Surface Atomic Structure and Growth Mechanism of Monodisperse {1

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Surface Atomic Structure and Growth Mechanism of Monodisperse {1 0 0}-Faceted Strontium Titanate Zirconate Nanocubes Hongchu Du,*,†,‡,£ Chun-Lin Jia,†,§ and Joachim Mayer†,‡ †

Ernst Ruska-Centre for Microscopy and Spectroscopy with Electrons, Forschungszentrum Jülich GmbH, Jülich 52425, Germany Central Facility for Electron Microscopy (GFE), RWTH Aachen University, Aachen 52074, Germany § Peter Grünberg Institute, Forschungszentrum Jülich GmbH, Jülich 52425, Germany £ Institute of Inorganic Chemistry, University of Bonn, Bonn 523117, Germany ‡

S Supporting Information *

ABSTRACT: The highly sensitive and selective properties of monodisperse faceted nanocrystals inherently stem from the atomic and electronic structures on the faceted surfaces. For elemental nanocrystals, the atomic structure on the surfaces is merely determined by the geometric shape itself. However, for compound materials such as alloys and complex oxides, atomic details on the faceted surfaces need to be studied on the atomic level. Here, we demonstrate that the surface atomic structure of faceted nanocrystals of complex oxides, {1 0 0}-faceted strontium titanate zirconate nanocubes, can be unambiguously resolved by aberration-corrected scanning transmission electron microscopy. The resolved surface atomic details reveal a layerwise growth process of the nanocubes, thereby allowing an in-depth understanding of the growth mechanism.

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provides one more degree of freedom to tailor the properties. Moreover, monodisperse nanocrystals can serve as building blocks for self-assembled superlattices, contributing to a new level of diversity for preparation of functional materials. In recent years, extensive research has been reported on the wet chemical synthesis of perovskite ABO3 nanocrystals, including BaTiO 3 , SrTiO 3 , BaZrO 3 , SrZrO 3 , and Ba1−xSrxTiO3.18−28 A successfully used method to grow {1 0 0}-faceted BaTiO3, SrTiO3, and Ba1−xSrxTiO3 nanocubes is the oil− (or surfactant−) water two-phase solvothermal method.24−28 It takes the same straightforward strategy as an oil− water two-phase method based on the growth of nanocrystals at the oil−water interface by the reaction between reactants in the oil and water phases but conducted under solvothermal conditions (Figure 1b). The oil−water two-phase solvothermal method allows combination of oil-soluble surfactants (oleic acid) and strongly alkaline aqueous solutions (NaOH (aq)) in one synthesis.18 The surfactants can passivate the nanocrystal surfaces and thereby protect the nanocrystals from coalescence, being essential for the formation of monodisperse nanocrystals. On the other hand, the strongly alkaline aqueous solutions act as mineralizers that are generally required in the synthesis of complex oxides to overcome the difficulties in crystallization. However, whether the {1 0 0} facets of the nanocubes are terminated with AO (SrO) or BO2 (TiO2) is a question which

onodisperse faceted nanocrystals, with controllable shapes and sizes, have been becoming increasingly important for applications in catalysis, gas sensing, and energy conversion.1−5 Such highly shape-sensitive and selective physical and chemical properties inherently stem from the atomic and electronic structures on the faceted surfaces. For elemental nanocrystals, the atomic structure on the surfaces is determined by the geometric shape itself.6 However, for compound materials such as alloys and complex oxides, the compositional segregation and different terminating lattice planes on the surfaces have to be taken into account.7 In order to understand the unique property and growth mechanism of these nanocrystals, atomic details on the faceted surfaces need to be studied on the atomic level. Strontium titanate (SrTiO3), strontium zirconate (SrZrO3), and their solid solutions (SrTi1−xZrxO3) are important members in the class of perovskite structures with a general formula ABO3 (Figure 1a). These materials are of great technological and fundamental importance not only because of their interesting properties, such as high dielectric permittivity,8 switchable resistance,9 ionic conductivity,10 and ferroelectricity,11,12 but also because of their ability to combine and to adjust these properties by chemical substitution with a wide variety of cations. Since the Ti 3d states dominate the conduction band in SrTiO3,13 the conduction band can be fine-tuned to the required electron affinity by the substitution of Zr4+ for Ti4+.14 On the other hand, ferroelectricity, dielectric permittivity, and nonlinear optical properties appear to have strong size effects,15−17 The size of SrTi1−xZrxO3 nanocrystals © XXXX American Chemical Society

