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Structure of small II-VI semiconductor nanoparticles:A new approach based on powder diffraction

Franziska Niederdraenk1, Knud Seufert1, Pawel Luczak1, Sulabha K. Kulkarni2, ChristineChory3, Reinhard B. Neder4, and Christian Kumpf1

1 Universitat Wurzburg, Lehrstuhl fur Experimentelle Physik II, 97074 Wurzburg, Germany2 University of Pune, Department of Physics, Pune-411007, India3 Universitat Wurzburg, Lehrstuhl fur Silikatchemie, 97070 Wurzburg, Germany4 Universitat Wurzburg, Institut fur Mineralogie, 97074 Wurzburg, Germany

Received 12 March 2007, revised 7 June 2007, accepted 11 June 2007Published online 10 July 2007

In many cases of nanostructures a better understanding of the optical, electronic, and catalytic properties ishindered by a lack of detailed structural knowledge of sufficient quality. This is in particular true for verysmall nanoparticles (< 5 nm) where quantum confinement effects play a dominant role. We introduce anovel method for the detailed structural analysis of such small nanoparticles, which includes not only themodeling of basic parameters like size, shape, and structure, but also allows to determine impurities anddefects (like, e.g., stacking faults), stress and relaxation effects, etc. Distributions of basic parameters as thesize distribution are also implemented by enabling ensemble averaging. Different examples obtained fromCdS, ZnS and ZnO particles are presented and demonstrate the superiority of the new approach comparedto simpler methods.

PACS 61.10.Nz, 61.46.Df, 61.72.Nm, 81.05.Dz

phys. stat. sol. (c) 4, No. 9, 3234–3243 (2007) / DOI 10.1002/pssc.200775423

phys. stat. sol. (c) 4, No. 9, 3234–3243 (2007) / DOI 10.1002/pssc.200775423

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Structure of small II-VI semiconductor nanoparticles:A new approach based on powder diffraction

Franziska Niederdraenk1, Knud Seufert1, Pawel Luczak1, Sulabha K. Kulkarni2, ChristineChory3, Reinhard B. Neder4, and Christian Kumpf∗1

1 Universitat Wurzburg, Lehrstuhl fur Experimentelle Physik II, 97074 Wurzburg, Germany2 University of Pune, Department of Physics, Pune-411007, India3 Universitat Wurzburg, Lehrstuhl fur Silikatchemie, 97070 Wurzburg, Germany4 Universitat Wurzburg, Institut fur Mineralogie, 97074 Wurzburg, Germany

Received 12 March 2007, revised 7 June 2007, accepted 11 June 2007Published online 10 July 2007

In many cases of nanostructures a better understanding of the optical, electronic, and catalytic properties ishindered by a lack of detailed structural knowledge of sufficient quality. This is in particular true for verysmall nanoparticles (< 5 nm) where quantum confinement effects play a dominant role. We introduce anovel method for the detailed structural analysis of such small nanoparticles, which includes not only themodeling of basic parameters like size, shape, and structure, but also allows to determine impurities anddefects (like, e.g., stacking faults), stress and relaxation effects, etc. Distributions of basic parameters as thesize distribution are also implemented by enabling ensemble averaging. Different examples obtained fromCdS, ZnS and ZnO particles are presented and demonstrate the superiority of the new approach comparedto simpler methods.

