GaAs

PHYSICAL REVIEW B 75, 115317 共2007兲

Atomic scale structure and optical emission of AlxGa1−xAs/ GaAs quantum wells C. Ropers,* M. Wenderoth,† L. Winking, T. C. G. Reusch, M. Erdmann, and R. G. Ulbrich IV. Physikalisches Institut der Universität Göttingen, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany

M. Grochol, F. Grosse, and R. Zimmermann Institut für Physik der Humboldt-Universität zu Berlin, Newtonstrasse 15, 12489 Berlin, Germany

S. Malzer and G. H. Döhler Max-Planck Research Group, Institute for Optics, Information, and Photonics, University of Erlangen-Nürnberg, Günther-Scharowsky-Strasse 1, 91058 Erlangen, Germany 共Received 6 December 2006; published 15 March 2007兲 A combined study of the optical and structural properties of AlGaAs/ GaAs quantum wells is presented. Microphotoluminescence experiments, magnetomicrophotoluminescence, and atomically resolved crosssectional scanning tunneling microscopy were performed on the same quantum well sample. Constant-current topographs with aluminum and/or gallium sensitivity are used to directly extract disorder potentials. Using these potentials, exciton absorption spectra, microphotoluminescence spectra, and diamagnetic shifts of individual exciton states are calculated in an envelope function approximation. Very good agreement between the theoretical and experimental results is found. DOI: 10.1103/PhysRevB.75.115317

PACS number共s兲: 68.37.Ef, 78.55.Cr, 78.67.De

I. INTRODUCTION

Semiconductor quantum wells, and, in particular, their optical properties, have been the subject of an enormous amount of research. The deviation from in-plane translational invariance in these systems and the disorder problem have attracted significant interest.1 Early on, it was recognized that disorder at interfaces leads to electron and hole localization, which influences the light emission from a quantum well. Nowadays, optical spectroscopy is very commonly used to infer upon a number of a quantum well’s structural properties such as its interface quality. Especially, enhanced resonant backscattering,2 resonant Rayleigh scattering,3 and correlation spectroscopy4,5 provided relevant information about disorder. Magneto-optical properties of individual localized exciton states have recently been shown to depend critically on the specifics of the disorder.6,7 Optical experiments thus complement data obtained by direct structural techniques such as, e.g., transmission electron microscopy,8 x-ray diffraction,9 or scanning tunneling microscopy.10,11 In order to obtain quantitative structural information from an optical experiment, some degree of theoretical modeling is required. The spatial confinement of electrons and holes in these structures is usually described by envelope functions in the effective-mass approximation.12 Given the broad applicability and the frequent use of this approach, it is surprising how little experimental work exists that directly confirms the connection between structure and optics in simple situations. By “directly” we mean that the structural properties of a given sample are measured as precisely as possible, predictions on the optical properties are made from the structural data, and that those are contrasted by an optical experiment. In this paper, we report on a study of AlGaAs/ GaAs quantum wells 共QWs兲 representing the prototype III-V heterostructure. A direct connection between the underlying atomic structure and linear optical properties is established: First, the structure was imaged by cross-sectional scanning 1098-0121/2007/75共11兲/115317共7兲

tunneling microscopy 共XSTM兲. Experimental details are given in Sec. II. Second, the XSTM data are used to extract the actual QW disorder potentials 共Sec. III兲. Absorption and microphotoluminescence 共␮PL兲 spectra as well as diamagnetic line shifts of localized excitons were computed using structural information directly obtained from the XSTM measurements. In Sec. IV, the four-dimensional Schrödinger equation of the electron and hole motion is solved for potentials generated directly from the measured topographs. Finally, the results are compared to ␮PL and magneto-␮PL measurements of the same sample in Sec V. II. EXPERIMENTS

