with alloy disorder and interface roughness scattering that gives rise to electron dephasing. The larger degree of electron localization near the interfaces in the presence of a DC electric eld is responsible for this eect. ? 2000 Elsevier Science B.V. All rights reserved. Keywords: Electro-absorption; Electro-retardation; Intersubband transitions; Interface roughness
The large second-order optical nonlinearities associated with intersubband transitions (ISBTs) in quantum wells (QWs) can be utilized to develop a new class of infrared (IR) nonlinear devices. In particular, IR electro-optical (EO) modulators that are based on these transitions were extensively investigated [1–7]. These modulators have the advantage of being compatible with other ISBT devices such as QWIPs and quantum cascade lasers, for integration purposes. ∗ Corresponding author. Fax: 972-2-5663878. E-mail address: [email protected]
(A. Sa’ar) 1 This paper is dedicated to the memory of Richard Planel, deceased 29 June 1999.
In this work we investigate experimentally EO modulation in an asymmetric coupled quantum wells (CQWs) structure. In a previous work  we found that this structure should deliver ecient EO modulation at the IR wavelengths. However, a careful analysis of the modulation process is essential to resolve both electro-absorption (EA) and electro-retardation (ER) in our sample since both the amplitude and the phase of the optical eld are aected by the DC-electric eld thus altering the state of polarization of the outcoming IR beam. The structure used for our experiments consists of 50 periods of CQWs. Each period consists of two
1386-9477/00/$ - see front matter ? 2000 Elsevier Science B.V. All rights reserved. PII: S 1 3 8 6 - 9 4 7 7 ( 9 9 ) 0 0 2 7 7 - 5
R. Kapon et al. / Physica E 7 (2000) 250–254
the sample were polished in 45 in the commonly used multi-pass waveguide geometry  to allow optical transmission The experimental set-up used to measure EO modulation is shown schematically in Fig. 2. IR-radiation from a black body source is focused onto the sample facet using ZnSe lenses. The sample, which is held at a constant temperature of 77 K inside an optical cryostat, is placed between a polarizer (P) and analyzer (A) whose angles can be rotated. In addition a quarter-wave plate (=4) is placed in the optical path. Radiation that emerges from the sample is collected using another ZnSe lens onto a 18 spectrometer followed by a HgCdTe detector. Fig. 1. FTIR linear absorption spectrum of our sample showing the Two types of measurements were made. In one type |1i → |3i intersubband transition at 137 meV. The inset shows the band structure of the sample together with the calculated energy of experiments a square wave voltage between 0 and levels and wave functions. V was applied across the sample at a frequency of 4 kHz and the dierential transmission through the sample, T , was measured using a lock-in ampli er. In wide, respectively, sepaGaAs QWs, 60 and 20 A the second set of measurements the linear transmis thick Al0:4 Ga0:6 As barrier. A 375 A rated by a 15 A sion, T , was measured using a chopper to modulate Al0:4 Ga0:6 As barrier, modulation doped (n-type) at the 11 −2 the incoming beam while the outcoming signal was center to a level of n2D = 4 × 10 cm , separates measured, again, using a lock-in ampli er. The frethe periods from each other. The entire structure was quency of modulation was the same for both expergrown on the top of a semi-insulating (1 0 0) GaAs iments to eliminate eects of frequency response of thick heavily substrate with top and bottom 5000 A 18 −3 the measuring system. doped n-GaAs layers (n = 1 × 10 cm ). It can be shown that full characterization of a The potential structure together with the allowed given anisotropic crystal requires a set of nine energy levels and wave functions, that were calculated measurements  at dierent orientations of the using a numerical