Received: November 17, 2015 Revised: December 29, 2015

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DOI: 10.1021/acs.chemmater.5b04486 Chem. Mater. XXXX, XXX, XXX−XXX

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Chemistry of Materials

Figure 1. Synthesis of perovskite oxide nanocubes by an oil−water two-phase solvothermal synthesis. (a) Structural model of ABO3 perovskite structure. (b) Schematic illustration of the strategy for the oil−water two-phase solvothermal synthesis. (c,d) Bright-field TEM images, (e,f) HAADF-STEM images of the nanocubes (SrTi0.5Zr0.5O3), at lower and higher magnification, respectively.

Figure 2. Surface atomic structure of the nanocubes. (a) HAADF STEM image of a SrTi0.75Zr0.25O3 nanocube in the [0 0 1] zone axis, averaged over four frames and denoised by a nonlinear filter.35 The Sr columns appear brighter than the Ti/Zr−O columns. (b) HAADF STEM image overlaid with color-scale two-dimensional Gaussian peaks from fitting the intensity distribution of each column. (c,d) Maps of integrated peak intensity of the Sr and Ti/Zr−O columns, respectively. Arrows in (d) indicate Zr-rich columns showing exceptional brightness. (e,f) Histograms of the intensities of the Sr and Ti/Zr−O columns, respectively.

In this work, we report on detailed studies of monodisperse {1 0 0}-faceted nanocubes of SrTi1−xZrxO3 (x = 0.25 to 0.5), which were synthesized using the oil−water two-phase solvothermal method. The surface atomic structure of the monodisperse faceted nanocrystals is determined by means of aberration-corrected high-angle annular dark-field scanning

still remains open for speculation and investigation. A comprehensive understanding of the growth mechanisms of these faceted nanocubes has not been achieved.26,27 Direct experimental evidence for the atomic structure on these nanocube surfaces has become one of the key steps in exploring the growth mechanisms. B


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surfaces and a high intensity region for the columns in the bulk. In contrast, those for the Ti/Zr−O columns are in a broad and continuous region (Figure 2d,f). The presence of residue surfactant molecules and the amorphous carbon support as discussed above may unavoidably broaden the distribution of the column intensity, but this should affect both the Sr and the Ti/Zr−O columns simultaneously to a similar extent. Therefore, the broad and continuous distribution of the intensity values for the Ti/Zr−O columns can be understood as the result of the inhomogeneity in the number of the Zr atoms occupying the B-sites in individual columns. Further efforts are underway in order to quantify the Zr in each individual Ti/Zr− O column. The bimodal distribution of the intensity for the A-site Sr columns could be mainly attributed to that the A-site atomic columns in the surface-terminating plane are evidently much less bright than those in the bulk of the particle. The low intensity of the image for the outermost layers of SrO can be understood by the following mechanisms. On one hand, the assynthesized nanocubes are more or less truncated, and as a consequence, the SrO columns terminating at the surfaces include fewer atoms than the bulk counterpart along the projection direction. On the other hand, these atoms on the surfaces have a high possibility to deviate from the ideal lattice sites and thus are not exactly in a line along the zone axis. This arrangement will result in a dechanneling effect and hence columns at the surfaces will show less intensity than those columns in the bulk. This is similar to the dechanneling effect found for the atomic columns at the dislocation cores.29 Moreover, surface diffusion dynamical processes under electron beam irradiation are unavoidable. All these mechanisms could therefore lead to a decrease of the image intensity for the atomic columns on the surfaces. Imaging the nanocubes along a direction away from the zone axis is expected to minimize the channeling and dechanneling effects on the formation of HAADF image. This in turn allows identification of the terminating layers. In order to avoid the channeling conditions, we tilted the crystal around the [0 1 0] axis away from the [0 0 1] zone axis and obtained an atomic plane image with the beam parallel to the (1 0 0) plane. From this image, the alternating stacking of SrO and Ti(Zr)O2 atomic planes can be clearly seen with a higher intensity for the SrO planes than that for the Ti(Zr)O2 planes (Figure 3a). The difference in intensity for the two different atomic planes is revealed by line profiles of the image intensity from the marked region in Figure 3a. As seen from the line profile (Figure 3b), the intensity for both SrO and Ti(Zr)O2 layers shows a gradual decrease from the inner layers to the outer terminating layers. This can be attributed to the truncation nature of the nanocubes. An important feature is that the terminating SrO layers at the (1 0 0) surfaces appear with evidently brighter intensity than the inner neighboring Ti(Zr)O2 layers, thereby providing obvious evidence for the SrO termination on the nanocube surfaces. We should also consider the possibility that the surface layer could be different from the bulk stacking sequence, or reconstruction of Ti(Zr)O2 double layers could occur. It is known that common phases of Ti(Zr)O2 have edge sharing Ti(Zr)O6 octahedra as observed in the reconstructed {1 0 0} surface of SrTiO3.30 In the projection along axis for the perovskite phase, the presence of edge sharing Ti(Zr)O6 octahedra will introduce extra Ti/Zr−O columns at the O column site in the normal perovskite lattice. The Ti/Zr−O