The investigated heterostructure contained five intrinsic GaAs QWs of 4, 6, 8, 10, and 20 nm nominal widths, separated by 15-nm-wide barriers of intrinsic Al0.3Ga0.7As. The structure was grown by molecular-beam epitaxy on an exactly 共001兲-oriented n-doped GaAs substrate with no growth interruptions at the interfaces. The quantum well sequence was sandwiched between n-doped GaAs layers. The doping profile was designed such that the requirements on the concentration of free charges for both experiments were met. At the low temperatures of the ␮PL measurements 共3 K兲, there were virtually no free charges from the doping in the quantum well region, whereas the room-temperature scanning tunneling probe had a sufficient free-electron density available to carry the tunneling current. The XSTM measurements were performed on an atomically smooth 共110兲 surface, which had been prepared by in situ cleavage of the sample in an ultrahigh vacuum 共UHV兲 chamber with a base pressure of less than 5 ⫻ 10−11 mbar. The XSTM tips were chemically etched from polycrystalline tungsten. After transfer into UHV, the tips were annealed at 1200 K, sputtered with 4 kV Ar ions, and characterized by field emission. Cross-sectional constant-current topographs of all five QWs

115317-1

©2007 The American Physical Society


ROPERS et al.

were taken with atomic resolution over lateral lengths of typically 200 nm. After the XSTM measurements, the sample was transferred into a confocal microscope setup with a cassegrain mirror objective, allowing a lateral spatial resolution of 600 nm at a wavelength of 800 nm. Optical excitation 共⬃1 ␮W兲 was provided with a dye laser operating at 1.9 eV. The luminescence was dispersed by a 60 cm monochromator with a spectral resolution of 150 ␮eV and collected with a liquid-nitrogen-cooled charge-coupled device. The accuracy with which the optical focus and the position of the XSTM measurements coincided was better than 200 ␮m.13 It was checked that neither the emission energy nor the inhomogeneous linewidth changed over such a distance. In addition to these ␮PL experiments, magnetic-fielddependent ␮PL spectra were recorded with fields up to B = 10 T applied perpendicular to the QW plane in a Faraday configuration. The details of those experiments are described in Ref. 6. A section of the constant-current topograph of the 4 nm QW is displayed in Fig. 1 共entire topograph is ⬃200 nm long兲. The image contains the topographic height h共x , z兲 in linear gray scale, and it shows the variations of the aluminum concentration with atomic precision and chemical sensitivity. Strictly speaking, single aluminum atoms were not imaged, since at the applied negative bias voltages, only the group-V sublattice is visible.14 Nonetheless, the topographic variation on an atomic scale demonstrates the sensitivity to the aluminum distribution. There is a clear difference in the average height 具h共x , z兲典x between the well and the barrier regions of about 25 pm. We presume that this height difference not only reflects the global difference in the average aluminum concentrations but that it can also serve as a local gauge for the aluminum distribution in the barriers and at the interfaces. III. POTENTIAL EXTRACTION AND GENERATION

It is the aim of this work to predict optical properties such as absorption or luminescence spectra of bound electron-hole pairs 共excitons兲, based on topographs as the one shown in Fig. 1. The atomic scale contrast of the images allows us to extract the structural properties in the cleavage plane that are needed as input in the calculation of optical spectra. The topographs are assumed to be a direct representation of the local aluminum distribution and, therefore, the alloy band gap in the structure. A linear scaling of the topographic height h共x , z兲 to the local band gap is applied as EG共x,z兲 = EG,GaAs + b关h共x,z兲 − h0兴,

FIG. 1. Top: Constant-current XSTM topograph of the 4-nm-wide QW, recorded at a bias voltage of UB = −2 V and a tunneling current IT = 100 pA. The crystallographic axes are indicated. The z-confinement wave functions ua共z兲2 for the electron 共black curve兲 and the heavy hole 共white curve兲. Bottom: Deduced bandgap profile as a function of z, averaged along the x direction and after removal of the atomic corrugation.

共1兲

with b = −19.8 mV/ pm 共Ref. 15兲 and h0 being the topographic height in the center of the quantum well. The scaling factor b is determined by the 494 meV band-gap difference16 between the well and the Al0.3Ga0.7As barrier. The local band gap is distributed with the standard band offset ratio17 f e / f h = 0.65/ 0.35 among the single particle electron and hole envelope function potentials Va共x , z兲. Here and in the following, the index a denotes the particle 共e and h for electron and heavy hole, respectively兲.