1 Introduction

In many different fields of basic and applied research semiconductor nanoparticles and quantum dots playa very important role. On the one hand, there are numerous applications, ranging from cell-labeling inmedicine and biology over the function as catalysts in chemistry and engineering to the use for solar cells,light-emitting diodes or lasers in physics and engineering [1]. All of these examples benefit from theunique optical and electronic properties of nanoparticles, which can drastically differ from the correspond-ing bulk properties. On the other hand, nanoparticles with diameters below 5 nm also represent modelsystems in the interesting intermediate field between solid-state and molecular physics. This makes themvery attractive for basic research [2]. Even though size-tunable properties are already widely used, they arestill not fully understood. Attempts to describe small nanoparticles are either based on solid-state calcula-tions adapted for smaller sizes (e.g., effective-mass approximation [3]) or on molecular orbital calculations(see e.g., [4]). However, both approaches imply strong simplifications and partly yield questionable orjust qualitative results. One lack of any calculation is the exact knowledge of the geometric structure ofthe nanocrystal. Usually a bulk-like structure is used as input for the calculations, an assumption whichis almost never fulfilled for small particles. With a precise information of all geometric properties, thetheoretical approaches could be significantly improved and would give much more accurate predictions fornanoparticles.

Thus, a precise structure determination would represent an essential step forward towards a reasonableunderstanding of the optical, electronic, magnetic, and catalytic properties of the particles. However, eventhe exact size determination is already a challenge in many cases. Most commonly microscopy methodsare used for analyzing the particle size. Rather precise information on sizes and shapes can be obtained for

∗Corresponding author: e-mail: [email protected], Fax: +49 931 888 5756

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

PACS 61.10.Nz, 61.46.Df, 61.72.Nm, 81.05.Dz

individual (or a relatively small number of) particles, down to diameters of 4−5 nm. However, statisticallyrepresentative data are difficult to obtain. Furthermore, imaging of particles below approximately 3 nm isextremely difficult, or even impossible at present. At least in this size regime, other methods are urgentlyneeded for a realistic structure analysis.

One rather indirect alternative for size determination has become very popular because of its simplic-ity: UV/VIS spectroscopy. The absorption maximum, which is assigned to the optical transition of thefirst excitonic state, is associated with the particle size. This is done via theoretical calculations like theeffective-mass approximation (EMA) or the tight-binding (TB) approach [4]. Beside the problem that themaximum in the absorption curves is often reduced to a shoulder and not clearly identifiable, an evenhigher uncertainty is caused by the simplification of the structure which has to be assumed in order toperform the calculations. The consequences are rather different results obtained with different techniquesfrom identical particles (see e.g. [5–10]).

A third method frequently used for size determination is powder X-ray diffraction. The finite size of theparticles is derived from the broadening of the Bragg reflections, which is often determined by, e.g., a Ri-etveld refinement and subsequent application of the Scherrer equation [11]. This works very well for crys-tal sizes down to approximately 8-10 nm. However, for diameters below this value, standard data analysismethods, like the Rietveld refinement, are not suitable any more since they are based on the assumption ofknown and perfectly periodic structure, which is not fulfilled for small particles. Such nanocrystals rathershow stacking faults, relaxation effects or anisotropic shapes, which all prevent a reliable analysis usingthe considered method.

These difficulties can be overcome by explicitly modeling the structure of the nanocrystals, since thisenables the direct implementation of any structural detail wanted or needed. The corresponding diffrac-togram is computed from the atomic model of the entire crystal using the Debye-formula [11] and com-pared with the data. Then the model is refined until a convincing agreement of calculated and measureddata is achieved. Such an approach includes the size and shape of the particles, (surface-)stress and relax-ation effects as well as stacking faults. We show exemplary results for II-VI semiconductor particles (ZnS,CdS and ZnO) and discuss further developments like the consideration of parameter distributions whichare implemented in the code by an ensemble-averaging procedure.

2 Experimental methods

2.1 Synthesis of the particles

The data presented in this paper were obtained from nanoparticles which were synthesized using wet-chemical routes. Precursors with different concentrations were mixed in solution, and organic stabilizermolecules were added in order to control the growth process and prevent agglomeration. After washing,centrifuging, and drying the nanoparticles were available as a powder. For details of the synthesis processessee [6, 7, 12].