transmission electron microscopy (HAADF-STEM). On the basis of the structural features on the faceted surfaces, a deeper insight into the growth mechanisms could be obtained. Pure perovskite nanocrystals of SrTi1−xZrxO3 were obtained by syntheses over the compositional range for x from 0.0 to 0.5 (see Supporting Information (SI), Figure S1). Nanocrystals of ZrO2 appeared when x reaches about 0.6 and gradually become the dominant products with further increase of Zr content. For cubic perovskites such as SrTiO3, it is well-known that the {1 0 0} faces have the lowest energy, however, the {1 0 0} faces were not well developed at low-level (x < ∼ 0.25) Zr doping under the studied synthesis conditions (SI, Figure S2). For x from 0.25 to 0.5, the synthesized SrTi1−xZrxO3 nanocrystals appeared to be monodisperse nanocubes with edge length of about 10 nm. Figure 1c,e show the TEM and HAADF-STM images of the nanocubes by taking SrTi0.75Zr0.25O3 as a representative example. Analysis of the broadening of XRD diffraction peaks showed that the growth rate of nanocubes decreases with the Zrsubstitution level (see SI, Figure S3). However, the number of the as-grown nanocubes, as indicated by the XRD diffraction peak height, increases with the reaction time. This implies a multistep nucleation and growth process. Particularly, this is more evident for the SrTi0.5Zr0.5O3 nanocubes, which show a size-saturated dependence on the reaction time. The increase of peak intensity therefore can be mainly attributed to the increase of the number of the nanocubes. Similar results were also reported for SrTi0.6Zr0.4O3 nanocubes.27 The surficial-terminating atom planes were investigated by HAADF-STEM imaging. Here, special care should be taken in the experiments to avoid electron-beam-induced amorphization of the nanocubes from their surfaces during the HAADF-STEM imaging. The small screen-detected electron beam current was therefore adjusted to a value between 0.10−0.01 nA for our experiments to allow acquiring a number of frames without evident beam-induced modifications of the nanocubes. Figure 2a shows a HAADF-STEM image of a nanocube taken in the [0 0 1] zone axis, which clearly reveals cation columns and the single crystalline nature of the nanocube. The majority of the surfaces of the nanocubes are parallel to the {0 0 1} planes. Owing to its Rutherford scattering nature, the intensity of the HAADF images depends on the composition through Zζ of the scattering cross section, where Z is the atomic number and ζ is close to 2 depending on the actual value of the collection angle of the HAADF detector. As a result, in the image of SrTi0.75Zr0.25O3 nanocubes, the intensity for the Ti/Zr−O atomic columns appear weaker than that for the Sr columns. The presence of residue surfactant molecules and the amorphous carbon support mainly contributed to the background, because on the one hand, these amorphous phases unlikely met a channeling condition, and on the other hand, they consist of low Z atoms. By fitting the intensity distribution of the two types of atomic columns with two-dimensional Gaussian functions, the integral peak intensity for each column was obtained. For convenience of identification, the Gaussian peaks from fitting were encoded in green and red colors for Sr and Ti/Zr−O columns, respectively, and overlaid over the HAADF image. As shown in Figure 2b, the surface-terminating cationic columns of the nanocubes appear to be the Sr columns in all the four {1 0 0} facets. It is noted that the intensities for the Sr atomic columns distribute statistically into two separate regions of values (Figure 2c,e), a low intensity region for the columns on the C