In the thin QWs considered here, electrons and holes are confined much stronger along the growth direction 共z兲 than in the plane of the QW 共x − y plane兲. For example, in the case of electrons in the 4 nm QW, the z-confinement energy amounts to ⬃100 meV, whereas the in-plane disorder confinement is on the order of only 10 meV. We therefore apply a single-sublevel approximation for the z direction.18 This well-justified separation reduces the exciton equation to a four coordinate problem, reducing the computational effort enormously. In the single-sublevel approximation, the effec-

115317-2


ATOMIC SCALE STRUCTURE AND OPTICAL EMISSION…

FIG. 2. 共a兲 One-dimensional effective disorder band gap W共x兲 generated from the XSTM topograph. 共b兲 The histogram of potential values on the atomic grid is roughly Gaussian with a 33 meV standard deviation. 共c兲 The power spectrum of the spatial fluctuations of W 共black line兲 with a fit for an exponential disorder model 共gray line, see text兲.

tive in-plane band-gap fluctuations along the x direction can be expressed as19 W共x兲 =

兺

a=e,h

fa

冕

dz兩ua共z兲兩2关Va共x,z兲 − 具Va共x,z兲典x兴.

共2兲

Here, the ua共z兲 are the single particle wave functions in the one-dimensional potential derived from the x-averaged confinement potential 具EG共x , z兲典x 共Fig. 1, bottom兲. The potential fluctuations W共x兲关plotted in Fig. 2共a兲兴 are the origin of lateral electron and hole localization in this QW, resulting in the inhomogeneous broadening of its optical spectra. Quantitatively, the effective potential W共x兲 is characterized in terms of its strength 关standard deviation of potential values ␴, see Fig. 2共b兲兴 and its spatial correlations. Specifically, we analyze here the potential extracted from the 4 nm QW over a length of 160 nm. First, the potential W共x兲 is binned on the atomic grid in the x direction 共atomic row distance ⌬ = 0.4 nm兲, leading to a row vector W共x j兲 with j⑀兵1 , 2 , . . . , 400其. The statistical information of the potential is then fully contained in the autocorrelation

be done, e.g., by randomly shifting the individual atomic rows in the barriers along the x direction with respect to each other. For this situation, one obtains a ␴ of about 30% smaller than the actual one. Therefore, we assert that z correlations contribute significantly to the inhomogeneous broadening of the optical spectra. In fact, certain z correlations are already discernible by close inspection of the topograph in Fig. 1, where weak stripelike contrasts in the z direction are visible, especially in the right barrier regions. The XSTM measurements can certainly only give the disorder potentials in the cleavage plane, y = y 0, leading to the one-dimensional in-plane potential W共x兲, which contains the information on the strength and the correlations of the lateral disorder in the x direction. Relevant for the inhomogeneous broadening of excitonic transitions are, however, the disorder properties of the quantum well also in the y direction 共perpendicular to the plane of the XSTM image兲. Although both directions are crystallographically equivalent in zinc-blende crystals, the growth takes place on reconstructed surfaces, leading to a possible growth anisotropy in the quantum well plane. Such an anisotropy has previously been observed in optical measurements that showed linearly polarized doublets of states with a preferred polarization direction.20 That sample had been grown with long interruptions at the interfaces, leading to the formation of large elongated islands at the interfaces. In our case, similar fine-structure splittings of individual states are present, but we did not find a strongly preferred orientation for the investigated sample. Furthermore, the formation of large anisotropic islands at the interfaces is not expected for this sample, as it was grown without any growth interruptions. In addition, island formation is not evident in our XSTM images, as we primarily find very short-range correlations. Finally, it also does not appear in the form of a monolayer splitting of the PL spectra. Consequently, we presume at this stage that the correlations in the y direction do not differ from those in the x direction. This allows us to generate two-dimensional in-plane potentials W共x , y兲, which contain in all directions the disorder properties found by the XSTM measurement. Specifically, such in-plane potentials can be obtained by a convolution between a suitable averaging function A共r兲 and uncorrelated random fluctuations U共r兲:18 W共r j兲 = 兺 A共r j − rl兲U共rl兲,

N

1 C共x j兲 = 兺 W共x j + xl兲W共xl兲. N l=1

共3兲

The information on the correlations in the growth direction 共z兲 is qualitatively contained in the value of ␴ = 冑C共0兲. This can be seen as follows: The value of ␴ is completely insensitive to lateral 共x兲 correlations as it only represents the standard deviation of the potential. If there are, however, correlations of the local band gap in the growth direction, this will result in increased fluctuations of the weighted z average in Eq. 共2兲, compared with a completely uncorrelated aluminum distribution. The effect of z correlations in the aluminum distribution can be quantified by comparing the determined ␴ value with that obtained after an intentional removal of all z correlations from the XSTM image. This can

具U共rl兲,U共rk兲典 = ␦lk .