2.2 X-ray diffraction

All X-ray diffraction experiments were performed at the high-flux wiggler beamline BW2 of the Ham-burger Synchrotronstrahlungslabor (HASYLAB). A special setup was used in order to efficiently avoidbackground scattering from amorphous compounds close to the sample. This is necessary since the widthof Bragg reflections of very small nanoparticles can reach magnitudes comparable to the width of thebroad peaks from amorphous materials. Therefore we used a Si single-crystal wafer as sample holder, onthe surface of which the nanoparticles were drop-coated. Additionally, the beampath close to the samplewas He-flooded during the measurement in order to avoid air-scattering and to reduce the beam damageof the sample due to radiation-induced oxidation. Scattering from the Si Bragg reflections was avoided bycarefully aligning the sample. For further details see Ref. [13].

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At a photon energy of 9.6 keV (λ ≈ 1.29 A) we performed powder-scans in the angular range of2θ ≈ 1...90◦. This corresponds to a scattering vectors in the range of |q| = 4π sin(θ)/λ = 0.1... 7.0 A−1.Several short, identical scans were performed in order to detect and (if necessary, avoid) beam damage.For the “single particle modeling” (see Section 3.1) the data was background corrected after summation ofthe individual scans, and then modeled as described below. In the “ensemble modeling” code (see Section3.2) a background fit is included, hence no explicit background correction was performed.

3 Modeling, experimental results and discussion

At first, the “single particle modeling” approach (Section 3.1) was developed in order to fit the measureddata. Herewith, for one individual particle the diffractogram is calculated and fitted to the experimentaldata, i.e., one searches for one individual particle that fits best. As long as the size distribution of themeasured nanoparticles is narrow and the particles are mostly free of crystal defects, this simple approachoften yields very good results. In many cases, however, in particular for particles synthesized using a wet-chemical route, the chance to fulfill these criteria is small. Therefore the method was further developedand an ensemble averaging as well as an automated fitting algorithm were implemented (Section 3.2).

3.1 Single-particle modeling

3.1.1 Method

This modeling approach tries to find an individual atomic model for the nanoparticle, the powder diffrac-togram of which fits the XRD data best. The particles were assembled by zincblende- or wurtzite-likestacking of atomic layers, so that different polytypes of both structures could be simulated. Then theoutermost atoms were removed in order to generate particles of a certain shape and size. Local strainwas accounted for by allowing the lattice parameters to relax to some extent. For the calculation of thediffraction pattern the atoms of the organic stabilizer molecules were neglected (except the atoms whichdirectly bond to the particle surface) since they consist of disordered low-Z compounds and thus do notsignificantly contribute to the powder diffractogram. From the explicit list of atomic coordinates for sucha model the corresponding powder diffractogram was calculated using the Debye formula

I(q) =∑

m

∑

n

fmfnsin(q rmn)

q rmn(1)

(for details see [13, 14]). The equation involves only the atomic form-factor fn of the atoms and theabsolute values of the distances rmn from one atom to every other. q = |q| is the absolute value of thescattering vector. Equation 1 can hence be used to calculate the diffractogram for any atomic structure.

This method is superior to standard analysis methods for powder diffraction data (like, e.g., the Rietveldrefinement) since, in principle, any desired structural feature can intrinsically be implemented in the model.This enables the investigation of standard parameters (size, crystal structure, shape, etc.), and also theevaluation of more dedicated parameters like, in particular, surface relaxation effects, imperfect crystals,stacking faults (e.g., a mixture of wurtzite and zincblende structures, as it is necessary for the system underinvestigation), etc. This is maybe the key-advantage of our approach, since small nanoparticles very oftentend to form polytypes or stacking faults, i.e., they exhibit significant deviations from the bulk structure,an aspect which cannot be tackled by standard approaches.