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Figure 3. Direct evidence of SrO termination of the surfaces of the nanocubes. (a) HAADF-STEM image of a SrTi0.75Zr0.25O3 nanocube tilted around one of its lattice axis, keeping the {1 0 0} lattice planes edge-on, but out-of the [0 0 1] zone axis to reduce the channeling effects. (b) Profile along the marked region in (a). A magnified view from the region is shown at the lower part of the profile. (c) ABFSTEM image of a SrTi0.75Zr0.25O3 nanocube in the [0 0 1] zone axis. Atomic columns were labeled according to (d) the simultaneously acquired HAADF-STEM image. Scale bar: 2 nm.

Figure 4. EDX spectrum imaging of the nanocubes. (a) HAADFSTEM image of SrTi0.75Zr0.25O3 nanocubes and amorphous particles. The contrast was adjusted for enhancing the visibility of the amorphous particles of low brightness. (b−d) EDX spectrum images of Sr-L, Ti-K, amd Zr-L lines, respectively. (e) Composite image of the spectrum images. (f) HAADF-STEM image overlaid with the composite image (e). (g) EDX spectra from the four nanocubes marked in (f), quantified by the Cliff−Lorimer method with series fitting for deconvolution of the overlapping peaks, blue: experimental, red: deconvoluted, green: background.

columns should be well resolved by HAADF. However, we did not find such extra columns at the surfaces by HAADF imaging. In addition, we further used annular bright-field STEM (ABFSTEM) as a complementary technique to verify the possible surface reconstructions. Figure 3c shows an ABF-STEM image of a nanocube in its [0 0 1] zone axis, in which all the Sr, Ti/ Zr−O, and O columns were resolved. The O columns appear to be less dark than the metal-containing columns. The atomic columns were labeled according to the simultaneously acquired HAADF image (Figure 3d), again supporting SrO-terminating {1 0 0} planes. In the ABF image, both the A-site Sr columns and O columns (less darker) appear to be at their normal positions in the terminating layers, which implies no significant surface reconstruction. Quantification of the spatial resolved energy-dispersive X-ray (EDX) spectra reveals that the atomic ratio between Sr, Ti, and Zr of the nanocubes basically agrees with the designed composition within the error limits (Figure 4). We should note that the channeling effect would also play a non-negligible role in the EDX measurements for the nanocubes projected along their low index zone axis. Therefore, we used a sample prepared from a bulk SrTiO3 as a calibration standard for correction in the quantification of the Sr and Ti in the same [0 0 1] zone axis. Correction is not made for Zr due to the lack of a standard sample. Moreover, we found fiber-like and granular particles with low contrast in the HAADF-STEM images (Figure 4a). STEM EDX spectrum mapping, as shown in Figure 4b − 4f, reveals that these amorphous particles also contain Sr, Ti, and Zr. These particles may be attributed to amorphous metal oxides formed by hydrolysis and dehydration of metal surfactant complexes. Attempts to acquire atomically resolved EDX and electron energy loss spectroscopy (EELS) were ended with no success due to severe movement of the nanocubes, particularly the rotation which could not corrected by the shift correction integrated with the instrument software. More efforts are underway to overcome the movement (rotation) of nanocubes in the atomically resolved EDX/ EELS spectrum imaging, which would lead to a deeper