code that solves the Ben Daniel– polarizer=analyzer=quarter-wave plate. However, in Duke and the Poisson equations self-consistently, are our semiconductor structure, the ISBT selection rules shown in the inset of Fig. 1. We found that the strucdictate the principal axis  so that a set of only ture has three subbands with transition energies of six measurements is sucient to characterize the 140 meV for the |1i → |2i ISBT and 200 meV for the structure. We denote each con guration of our ex|1i → |3i ISBT at 77 K. Linear-polarization-resolved perimental system by three parameters (p ; A ; 0 ) FTIR absorption measurements, shown in Fig. 1, where p and A are the angles of the polarizer and reveal a strong absorption line at 137 meV with a analyzer relative to the extraordinary axis (growth full-width at half-maximum (FWHM) of 10 meV. direction) and 0 is the retardation introduced by the This is in good agreement with our energy level quarter-wave plate. In Ref.  we showed that one calculations. The sample was processed into a can use the following relations to deduce the induced 2 2 × 5 mm rectangular mesa structure, using standard absorption and induced retardation from the following photolithography and wet chemical etching. Finally, set of measurements: ohmic metal contacts of AuGe=Ni=Au were de ned at 1 T (0; 0; 0) the bottom and the top mesas. Two parallel edges of = − (2) V; (1) T (0; 0; 0) 2 h i T (0; 0; 0) h i −2 T ; ; − T ;− ; − T ; ; −T ;− ; 4 4 2 4 4 2 2T (0; 0; 0) 4 4 2 4 4 2 i h (2) = (2) V; ; ;0 − T ;− ;0 T 4 4 4 4
R. Kapon et al. / Physica E 7 (2000) 250–254
Fig. 2. The experimental set-up: (P) input polarizer, (=4) quarter wavelength plate; (A) output analyzer-polarizer. p and A are the angles of the polarizer and the analyzer relative to the extra-ordinary direction, respectively.
where V is the applied voltage, and (2) and (2) are the electric- eld-induced EA and ER coecients, respectively. In our analysis we plotted (2) V and (2) V versus the applied voltage, for each photon energy, and tted the data to a linear approximation (see Fig. 2 of Ref. ). The slope of the resulting lines are (2) and (2) , respectively. The open circles in Fig. 3(a), show the induced EA. A rst positive peak at 125 meV and a second negative peak at 137 meV are observed. While the second peak at 137 meV is very close to the resonance of the linear absorption, the rst peak at 125 meV is shifted to a signi cantly lower energy. The results for the induced ER are shown in Fig. 3b in open circles. Here, the rst positive ER peak appears at 133 meV while the second negative peak appears at 142 meV. In order to correlate the experimental results to the physical mechanisms responsible for EO modulation we followed the approach described in Ref. . A rst-order expansion (with respect to the DC-electric eld) of the linear susceptibility, (1) (!) = 21 =( − 21 − i 21 ); yields (1) 9(1) 9(1) 9 (2) (!)= 21 + 21 + 21 ; 921 9 21 9 21 (3) where (2) =
21 = ; 921 ( − 21 − i 21 )2
9(1) 1 ; = 9 21 ( − 21 − i 21 )
Fig. 3. (a) EA versus the photon energy; (b) ER versus photon energy. Circles denote the measured values while the solid lines represent the best t of the experimental results to the model.
9(1) i 21 = ; 9 21 ( − 21 − i 21 )2
where (2) (!) is the second-order EO susceptibility,
21 is proportional to the transition dipole matrix elements times the population dierence of the lowest two subbands, 21 is the transition frequency and 21 is the line width of the transition. In the present analysis we took into account line-width modulation that was not considered in Ref. . In Fig. 4 we plot the real (Fig. 4b) and imaginary (Fig. 4a) parts of each term in Eq. (4) normalized
R. Kapon et al. / Physica E 7 (2000) 250–254
Fig. 4. Spectral behavior of: (a) the imaginary; (b) the real part of the EO susceptibility for modulation arising from changes in
21 (solid line), 21 (dashed line) and 21 (dotted line).