For modeling the data we followed a general strategy: First spherical particles with either a purezincblende or pure wurtzite structure were considered. Different sizes were tested and – in some cases– already a good agreement with the experimental data could be obtained. However, often this was notthe case, so that in the next step the lattice parameters were varied, and subsequently the effect of stackingfaults was checked. Finally, if necessary, an anisotropic shape was considered. In the “single particle mod-eling”, an automated fitting was not yet established, so that only manual “forward-calculation” could be

3236 F. Niederdraenk et al.: Structure of small II-VI semiconductor nanoparticles

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.pss-c.com

performed. Even though the effort was rather high, a large series of different atomic models with variousparameters were tested.

3.1.2 Modeling of ZnS and CdS particles

Figure 1 shows experimental data of thioglycerol-stabilized ZnS nanoparticles [6, 13] as well as threediffractograms that were calculated with our modeling method. All curves are based on models withspherical shape. The red dash-dotted line stems from a pure wurtzite particle and obviously cannot re-produce the data. The other three curves, which all fit the measured diffractogram relatively well, werecomputed from zincblende-models of different sizes and demonstrate, that a fairly precise determinationof the particle diameter is possible. The solid black curve (2.6 nm in diameter) reproduces the data best,especially at the first two peaks (q ≈ 2.0 A−1 and 3.3 A−1). Only the third reflection is reproducedslightly better by the green dashed line (2.1 nm). However, the correct diameter lies closer to 2.6 nm sincethe relevance of agreement between measured and calculated data decreases with increasing q-values (dueto the increasing number of overlapping Bragg reflections and other effects like, e.g., a slightly incorrecttemperature factor affecting mainly the peaks at higher q-values). There are no significant indications foran anisotropic shape as the peak widths are too close to each other. Hence, we assume spherical crystalliteswith a pure zincblende structure and a diameter of 2.5 ± 0.3 nm for these ZnS particles.

Fig. 1 Powder diffractogram of thioglycerol-stabilized ZnS-nanoparticles and calculated curves for three zincblendeand one wurtzite structural model.

Another example for the “single-particle modeling” approach is shown in Fig. 2. Experimental datafrom glutathion-stabilized CdS particles [7] are shown together with three attempts of reproducing themeasured curve. Neither pure wurtzite, nor pure zincblende models led to a good agreement (red dash-dotted and blue dashed curves, respectively) – not even when strongly distorted unit cells are applied (upto 15% for each direction). While the wurtzite model shows excrescent peaks at q ≈ 2.6 and 3.7 A−1,the peaks stemming from the zincblende model are shifted and show incorrect intensities compared to themeasured curve. Only the consideration of stacking faults, i.e., of a mixture of both structures, improvesthe fit. The black solid curve stems from a spherical particle with a diameter of 2.1 nm which possesseszincblende structure in one, wurtzite structure in the other half. With this assembly all main features in theexperimental data can be reproduced. Only the region around q ≈ 3.7 A−1 is not very well fitted, however,this is the best result that could be obtained for a spherical model. A further improvement by introducingother particle shapes is addressed in Section 3.2.2.

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Fig. 2 Powder diffraction data of glutathion-stabilized CdS nanoparticles and calculated curves for pure wurtzite andzincblende structural models as well as one model including stacking faults. In all cases a spherical shape is assumed,and lattice distortions of up to 10% were allowed in the fitting.

Our results from single-particle modeling suggest a different behavior of very small ZnS and CdSnanoparticles. While the ZnS bulk structure (zincblende) is preserved, even in the case of very smallZnS particles, the CdS particles exhibit significant deviations from a bulk CdS crystal. The experimen-tal CdS data could only be reproduced by a particle consisting of half wurtzite- and half zincblende-likestacked layers, i.e., they show a clear trend to form stacking faults. Both findings (for ZnS and CdS)are in agreement with earlier publications [15, 16]. This, in turn, means that at least small CdS particlescannot be fitted using one of the common XRD analysis techniques like, e.g., the Rietveld refinement.Following these “established” methods, even a size determination must fail due to the mixed structure. AUV/VIS absorption measurement in combination with a tight-binding calculation leads to an approximatesize of 1.8 nm whereas the use of a Rietveld refinement and the Scherrer equation yields 1.3 nm [4,13]. Incomparison, our spherical model results in 2.1 nm diameter when considering stacking faults.