共4兲

l

Here, r j is now a two-dimensional vector on a square grid in the x − y plane. The averaging function A共r j兲 is directly related to the one-dimensional correlation function C共x j兲, which in terms of their Fourier components21 can be expressed as ˜ = 1 兺兩A ˜ 兩2 . C j ji N2 i

共5兲

In the past, different models for the averaging function A and the resulting types of correlations have been proposed.1 We have obtained best results for an exponential ansatz of the form

115317-3


ROPERS et al.

A共r j兲 =

␴

␩

e−兩r j兩/lc,

␩2 = 兺 e−2兩r j兩/lc ,

共6兲

D共␻兲 = 兺兩M ␣兩2␲␦共ប␻ − E␣兲,

共10兲

P共␻兲 = 兺兩M ␣兩2N␣␲␦共ប␻ − E␣兲,

共11兲

␣

j

where ␴2 is the potential variance and lc the correlation length. These fitting parameters are determined by a fit of the one-dimensional correlation functions generated using Eq. 共5兲 to the experimentally determined correlation 关Eq. 共3兲兴. A fit in k space 关see Fig. 2共c兲兴 yields ␴ = 33.4 meV and lc = 0.39 nm by a least-squares fit. The value of lc on the order of the lattice constant indicates that in the investigated structure, mostly short-ranged correlations over a few lattice constants are present in the aluminum distribution along the x direction. The potential generated via Eq. 共4兲 is then distributed between the electron and the hole as Wa共x , y兲 = f aW共x , y兲. The four-dimensional Schrödinger equation with these potentials is solved for the in-plane exciton motion, including a magnetic field perpendicular to the QW plane.

␣

M␣ =

冕

dr⌿␣共r,r兲.

共12兲

N␣ is the occupation of the state ␣. The application of a magnetic field results in a Zeeman splitting and, more importantly in this context, in an upward energetic shift of the transition energies. The diamagnetic shift starts quadratically in Bz, and we define the diamagnetic shift coefficient ␥␣ of the state ␣ as

␥␣ =

冏

1 d2E␣共Bz兲 2 dBz2

冏

.

共13兲

Bz=0

IV. HAMILTONIAN

V. RESULTS AND DISCUSSION

The in-plane Hamiltonian for excitons in a disordered quantum well in effective-mass approximation under applied perpendicular magnetic field Bz in the Coulomb relative gauge, neglecting the small spin-orbit coupling, reads7

In this section, we compare ␮PL spectra and diamagnetic properties of the investigated QW with the results of solving the four-dimensional exciton equation for the potentials extracted from the XSTM topographs. We begin in the absence of an external magnetic field.

2 2 2 2 ˆ = − ប ⌬ − ប ⌬ + e Bz 共r − r 兲2 + W 共r 兲 + W 共r 兲 H h h re rh e h e e 2me 2mh 8␮

− VC共r兲 +

eBz eBz iប关共y e − y h兲⳵xe − 共xe − xh兲⳵ye兴 + iប关共y e 2me 2mh

− y h兲⳵xh − 共xe − xh兲⳵yh兴 + ␮BBz共g*e ␴ze + g*h␴zh兲,

共7兲

where ra = 共xa , y a兲 are in-plane coordinates, ␮B is the Bohr magnetion, g*a are the electron and hole g factors, and ␴za the spin Pauli matrices. All effective masses are in-plane ones 共ma ⬅ ma,储兲, and ␮ = memh / 共me + mh兲 is the reduced mass of the exciton. The material parameters used in the calculations are listed in the Appendix. We共re兲 and Wh共rh兲 denote the disorder potentials for electron and hole, while the effective Coulomb potential is given by VC共r兲 =

