An instability in the growth of nonperiodic InGaAs/GaAs multiple quantum well samples, ordinarily of high-quality when grown with equal periods of order of half the wavelength of light in the material, leads to a dramatic microscopic, self-organized surface grating. This effect was discovered while growing quantum wells with two unequal barrier lengths arranged in a Fibonacci sequence to form an optical quasicrystal. A laser beam incident normal to the surface of the sample is diffracted into a propeller-shaped pattern. The sample surface has a distinctly cloudy appearance when viewed along one crystal axis but is mirror-like when the sample is rotated 90°. The instability results in a five-fold increase in the absorption linewidth of the heavy-hole exciton transition. Atomic force microscopy, transmission electron microscopy, and scanning electron microscopy were used to study the samples.
© 2008 Optical Society of America
The growth of self-organizing semiconductor nanostructures has been the subject of intense research activity for the promising fundamental and technological opportunities that they present. Here we report on the curious all-epitaxial growth of self-organizing, microscopic gratings on standard (001)-cut GaAs substrates by molecular beam epitaxy (MBE). Changes in surface morphology, including the formation of surface corrugations, have been reported previously in strained InGaAs single layers exceeding the critical thickness [1,2,3]. Our growth instability manifests itself in InGaAs/GaAs multiple quantum well (MQW) samples with nonperiodic or unequal spacings between QW centers. We discovered the effect accidentally while growing MQW samples whose spacings follow a Fibonacci sequence thereby forming an optical quasicrystal. When such structures are grown with GaAs QWs between AlGaAs barriers, they display new resonant excitonic-polaritonic effects [4,5] that are interestingly different from those observed in traditional periodic MQW samples [6,7]. One might expect to see the same new effects using InGaAs QWs between GaAs barriers with spacings obeying the Fibonacci sequence, and, in fact, based on many years of experience growing narrow linewidth InGaAs QWs [8,7], we tried them first. However those samples exhibit a spontaneously organized surface grating that obscures such new effects. The ridges of the surface grating grow higher and wider as more QWs are grown. This surface grating is not observed in the GaAs/AlGaAs samples, where the lattice constants of the QWs and the barriers are nearly equal. It is also rarely observed in periodic InGaAs/GaAs MQW samples with equal length barriers, even when the number of QWs is very high. It has so far always been observed with sequences of nonperiodic spacings, even when the QW indium concentration is very low. These results will interest researchers fabricating and characterizing light-emitting one-dimensional photonic quasicrystals based on excitonic resonances and perhaps those growing lines of quantum dots .
In what follows we describe the properties of these structures, which we have investigated using atomic force microscopy (AFM) and fast Fourier transform (FFT) analysis of the surface profiles, as well as transmission electron microscopy (TEM) for imaging cross sections of the structures. Our TEM investigations were performed with a Philips CM 200 FEG/ST electron microscope with an electron energy of 200 keV. Plan-view TEM samples were prepared by chemical etching from the substrate side using a 5:1 solution of NaOH (1 mol/l) and H2O2 (30%). Standard procedures  were used for cross-section sample preparation. TEM dark-field (DF) images were taken under two-beam conditions using the composition-sensitive (002) reflection close to the -, [1–10]- or -zone axes. Our AFM, manufactured by Nanoscience Instruments Inc., has nominal lateral and z resolution of 1.1 nm and 0.21 nm respectively. All of the surface roughness values reported here are root-mean-square (rms) values computed from AFM measurements over a 1600 µm2 area (40 µm by 40 µm). The FFTs were executed on the same AFM images as the surface roughness measurements. Unless stated otherwise, all of the samples reported here were grown on a standard (001)-cut GaAs substrate with a maximum tilt of ±0.1° and a 500 nm GaAs buffer layer grown on the substrate before starting QW growth.
FIB1 is a periodic MQW sample with 10 In0.04Ga0.96As/GaAs QWs with center-to-center separation of ≅114 nm, i.e. about half the optical wavelength in the material, λ/2. The thickness of each InGaAs QW was ≅8 nm; a two minute growth interruption was made before each QW, but none after. This is a standard, high quality, narrow linewidth periodic MQW sample like those that have been well known and extensively studied for many years [7,8]. The sample surface was determined by AFM to be very flat, with rms roughness of 0.63 nm. For comparison, a new, clean, pre-growth GaAs substrate revealed an rms surface roughness of 0.59 nm. A typical 100 µm2 AFM image of FIB1 is shown in Fig. 1. Figure 2 shows an FFT of the real space image for the surface of FIB1. The spot at the center of the image reveals that the sample has only very low spatial frequencies on the surface. The vertical scale on the right hand side of all the FFT images indicates the relative magnitude of the corresponding spatial frequency component, and the peak value is related to the rms surface roughness. The vertical lines near the center of the image are artifacts of the AFM scan, and indicate the direction in which the scan proceeds. TEM measurements performed on cross sections of FIB1 confirmed that the QWs were uniformly flat throughout the growth. Figure 3 shows a typical image, with the InGaAs QWs displaying dark contrast.