3.2 Automated fitting and ensemble modeling

The results from the single particle modeling are not completely satisfactory due to two main reasons: (1)The high number of parameters which – in principle – can be considered makes the modeling procedurevery inefficient. (2) The assumption that one individual “best fitting” particle is representative for the entirepowder sample is unrealistic, since the particles – in particular those synthesized wet-chemically – are notexpected to have uniform parameters. Therefore we extended the method accordingly. At first, the singlemodeling code was embedded in an evolutionary algorithm [17], which enables a fast fit and a higherreliability of the findings regarding the convergence to the best fit result. Secondly, particle distributionswere considered, so that ensemble properties become accessible.

3.2.1 Method

The ensemble modeling method is based on an evolutionary algorithm. One ensemble is represented by 50individual atomic models which are based on one set of parameters. These parameters cover values for (1)the size of the particles (or several size-parameters when certain anisotropic shapes are considered), (2) thelattice parameters of the basic structure as well as the position of the atoms in the unit cell, (3) parameters



for surface strain relaxation of the particles, and (4) a certain stacking fault probability, i.e., the likelihoodthat the stacking of neighboring layers differs from that of bulk-like stacking. Furthermore, an atomicdisplacement parameter (ADP) and parameters for a polynomial background in the experimental data areconsidered. A Gaussian (or other) distribution can in principle be considered for all of these parameters.In that case the individual parameter is replaced by its mean value and a distribution width.

In the beginning of the modeling procedure a certain range of values for the parameters has to bedefined. Within this range 35 different parameter sets (in the following referred to as “members”) areselected arbitrarily, and – for each of the members – 50 individual structural models are made, i.e., 50particles, according to the parameter set of the corresponding member. The individual particles withinone member differ in their individual structure only if parameter distributions or stacking probabilities areconsidered. They have, for example, a different number of stacking faults built in at different locations, butthey have the same probability for stacking faults. Therefore, one member represents a particle ensembleconsisting of 50 individual particles based on a well defined set of structural parameters.

The 35 (in the beginning arbitrarily) selected members form a “generation” in the fitting procedure.From the first generation (“parent members”), another 35 “child-members” are generated by applying thedifferential algorithm described in Ref. [17]. Now, for each of the parent- and child-members a diffractionpattern is computed by averaging over the calculated scattering curves of all 50 individual particles themember consists of (whereby the curves of each individual are calculated as described in Sec. 3.1). Fromthe resulting 70 diffractograms 35 members are selected by comparing each child with its parent (memberj of generation n with member j of generation n − 1) using a standard Rwp-value as a goodness of fitfunction. These 35 selected members represent the new parents and are used to generate 35 new children,etc. More details on the implementation of this differential evolutionary algorithm into the refinementprocedure will be published elsewhere [18]. A more detailed description of the ensemble fitting routinecan also be found in [19].

After a sufficiently high number of generations (in the range of 1000-5000) this procedure converges tothe member, which can be considered as that ensemble of particles describing the experimental data best.If reasonable starting values and boundaries (in which the parameters are allowed to vary) are defined, theroutine is relatively stable, even though the parameters are partly strongly correlated. The numerical effortfor this evolutionary procedure is quite high since, in each generation, the diffractograms of 35 × 50 newparticles have to be calculated. Depending on the size of the particles (i.e., the number of atoms whichhave to be considered) the modeling of one data set takes several hours up to several weeks on 40 Dual-Octaron PCs in parallel. However, compared to a conventional (e.g., Rietveld) refinement, the number offree parameters is usually not significantly higher, and therefore the risk of getting captured in a relativeminimum of the “goodness of fit” function is, in principal, not higher than for other refinement methods.