冕

dzedzhu2e 共ze兲u2h共zh兲

e2

4␲⑀0⑀S冑r2 + 共ze − zh兲2

, 共8兲

with the static dielectric constant ⑀S. The two-particle 共exciton兲 Schrödinger equation with the Hamiltonian Eq. 共7兲, ˆ 共r ,r 兲⌿ 共r ,r 兲 = E ⌿ 共r ,r 兲, H e h ␣ e h ␣ ␣ e h

共9兲

allows to obtain eigenstates and eigenvalues E␣ of the localized excitons. They are the ingredients for the subsequent calculation of absorption spectrum D共␻兲共or optical density兲, photoluminescence spectrum P共␻兲, and oscillator strengths M ␣,1

A. Microphotoluminescence

In the optical experiments, ␮PL with a spatial resolution of ⬃600 nm was measured. Figure 3共a兲 shows a ␮PL spectrum of the 4-nm-wide QW, recorded at a temperature of 4 K. Typical features of exciton spectra from disordered QWs can be seen.22,23 The spectrum shows an inhomogeneous broadening with individual bright peaks on the lowenergy side due to strongly localized excitons. On the highenergy side, the more continuous spectrum consists of many closely spaced peaks. In Fig. 3共b兲, numerically computed absorption spectra for an area corresponding to the experimental focus are shown as a black line.24 In order to obtain calculated PL spectra for a useful comparison between experiment and computation, the occupation number N␣ of the individual states should be known. As a proper calculation of all N␣ by solving kinetic equations1 is technically very demanding in this case, we assume a Maxwell-Boltzmann distributed population with an effective carrier temperature TX. From the high-energy tail of the experimental ␮PL, we obtain an effective carrier temperature of TX = 20± 1 K. It is well known that the effective carrier temperature in narrow quantum wells can be significantly above the lattice temperature,25 and the value obtained here is in good agreement with previous findings.26 Multiplying the calculated absorption spectrum with a MaxwellBoltzmann factor of the form P共␻兲 = Ce−ប␻/kBTXD共␻兲

共14兲

results in the gray curve in Fig. 3共b兲, which is Stokes shifted by about 2.5 meV with respect to the absorption. In Fig. 3共c兲,

115317-4



clearly a result of the deviation from an exactly thermal occupation of those tail states in the experiment.1,27 Finally, in Fig. 3共d兲, an experimental ␮PL spectrum at an elevated temperature of 50 K 共black兲 is compared with the calculated PL spectrum for TX = 50 K. Again, good agreement of both curves is found, which indicates that the carriers are well equilibrated among themselves and with the lattice at this temperature. The temperature-dependent change in transition energy due to band-gap reduction17 was taken into account in the calculations of the spectra at the higher temperature. A band-gap difference of 2.5 meV between the lattice temperatures TL = 4 K and TL = 50 K was precisely determined in the experiment from the shift of individual localized states when the temperature was continuously raised. It should be noted that the absolute energy of all theoretical curves was blueshifted by 4 meV in order to achieve the agreement shown above. As the absolute transition energy is a very sensitive function of the material parameters as well as the average QW thickness, possible strain contributions, image charge effects on the binding energy,28 etc., a more accurate prediction of the absolute energy cannot be expected, given the precision of all known input quantities to the calculation. In view of the limited number of simple assumptions made in the calculations, the rather accurate prediction of the inhomogeneous broadening and the temperature-dependent line shapes of the ensemble of excitonic states is already striking. In order to get a more detailed description and to further test the validity of our approach, we proceed with the study of diamagnetic properties of individual exciton states. B. Diamagnetic shift of individual states

FIG. 3. Comparison between measured ␮PL spectra and spectra calculated from the XSTM data. 共a兲 Experimental ␮PL spectrum measured at T = 4 K. 共b兲 Calculated absorption spectrum 共black兲 and predicted luminescence 共gray兲 for an effective carrier temperature of 20 K, which is a realistic value for a 4 nm QW at 4 K lattice temperature. 共c兲 For easier comparison, experimental ␮PL 共4 K, black兲 and calculated PL 共20 K, gray兲 are plotted in the same graph. 共d兲 Experimental ␮PL spectrum measured at T = 50 K 共black兲 and calculated PL 共gray兲 for a thermal carrier population at 50 K.