FIB3 is a 56 In0.04Ga0.96As/GaAs MQW sample grown like FIB1 except for the QW separations which are according to the Fibonacci sequence where the Fibonacci chain Fj contains QWs with two different separations A and B between the centers of the wells. The ratio of the optical pathlengths of B to A equals the golden mean (√5+1)/2; note that the physical length of A or B is the sum of the barrier length and the QW length (≅8 nm). For In0.04Ga0.96As wells and GaAs barriers, the two refractive indices are almost the same so the ratio of the physical lengths of B to A is also close to the golden mean. For FIB3, the short separation A is ≅82 nm and the long separation B is ≅126 nm, i.e., close to 0.36λ and 0.59λ. The Fibonacci recursion relation is found as follows: Fj+1 is formed by adding Fj-1 to the end of Fj: BABBABABBABBA… [5,4]. This structure is thus a nonperiodic, but deterministic, sequence of QWs with two unequal spacings. The same two minute growth interruption was employed as for FIB1.
Upon removing FIB3 from the MBE machine, it was immediately evident that the surface was different from the many hundreds of samples grown in our MBE machine in all previous years. At viewing angles of about 40° from normal, the surface has a distinctly cloudy appearance when viewed along one crystal axis but is mirror-like when the sample is rotated 90°. A helium-neon laser beam at 632 nm incident normal to the surface of the sample revealed strong scattering of the incident light in a propeller-shaped pattern. Additional reflection experiments with a 458 nm argon-ion laser and a 780 nm continuous-wave Titanium sapphire ring laser confirmed that the surface was acting like a grating, with an average grating period broadly centered around 800 nm. A picture of scattered helium-neon laser light from the surface of FIB7, which is representative of the scattering behavior, is shown in Fig. 4.
The quality of InGaAs QWs is often characterized by the absorption linewidth of the heavy-hole exciton. We performed low-temperature transmission measurements in order to compare the absorption linewidths of FIB3 (Fig. 5) and FIB1 (Fig. 6). Because the substrate absorption is nonzero in the vicinity of the heavy-hole resonance, it has been subtracted as shown in the figures. The shift of the exciton peak between the two samples is due to a small decrease in the QW indium concentration (still roughly 4%). The 0.6 meV absorption linewidth of FIB1 is typical for our high quality periodic QWs [7,8]. In contrast, the >3 meV absorption linewidth of FIB3 prevents the observation of the interesting resonant reflection and photoluminescence phenomena we reported for GaAs/AlGaAs Fibonacci-spaced QWs . The increase in linewidth likely results from the different thicknesses of the QW as the instability develops. AFM scans reveal that the last QW is growing on hills ≅100 nm high and ≅800 nm apart, so the slopes often exceed 10°. We speculate that this along with edge growth effects result in very different thicknesses on the ridges, in the valleys, and on the slopes, giving different quantum confinement energy shifts and thus a broad total linewidth.
AFM measurements of FIB3 confirmed a dramatic grating pattern on the surface of the sample. While a typical MQW sample grown in our machine, such as FIB1, has an rms surface roughness of less than a nanometer, the surface of FIB3 reveals an rms surface roughness of nearly 40 nm. The FFT analysis of the surface confirms a strong periodicity of the surface wave along the  crystal axis, with the grating period peaked at 800 nm. These characteristics of the surface are consistent with the visual observation of scattering at steep angles along preferential crystal axes. The AFM image and FFT of FIB3 are shown in Fig. 7 and Fig. 8, respectively. It is highly curious that the growth of nonperiodic QW spacings results in a grating on the surface that is almost periodic.
FIB4 and FIB5 were grown according to the same Fibonacci sequence as FIB3, but with 13 QWs and no growth interruptions. The results were similar to FIB3, namely an observable rotation dependent cloudiness of the sample surface, confirmed as an organized surface grating by AFM and TEM. One difference, however, was that the ridges of these two samples were neither as wide nor as long as the ridges of FIB3. Figures 9 and 10 show an AFM image and an FFT image of the surface of FIB4. The surface roughness (17.3 and 27.5 nm, respectively) was also less than on FIB3.
Figure 11 shows a scanning electron microscopy (SEM) image of the surface of FIB4, and Fig. 12 shows a TEM cross section. The latter shows that the first QW is flat, but that subsequent QWs develop a “waviness” that propagates and intensifies as more QWs are grown. This corroborates a general pattern we have observed, that with all else equal, growth of a larger number of QWs results in a more dramatic surface grating, with deeper grooves and longer and wider surface ridges.
FIB6 consists of 8 periodic QWs with 114 nm separations and with no growth interruptions. It was grown to check that equal separations still gave a smooth top surface. The result was a sample that looked normal to the eye, so the growth series was continued with unequal separations. Later, however, first TEM and then AFM measurements revealed a shallow and high-frequency surface grating. FIB6 illustrates that surface gratings are not strictly confined to samples with unequal or nonperiodic barriers, but can also be initiated by other growth factors. Figures 13 and 14 show AFM and FFT images of FIB6, respectively. The fact that the sample looks normal to the eye is explained by the frequency of the surface grating, which is too high to noticeably scatter visible light.