3.2.2 Modeling of ZnO particles

In the following we demonstrate the abilities of the method by means of data obtained from ZnO nanoparti-cles. A comparison between the results of a standard Rietveld refinement and our new ensemble modelingapproach as well as the variations of differently shaped models are shown.

Comparison of Rietveld refinement and ensemble modelingFigure 3 shows a comparison of the results of a typical Rietveld refinement and our ensemble modeling

method. The lower part contains the experimental data together with the corresponding Rietveld fit resultwhich was obtained using the code SIMREF [20]. We assumed a pure wurtzite structure in order to obtainthe positions of the (hkl) reflections and a Pseudo-Voigt function for their profile. An RwP -value of 6.2%demonstrates a reasonable agreement of data and fit. From the width of the peaks, the particle size of3.0 nm can be computed using the Scherrer equation [11].

A detailed comparison of measured and calculated curves reveals that there are mainly two peaks that arenot well reproduced by the Rietveld refinement, namely the (100) at q ≈ 2.3 A−1 and the (110) reflectionat q ≈ 3.8 A−1, the intensities of which are too small in the calculation. Smaller problems occur with

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Fig. 3 Comparison of the results of a Rietveld fit (bottom) and the new ensemble modelling procedure (top) for thesame XRD data obtained from ZnO particles. The dots represent the measured data.

B

C

topview sideview

A

Fig. 4 Shapes of three different polyhedra which were used in the modeling process. For details see text.

the (101) reflection at q ≈ 2.5 A−1, which is larger in the calculation, and with the peak at q ≈ 4.5 A−1

corresponding to the (200), (112), and (201) reflections, which is too wide. Such deviations often arise fromanisotropic particle shapes which – in our case – could be well confirmed using the ensemble modelingapproach. The result of that is shown in the upper part of Fig. 3, together with the same experimental data.The Rwp-value improved to 5.2%. This fit is based on a cylindrical atomic model with hexagonal basis asshown in Fig. 4 labeled “A”. This model has two size parameters, one for the height ([001]-direction, thedirection of the layer stacking) and one for the diameter of the layers. A closer look to the data reveals that



all peaks mentioned above fit better than in the Rietveld refinement, even though they do not yet perfectlyreproduce the experimental data. Higher calculated intensities for (hk0) reflections correspond to a largersize of the particle in lateral direction (4.1 nm in diameter), whereas the height of the “A”-model hardlydiffers from the Rietveld result (2.8 nm compared to 3.0 nm from the Rietveld fit). However, there isalso one reflection, the (103) at q ≈ 4.3 A−1, which is less well described by the ensemble modelingmethod. This could not (yet) be improved by considering different shapes, see below. But since the overallagreement improved significantly, we can conclude that considering the anisotropic shape represents a stepforward and enables the access to more detailed structural information on this nanoparticle system.

Particle modeling: Different shapes

In the entire modeling process the selection of the correct hape for the atomic model is the key-issue.Figure 5 shows a comparison of modeling results for four different selected shapes (out of eleven, for whichcalculations were performed). As mentioned above the simplest model used as starting point is a spherewithout any structural defects (lowest curve in the figure). For the ZnO nanoparticles under investigationthis model already yielded an acceptable result, only the (100) and (110) reflections (at q ≈ 2.2 and≈ 3.8 A−1, respectively) were inadequately reproduced. An Rwp-value of 7.4% was obtained and adiameter of 3.3 nm. However, testing anisotropic structural models improved the situation. In addition tothe “A”-shape discussed above, data obtained from two other models are shown in Fig. 5, “B” and “C”.Compared to “A” these polyhedra have additional {111}-oriented surfaces (in order to avoid 90°-edges atthe top and bottom of the cylinder, see Fig. 4). The difference between models “B” and “C” is that (besidethe {001} surfaces at the top and bottom of the particles) “B” has only {111}-oriented planes whereas “C”also has vertical surface planes (of the {100}-type, see Fig. 4). The number of size-related fit-parameterstherefore increases to three for polyhedron “C”.