the experimental and the predicted ␮PL curves are shown together. One can observe excellent qualitative and quantitative agreement of both curves. Due to the randomization procedure that is involved in the generation of the in-plane disorder potentials, the individual peak positions and peak heights are not to be considered significant. Instead, the overall spectral structure and the size of the inhomogeneous broadening should be characteristics for the disorder potential studied. These features are clearly reproduced by the calculations. Certain qualitative deviations are discernible at the low-energy tail of the luminescence, where the predicted luminescence strength of individual states in the calculations tends to exceed those found in the experiments. This is

In this section, the results of micromagneto-PL measurements are contrasted with the calculations described above, now including a magnetic field perpendicular to the QW plane. The application of a magnetic field results in a diamagnetic shift of exciton transition energies,23 with a quadratic field dependence for magnetic fields below several teslas. In the absence of disorder, the diamagnetic shift is given by the expectation value of the quadratic Bz term in Eq. 共7兲, which results in the shift coefficient

␥id =

e2 2 具r 典, 8␮

r = re − rh .

共15兲

The shift coefficient is, therefore, a measure of the spatial extension of the excitonic 共relative兲 wave function. For an exponential exciton wave function, the expectation value is 具r2典 = 共3 / 2兲aB2 , with aB being the Bohr radius of the QW exciton. In the simplest approximation, the dependence on reduced mass is aB ⬀ 1 / ␮, and, consequently, a scaling like ␥id ⬀ 1 / ␮3 is expected. As the hole is much heavier than the electron, the electron effective mass plays the decisive role in ␮, and a proper choice of me is important for a correct theoretical modeling of the diamagnetic shift coefficient29,30 共see the Appendix兲. In a disordered potential, the lateral localization may squeeze both the center-of-mass motion of the electron-hole pair as well as the relative motion, reducing the

115317-5


ROPERS et al.

TABLE I. Electron and hole effective masses, bulk band gaps at T = 4 K, and static dielectric constant used in the calculation.

me,z mh,z EG 共eV兲 me,储 mh,储 ⑀S

GaAs

Al0.3Ga0.7As

0.070a 0.36b 1.519b 0.078a 0.233d 12.5b

0.084b 0.39b 2.0128c

a

Reference 30. 17. c Reference 16. dReference 34. bReference

FIG. 4. Diamagnetic shift coefficients of single exciton states 共square boxes兲 in a 4-nm-wide QW 共experimental data from Ref. 6兲 and the shift of the entire broadened PL line 共full circle兲. The results from the present calculation are shown as dots. Both the experimental values and the calculated ones are below the shift coefficient for the absence of disorder 共dotted line兲.

diamagnetic shift coefficients of localized excitons. As the amount of lateral localization will vary from state to state, variations in the diamagnetic shift coefficients can be expected in an ensemble of states.23 Recently, we have studied theoretically7 and experimentally6 the influence of disorder on diamagnetic shifts. We have found a systematic dependence of the shifts of individual exciton states on the state energy, and we refer the reader to these references for further details. Here, we compare the experimentally found diamagnetic shift coefficients with those calculated, again with the use of the XSTM-determined disorder potential. It should be noted that these magneto-optical experiments are again performed on the QW as the XSTM measurements, although in this case not at exactly the same position. As all other optical properties were found to be very homogeneous across the sample, we do not expect this to have a noticeable effect on the results. In Fig. 4, the measured diamagnetic shift coefficient ␥␣ is plotted for several individual localized states as a function of their state energies E␣ 共squares兲, together with the overall shift coefficient of the broadened PL line ␥tot = 22 ␮eV/ T2 共full circle兲. This value is comparable to what has been obtained in previous magnetoluminescence experiments31–33 on a 5-nm-wide QW 共25 ␮eV/ T2兲. The individual states show a certain scatter and typically possess smaller shift coefficients than the overall line, pointing to stronger lateral localization of these states. Very accurate shift coefficients could only be obtained for states well separated from the broadened PL line and by averaging over the Zeeman doublet,6 which poses a limit on the statistics. The overall line and the individual states have shift coefficients smaller than the value obtained in the absence of disorder 共dotted line at ␥id = 37 ␮eV/ T2兲. The large number of small dots corresponds to the calculated shift coefficients for the states in the XSTM-determined disorder potentials. Both the absolute magnitude of the shift coefficients as well as the small increase of the coefficients

with increasing state energy are reproduced by the calculations. Thus, the disorder-induced localization reduces the shift coefficient dramatically due to a shrinkage of the exciton wave function. VI. CONCLUSIONS