understanding of the electronic structure and bonding at the nanocube surfaces, at the truncated corners, or even at the growth steps. The presence of amorphous particles surrounding the nanocubes raises a question whether the crystallization of the nanocubes occurs directly by hydrolysis and dehydration of the metal−surfactant complexes, or the crystallization process essentially involves more additional steps including dissolution of the dehydrated metal oxides with subsequent crystallization in the swelled water droplets containing NaOH at the oil− water interfacial region (Figure 1b). To clarify this open question, we prepared TiO2 nanorods according to ref 31, and we used the TiO2 nanorods as Ti-source. Interestingly, we found that the SrTiO3 nanocrystals dominate the products even for a reaction time of 12 h (see SI, Figure S4). Since TiO2 is unlikely to be soluble in sodium oleate and organic solvents, the results therefore indicate that NaOH water solution droplets in the oil−water interfacial region is involved in the nucleation and growth, providing evidence for the processes indicated by the green and blue arrows, as depicted in Figure 1b. Additionally, the marginal reaction-time dependence of the size of nanocubes for high Zr substitution level could be attributed to the slow growth rate. Under such a circumstance, the supplementary growth resulting from the above-mentioned processes appear to be negligible. Because the rod shape of the source TiO2 nanocrystals is not retained by the as-synthesized SrTiO3, an in situ crystallization mechanism can be ruled out under the studied conditions.26 It is evident that Zr-substitution plays an important role in reducing the growth rate of the nanocubes and stabilizing the {1 0 0} facets and thereby is indispensable for preparation of cubic-shaped nanocrystals under the studied conditions (see D


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Chemistry of Materials Figure 1 and SI, Figure S1 − S3). Correspondingly, one might consider the nanocubes to be terminated with Ti/Zr−O lattice planes on the surfaces to stabilize the {1 0 0} facets.27 However, HAADF-STEM imaging conclusively reveals the surfaceterminating cationic columns in the {1 0 0} facets of the nanocubes to be the Sr columns. This puzzle and the effects of Zr-substitution can be understood by a layerwise growth process. Along the axis, the structure of SrTi1−xZrxO3 (ABO3) (Figure 1a) can be described as a layered structure with alternating stacking sequence of a SrO (AO) layer and a Ti1−xZrxO2 (BO2) layer. Therefore, reduction of the nanocube growth rate, stabilization of the {1 0 0} facets, and thereby the formation of cubic-shaped nanocrystals can be realized by slowing down the growth of either the AO layer or the BO2 layer. We presume that the growth rate of the AO layers is much faster than the BO2 layers, which appears to be valid for the studied system. In a layerwise growth process, a growing BO2 layer (B1) will be covered by an AO layer (A1) before its finish (Figure 5a). However, growth of a new BO2 layer on the

terminating cationic columns at the surface of the nanocube are again the Sr columns (Figure 5c). The magnified images shown in Figure 5d,e clearly reveal the atomic details of the growth step, and hence the layerwise growth process. In conclusion, we have conclusively determined the terminating cationic columns in the {1 0 0} surface planes of the monodisperse {1 0 0}-faceted SrTi1−xZrxO3 (x = 0.25−0.5) nanocubes to be the Sr columns by HAADF-STEM imaging. The atomic details of the revealed growth step provide unambiguous evidence for a layerwise growth process of the monodisperse nanocubes in an oil−water two-phase solvothermal synthesis, with which the effects of the Zr substitution on the formation of nanocubes and the termination of the Sr columns at surfaces can be understood. The success of determination of the surface atomic structure of faceted nanocrystals provides experimental evidence helping in understanding the growth mechanisms of the nanocrystals. Because many properties depend on the surface atomic structure, we believe that this work will stimulate more efforts to reveal the surface atomic structure of faceted nanocrystals of alloys and complex oxides in order to achieve an in-depth understanding of their properties and growth.