FIB7 is an 8 QW sample grown in the Fibonacci sequence, but with larger barriers than the previous Fibanocci samples, and three-minute growth interruptions before each QW. The motivating idea was that the growth instability could be caused by a strain problem that would be relieved by growing larger barriers. The short separation is ≅216 nm and the long separation is ≅347 nm, i.e., 0.95λ and 1.53λ, with the ratio of the two lengths again equal to the golden mean . Its barriers are thus approximately two and one half times thicker than the barriers for FIB3-5, nonetheless FIB7 has a surface grating similar to FIB4 and FIB5, with rms roughness equal to 15.5 nm and average grating period of 455 nm. Figures 15 and 16 show an AFM image and FFT image for FIB7.
Figure 15 shows again ridge structures with two different periodicities oriented approximately perpendicular to each other which can also be seen in TEM cross-section samples which are prepared along the [1–10] (Fig. 17(a)) and  directions (Fig. 17(b)). The strong contrast variations in Fig. 17(b) are related to the bending of the TEM sample which leads to local changes of the excitation of the (002) reflection. The insets display magnified sections of the region containing the bottom QWs which clearly demonstrate that even the first InGaAs QW grown on the buffer layer surface is not flat.
FIB8 is an 8 QW sample grown with separations between the QWs in the sequence S, L, S, L, S, L, …., where S≅114 nm and L≅228 nm and with three-minute growth interruptions before each QW. The idea for FIB8 was to see if the growth instability could be affected simply by unequal separations, or if there is something inherent in the nonperiodic Fibonacci sequence that is responsible for the effect. The resulting sample showed a surface grating with average period 475 nm and rms surface roughness 24.4 nm. The AFM and FFT images are shown in Figs. 18 and 19.
If the observed instability propagates and intensifies with growth, is it always present, perhaps to an extent more noticeable with unequal separations? To answer this question, we looked at several previously grown periodic MQW samples, both with and without growth interruptions. DBR34 is a 200 In0.04Ga0.96As/GaAs QW sample grown with 114 nm separations and a three-minute growth interruption before each QW. This is the largest number of periodic QWs grown on a single sample in our MBE machine, and we assume that if there is even a small intensifying surface instability it will be evident after 200 QWs. The surface, however, is flat, with an rms surface roughness of 0.8 nm. The AFM and FFT images are shown in Figs. 20 & 21.
MOD16 is a periodic 30 In0.26Ga0.74As QW sample with high indium concentration, thin barriers and no growth interruptions. The surface is not flat, but is more highly disorganized than the surface structures of the Fibonacci samples. This result, along with the result for FIB6, suggests that the lack of a growth interruption before the QWs may trigger the surface instability even with equal barrier samples like MOD16 and FIB6. The AFM and FFT images of MOD16 are shown in Figs. 22 and 23.
To find out whether the lack of a growth interruption is sufficient to trigger the surface instability, we examined a periodic 20 In0.06Ga0.94As QW sample (NMC11). These QWs were grown without growth interruptions, yet the surface is flat (0.7 nm rms), indicating that the absence of growth interruption is not by itself sufficient to trigger the observed surface instability. We also examined two other possible causes of the growth instability. We grew 13 QW Fibonacci structures on a brand new substrate (FIB9), thinking that perhaps deleterious surface effects on an older substrate could cause the observed surface instability, and on a substrate with polish tilted 1° toward the [1–10] axis (FIB15). The latter substrate was used to increase the step density because the preferential alignment of the ridges along the two <110>-type directions suggests that steps at the substrate or buffer-layer surface could contribute to the initiation of the growth instability. However, in both cases, the waviness and roughness of the sample surfaces were not affected noticeably.
In conclusion, in an attempt to grow Fibonacci-spaced InGaAs/GaAs QWs, we have discovered an all-epitaxial technique for growing self-organized gratings on the surface of MQW samples using unequal barriers. The observed growth instability is sometimes, though rarely, present on periodic structures, and may be triggered by the absence of a growth interruption before the QWs. Even with a growth interruption, however, so far the instability has always occurred if unequal barriers are grown. Furthermore, the use of unequal barriers to trigger the instability results in a more organized surface structure than when a disturbance is observed with equal barrier structures. The height, period, and length of the ridges of the grating increase with the number of layers in the structure. Although InGaAs/GaAs QWs have been one of the preferred systems for studies of periodic QWs because transmission measurements can be made without substrate removal, so far the GaAs/AlGaAs system has yielded much better results for nonperiodic QW spacings. If growth conditions can be found that eliminate this instability, it is likely that InGaAs/GaAs QWs will play a major role in the emerging field of complex nonperiodic nanophotonics.
We thank S. W. Koch for helpful discussions. The Tucson group thanks AFOSR, NSF, JSOP, and Arizona Technology & Research Initiative Funding (TRIF) for support. The Karlsruhe groups acknowledge financial support from the Deutsche Forschungsgemeinschaft (DFG) and the State of Baden-Württemberg through the DFG-Center for Functional Nanostructures (CFN) within subproject A1.4 and A2.5. The St. Petersburg work was supported by RFBR and the “Dynasty” Foundation — ICFPM.
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