In summary, the correlation between microphotoluminescence and diamagnetic properties and the results of a highresolution postgrowth structural method have been combined to investigate a typical heterostructure with disorder. XSTM constant-current topographs were scaled into envelope function potentials, and it was demonstrated that this method allows an accurate prediction of fundamental optical properties of the structure. These predictions cover photoluminescence and absorption line shapes and linewidths, as well as the diamagnetic properties of individual localized exciton states. The extracted small correlation length suggests that spatial correlations, which are present in both the growth direction 共z兲 as well as in the x direction, extend in the investigated sample mostly over only a few atomic distances. This correlation length may seem unexpectedly short, and at this point we cannot exclude certain longer-ranged contributions to the correlation. However, the dominant contribution to the disorder in this structure, grown without growth interruptions, is certainly on the order of 1 nm and below. Additional experimental and theoretical work will be necessary to resolve finer details of the correlation. On the theoretical side, one could use a multimodal approach to the correlation function or employ a more sophisticated interpretation of the XSTM images beyond the linear scaling, e.g., with a densityfunctional approach. Experimentally, XSTM measurements for two perpendicular cleavage planes are desirable, and even larger topographs could in the future improve the statistics on longer correlations. For the present work, a single exponential averaging function was sufficient to obtain convincing agreement between the theoretical computations and the experimental data. Within the present approach, other types of disorder could be studied, e.g., in QWs produced with growth interruptions or with AlAs/ GaAs interfaces. The experimental and theoretical methods taken here are applicable to a number of

115317-6



heterostructure systems. The combination of both allows us, in principle, to fully characterize and quantify the disorder in the sample. It will yield additional valuable information also on other optical properties and may provide access to a more detailed understanding of the structure-function relation in optoelectronic devices. ACKNOWLEDGMENTS

We acknowledge fruitful discussions with E. Runge and P. Vogl. This work is supported by the Deutsche Forschungsgemeinschaft 共DFG兲 within SFB 296 and SFB 602. M.G. ac-

*Present address: Max-Born-Institut fur Nichtlineare Optik und Kurzzeitspektroskopie, Max-Born-Strasse 2A, 12489 Berlin, Germany. Electronic address: [email protected] † Electronic address: [email protected] 1 E. Runge, Solid State Phys. 57, 149 共2002兲. 2 W. Langbein, E. Runge, V. Savona, and R. Zimmermann, Phys. Rev. Lett. 89, 157401 共2002兲. 3 G. Kocherscheidt, W. Langbein, U. Woggon, V. Savona, R. Zimmermann, D. Reuter, and A. D. Wieck, Phys. Rev. B 68, 085207 共2003兲. 4 F. Intonti, V. Emiliani, C. Lienau, T. Elsaesser, V. Savona, E. Runge, R. Zimmermann, R. Nötzel, and K. H. Ploog, Phys. Rev. Lett. 87, 076801 共2001兲. 5 G. von Freymann, U. Neuberth, M. Deubel, M. Wegener, G. Khitrova, and H. M. Gibbs, Phys. Rev. B 65, 205327 共2002兲. 6 M. Erdmann, C. Ropers, M. Wenderoth, R. G. Ulbrich, S. Malzer, and G. H. Döhler, Phys. Rev. B 74, 125412 共2006兲. 7 M. Grochol, F. Grosse, and R. Zimmermann, Phys. Rev. B 71, 125339 共2005兲. 8 A. Ourmazd, D. W. Taylor, J. Cunningham, and C. W. Tu, Phys. Rev. Lett. 62, 933 共1989兲. 9 M. Fleming, J. Appl. Phys. 51, 357 共1980兲. 10 A. Y. Lew, S. L. Zuo, E. T. Yu, and R. H. Miles, Phys. Rev. B 57, 6534 共1998兲. 11 R. Grousson, V. Voliotis, N. Grandjean, J. Massies, M. Leroux, and C. Deparis, Phys. Rev. B 55, 5253 共1997兲. 12 G. Bastard, Wave Mechanics Applied to Semiconductor Heterostructures 共Wiley, New York, 1991兲. 13 The position in the STM measurements was determined by an optical microscope while the experiment was performed. In this way, the distance to the sample edge was measured. In the subsequent optical experiments, the same sample position was placed in the optical focus by manual translation stages. 14 H. W. M. Salemink and O. Albrektsen, Phys. Rev. B 47, 16044 共1993兲. 15 The negative sign arises from the fact that the barrier regions are lower 共darker兲 than the central well region. 16 R. Scholz and P. Vogl 共private communication兲. 17 S. Adachi, GaAs and Related Materials 共World Scientific, Singapore, 1994兲. 18 R. Zimmermann, F. Grosse, and E. Runge, Pure Appl. Chem. 69,