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METHODS

Synthesis. For a typical synthesis, a 0.2 M strontium oleate solution in octadecene was prepared as follows: 2.1472 g of Sr(OAc)2· 0.5H2O and 12.0 g of oleic acid were mixed and evacuated in a 100 mL three neck round-bottom flask at 100 °C for 30 min. Afterward the mixture was heated to 250 °C under magnetic stirring in Ar atmosphere and maintained for 30 min. Then the solution was cooled to 100 °C and evacuated for 30 min. After it was cooled to room temperature, the solution was diluted with octadecene to 50 mL. For the typical preparation of SrTixZr1−xO3 (x = 0.0−0.75), 1.6 g of NaOH was dissolved in 10 mL of water, and the solution was transferred into a 45 mL Teflon-lined stainless steel autoclave. Strontium oleate solution (5.0 mL), Zr(n-OBu)4 (x mmol), Ti(n-OBu)4 (1 − x mmol), and ocatedecene (5.0 mL) were mixed and stirred for 30 min, and the solution was transferred into the autoclave above the NaOH water solution without any stirring. The autoclave was sealed and heated at 200 °C for 24 to 72 h. After the reaction, 10 g of ethanol was added into the crude solution. The mixture was centrifuged at 3500 rpm for 10 min and resulted in two phases. The nanocubes in the transparent upper phase were precipitated by ethanol (5.0 g) and collected by settlement (∼5 min) and decantation. These steps were repeated two times. Then the nanocubes were redispersed in heptane (3.0 g), precipitated by ethanol (10 g), and collected by centrifugation and decantation. The redispersion, precipitation, and centrifugation steps were repeated two more times. After that, the purified nanocrystals were redispersed in 3.0 g of heptane. For syntheses with TiO2 nanorods as Ti-source, first, TiO2 nanorods were synthesized according to ref 31 with slight modification: 10.0 mL of octadecene, 5.5 g of oleic acid, and 5.0 mL of triethylamine were mixed in a flask by magnetic stirring for ∼5 min, then 1.0 mL (∼3.0 mmol) of Ti(n-OBu)4 was added dropwise forming an orange-red color solution. After it was stirred for another ∼ 5 min, the solution was transferred into a 45 mL Teflon-lined stainless steel autoclave. The autoclave was sealed and heated at 200 °C for 24 h. The synthesized TiO2 nanorods were collected by centrifugation and then redispersed in heptane and precipitated with ethanol for purification. After several cycles of purification, the TiO2 nanorods were dispersed in 5.0 mL of heptane. Then, 1.6 mL of the prepared TiO2 nanorods dispersion was precipitated by 10 mL of ethanol and redispersed in a mixture of 5.0 mL of Sr-solution and 5.0 mL of octadecene by magnetic stirring for 30 min, which was used as oilphase. The rest steps were identical to a typical synthesis. Characterization. X-ray diffractograms were recorded on a Philips PW1050 diffractometer using Co Kα radiation (λ = 1.7903 Å).

Figure 5. Layerwise growth process of the nanocubes. (a) Sketch of a layerwise growth process for a {1 0 0} facet of the perovskite ABO3 structure presuming a faster growth rate for the AO layer (green) than that for the BO2 layer (blue). The gray, yellow, and red symbols indicate the possible atom sites for the BO2 layer growth in the order of increasing preference. Oxygen was omitted in the model for clarity. (b) HAADF-STEM image of a SrTi0.75Zr0.25O3 nanocube with a growth step, averaged from 2 frames and denoised by a nonlinear filter.35 (c) HAADF-STEM image overlaid with color-scale twodimensional Gaussian peaks from fitting the intensity distribution of each column. (d) Magnified image of the growth step. (e) Magnified image of the growth step overlaid with structural model.

terminating AO layer (A1) is less favorable when a growing layer with steps (B1) is available. As a result, the layer with a faster growth rate (AO) will terminate the surface rather than the layer with a slower growth rate (BO2). The observation of growth steps at the surfaces of nanocubes provides direct evidence supporting the layerwise growth process for the nanocube facets. Figure 5b shows the HAADF-STEM image of a SrTi0.75Zr0.25O3 nanocube with a growth step. Another example is shown in SI, Figure S5. The E


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Chemistry of Materials Conventional bright-field TEM imaging of the nanocubes was conducted on a Philips CM30 (300 kV) electron microscope with a 1k × 1k MSC−CCD-camera. Aberration-corrected HAADF-/ABFSTEM and spatial resolved EDX spectrum imaging were carried out at 200 kV on an FEI Titan G2 80−200 ChemiSTEM microscope equipped with a high-brightness Schottky field emission electron gun, a probe Cs corrector, and a Super-X EDX system. A homemade software package DMPFIT,32 supplied within the Gatan GMS2 software based on the Levenberg−Marquardt technique from the MPFIT C code,33 a MINPACK-1 Least Squares Fitting Library in C,34 was used for fitting the atomic columns in the HAADF images with two-dimensional Gaussian peaks. A nonlinear filtering algorithm was used for noise reduction in the HAADF and ABF images.35

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.chemmater.5b04486. XRD and additional TEM results (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Author Contributions

H.D. designed and performed the experiments. All authors analyzed the data, discussed and interpreted the results. H.D. drafted the paper. C.-L.J and J.M. commented on the manuscript. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work has been supported in parts by the Deutsche Forschungsgemeinschaft (SFB 917). REFERENCES

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