Fig. 5 Modeling of the data from Fig. 3 using the models depicted in Fig. 4 and one spherical model. The dotsrepresent the measured data. For details see text.

phys. stat. sol. (c) 4, No. 9 (2007) 3241


Both polyhedra “B” and “C” were refined to very similar Rwp-values of ≈5.8%. They represent noimprovement compared to “A”, even though the analysis of these structures is still in progress; therefore “A”is still the favorable shape. As next step in the modeling, stacking faults were considered. We performedsuch calculations for all four shapes presented. Only for the sphere a significant improvement could beobserved, which is not very surprising since stacking faults can partly compensate for an anisotropic shapeof the nanoparticles. They reduce the coherence of the scattering process in a sophisticated way (i.e.,differently for different (hkl) reflections), which can have similar effects like specific particle shapes.Therefore anisotropic shape-parameters and stacking faults are often highly correlated. In our case wecannot exclude that the true structure of the nanoparticles under study is a mixture of both, anisotropicshape and stacking faults. This aspect is under further investigation. Since the computational effort forcalculations of particles with stacking faults is very high, the calculations are not yet finished. The resultswill be published in a forthcoming paper [21].

In conclusion, the cylindrical “A”-model with an average height of 2.8 nm and a diameter of 4.1 nm, i.e.,a moderate aspect ratio of 0.7, matches the experimental data best. Stacking faults do not play a significantrole for these particles. This finding is remarkable since differently synthesized ZnO particles exhibit avery high number of stacking faults [13, 19]. The unit cell dimensions and positions of the oxygen atomsin the unit cell refined to a = 3.275 A, c = 5.254 A, and zO = 0.387 A. These values are ≈ 1% largerthan the bulk values (a = 3.2494 A, c = 5.2038 A, zO = 0.3830 [22]) indicating a small lattice relaxationof the particles. This is consistent with results obtained from other ZnO particles [13, 19]. Therefore, theparticles conserve an almost undisturbed ZnO bulk-structure, but tend to a flat shape.

4 Summary and outlook

We have demonstrated the recent progress in the structural investigation of very small nanoparticles (<5 nm), mainly by improving the methods for modeling X-ray powder diffraction data. Besides the ba-sic properties like size, shape, and lattice constants, other parameters like stress, relaxation effects, andstacking faults, which are usually difficult to access, can be obtained. The new method is based on theDebye formula, which is used to calculate the diffracted intensities of ensembles of particles representedby specific atomic models. Ensemble averaging was implemented in order to study parameter distributionsfor, e.g., size and shape, and for considering inhomogeneities like stacking faults, which occur with certainprobabilities. The power of the technique is demonstrated by exemplary data obtained from ZnS, CdS, andZnO particles. While ZnS and ZnO nanocrystals exhibit almost undisturbed crystal structures, even forvery small sizes, the necessity of considering stacking faults was demonstrated for CdS particles.

At present, the potential of the code regarding parameter distributions is not yet fully utilized, partlydue to problems with highly correlated parameters. This aspect will be improved within the near futureand is expected to improve the fit results since there are some indications for particles having two ormore preferred particle sizes. However, a careful handling of constraints for some of the parameters willalways be necessary since strongly correlated parameters often occur when nanoparticles are modeled.Furthermore, the consideration of gradual surface strain will be enabled. In general, we hope that quantumchemical calculations benefit from our detailed structural results, and that we herewith contribute to animprovement of the general understanding of quantum size effects in very small nanoparticles.

Acknowledgements The authors acknowledges technical assistance by the HASYLAB staff. This work was sup-ported by the Deutsche Forschungsgemeinschaft through the Sonderforschungsbereich 410 (TP C5, B8 and C6), theVolkswagen-Stiftung (Project I/74922), and the IHP program “Access to Research Infrastructures” of the EuropeanCommission (HPRI-CT-1999-00040).



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