knowledges the financial support from the Graduate School No. 1025 of the DFG. APPENDIX: MATERIAL PARAMETERS

Table I lists the material parameters used in the calculations. It is important to note that due to the quantum well confinement, a correction to the effective masses, in particular, for the conductin band, has to be taken into account.16,30,35 We take results from Ekenberg’s seminal paper.30 These are very similar to recent tight-binding calculations of the QW band structure,16 Which also provided us with the Al0.3Ga0.7As band gap.

1179 共1997兲. have checked that this description in terms of a combined 共summed兲 band-gap potential does not differ from an individual treatment of the single particle potentials at this point, because the electron and hole potentials are almost perfectly correlated. 20 D. Gammon, E. S. Snow, B. V. Shanabrook, D. S. Katzer, and D. Park, Phys. Rev. Lett. 76, 3005 共1996兲. 21 ˜ 共1兲 We use the standard definitions for the Fourier coefficients: C 19 We

j

= 兺kC共xk兲exp关−i2␲共jk / N兲兴 and ˜A jk = 兺m,nA共xm , y n兲exp关−i2␲共jm + kn / N兲兴. 22 A. Zrenner, L. V. Butov, M. Hagn, G. Abstreiter, G. Böhm, and G. Weimann, Phys. Rev. Lett. 72, 3382 共1994兲. 23 H. F. Hess, E. Betzig, T. D. Harris, L. N. Pfeiffer, and K. W. West, Science 264, 1740 共1994兲. 24 Calculations were performed for 100 areas of 100⫻ 100 nm2, yielding a total area of 1 ␮m2. Spectra were obtained by representing the individual states as Lorentz curves of 200 ␮eV width. Subsequently, these spectra were added with a spatial weight corresponding to the experimental focus 共spot size: 600 nm; focus shape: Airy function兲. 25 M. Gurioli, A. Vinattieri, J. Martinez-Pastor, and M. Colocci, Phys. Rev. B 50, 11817 共1994兲. 26 C. Colvard, D. Bimberg, K. Alavi, C. Maierhofer, and N. Nouri, Phys. Rev. B 39, 3419 共1989兲. 27 G. Mannarini, R. Zimmermann, G. Kocherscheidt, and W. Langbein, Phys. Status Solidi B 238, 494 共2003兲. 28 D. B. Tran Thoai, R. Zimmermann, M. Grundmann, and D. Bimberg, Phys. Rev. B 42, 5906 共1990兲. 29 D. D. Smith, M. Dutta, X. C. Liu, A. F. Terzis, A. Petrou, M. W. Cole, and P. G. Newman, Phys. Rev. B 40, 1407 共1989兲. 30 U. Ekenberg, Phys. Rev. B 40, 7714 共1989兲. 31 S. Tarucha, H. Okamoto, Y. Iwasa, and N. Miura, Solid State Commun. 52, 815 共1984兲. 32 D. C. Rogers, J. Singleton, R. J. Nicholas, C. T. Foxon, and K. Woodbridge, Phys. Rev. B 34, 4002 共1986兲. 33 T. Someya, H. Akiyama, and H. Sakaki, Phys. Rev. Lett. 74, 3664 共1995兲. 34 A. Siarkos, E. Runge, and R. Zimmermann, Phys. Rev. B 61, 10854 共2000兲. 35 M. Städele and K. Hess, J. Appl. Phys. 88, 6945 共2000兲.

115317-7