to their maximum value. Note that each of the contributions to modulation yields a dierent spectral dependence. Furthermore, since EA is related to the imaginary part of the susceptibility and ER to the real part of the susceptibility, we can t the experimental results for both the ER and the EA coecients to Eq. (4). The tting procedure yields the relative contribution of each term in Eq. (4) to the experimental results. The solid lines in Fig. 3 show the results of the t where the values of 21 ; 21 and 21 in the absence of a DC-electric eld were taken from the linear absorption measurements. The tting procedure yields: = 4 × 10−3 =V, 1=V ( 21 = 21 ) ∼ = 7:6 × 1=V (21 =21 ) ∼ −2 = 1 × 10−3 =V. Thus, 10 =V and 1=V ( 21 = 21 ) ∼ close to resonance, EO modulation in our structure is mainly due to line-width broadening. Let us now compare the experimental results to a model based on a numerical solution of the Ben Daniel–Duke and Poisson equations for our structure. We treated the biased QW as a new structure and numerically calculated 21 and 21 as a function of the DC-electric eld, F. First-order expansion of the form 921 F + ···; 21 (F) = 21 (0) + 9F 9 21
21 (F) = 21 (0) + F + ··· 9F
was used to derive the following relations for the modulation coecients 21 21 9 ln 21 9 ln 21 F and F: (6) = = 21 9F
21 9F The second term in our expansion is related to quantum interference and carrier density modulation as discussed in Ref. . However, since in our experiment 2 − 1 ≈ 137 meV/kT ≈ 7 meV we expected carrier density modulation to be negligible. Line-width broadening may originate in several processes that give rise to electron dephasing [13,14]. In our structure, because of the large number of interfaces and the thin intermediate barrier, we assign line-width broadening to interface roughness and alloy disorder scattering at the interfaces. Following Ando et al.  we write line-width broadening due to interface roughness as follows: Z 9V 2 ; where Fe = d z| (z)|2
= KFe 9z P ∼ 2 (7) = V0 | (zj )| ; j
where the summation is over all interfaces, V0 is the conduction band discontinuity at the interface, (z) is the envelope wave function and for each interface at z = zj we took 9V=9z = V0 (z − zj ). Hence, we conclude that the strength of interface roughness scattering has a quadratic dependence on the probability to nd the electron at a given interface. Therefore one should expect to nd a weak contribution of this scattering process in a rectangular QW where the ground envelope state tends to vanish near the interfaces and a large contribution for QW structures with many interfaces. In our CQW a large probability to nd the electron near the interfaces is expected, particularly for the interfaces of the thin intermediate AlGaAs barrier. Taking now, again, the envelope-state to be a function of the DC-electric eld and applying the linear expansion procedure (as in Eqs. (5) and (6)) for line-width modulation yields ∼ 9 ln(| |4 ) = : (8)
9F F=0 Eqs. (6) and (8) can be now used to calculate the various modulation coecients. Applying the numerical code for our structure yields: = 6 × 10−3 1=V; 1=V ( 21 = 21 ) ∼ =1 × 1=V (21 =21 ) ∼ −1 = 8:5 × 10−4 1=V in 10 1=V and 1=V (( 21 = 21 ) ∼
R. Kapon et al. / Physica E 7 (2000) 250–254
reasonably good agreement with our experimental results. In conclusion, we have presented an experimental technique for measuring the EO modulation based on ISBTs in QWs. Our experiment reveals that both electro-refraction and electro-absorption contribute to EO modulation. Furthermore, we have found that in QW structures with several interfaces the modulation stems mostly from line-width broadening under the application of an external electric eld. References  P.F. Yuh, K.L. Wang, IEEE J. Quantum Electron. 25 (1989) 1671.  E.B. Dupont, D. Delacourt, M. Papuchon, IEEE J. Quantum Electron. 29 (1993) 2313.  R.P.G. Karunasiri, Y.J. Mii, K.L. Wang, IEEE Electron Dev. Lett. 11 (1990) 227.
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