domain: spi chim phys

text:
Dr Ioanna Dimkou

Dr Anjali A Harikumar

Akhil Ajay

Dr Fabrice Donatini

Dr Edith Bellet-Amalric

Adeline Grenier

Dr Martien I Den Hertog

Dr Stephen T Purcell

Eva Monroy
email: eva.monroy@cea.fr

Dr E Ajay

Monroy

Design of AlGaN/AlN dot-in-a-wire heterostructures for electron-pumped UV emitters

Keywords: ultraviolet, nanowires, AlGaN, quantum dot, lamp

This paper describes the fabrication of nitrogen-polar AlxGa1-xN/AlN (x = 0, 0.1) quantum dot superlattices integrated along GaN nanowires for application in electron-pumped UV sources.

The nanowires are grown using plasma-assisted molecular-beam epitaxy on n-type Si(111) wafers using a low-temperature AlN nucleation layer. Growth conditions are tuned to obtain a high density of non-coalesced nanowires. To improve the uniformity of the height along the substrate, the growth begins with a base long nanowire (~900 nm), with a diameter of 30-50 nm. The AlxGa1-xN/AlN active region is 400 nm long (88 periods of quantum dots), long enough to collect the electron-hole pairs generated by an electron beam with an acceleration voltage £ 5 kV. The spectral response is tuned in the 340 to 258 nm range by varying the dot/barrier

thickness ratio and the Al content in the dots. Internal quantum efficiencies as high as 63% are demonstrated.

Introduction

UV light emitting diodes (LEDs) are in demand for disinfection purposes [1] at specific wavelengths. They also offer lower power consumption, the potential for longer lifetime, lower voltage requirement and non-toxic waste after use, as opposed to their mercury lamp counterparts. AlGaN is the material of choice for this technology because of its direct bandgap and the possibility to dope it both n-type and p-type [2,3] . Having said that, such devices have not reached their industrial maturity yet due to drawbacks in terms of electrical injection and light extraction, which limit their wall-plug efficiency (WPE) [4] . Let us remind that the WPE is the ratio between the optical output power and the electrical input power. WPE can be written as WPE = LEE × IQE×EIE, where LEE is the light extraction efficiency, IQE is the internal quantum efficiency (number of photons generated per electron-hole pair injected in the active medium), and EIE is the electron injection efficiency.

A promising alternative for the fabrication of highly efficient, eco-friendly UV lamps is based on the injection of electrons into AlGaN/AlN nanostructures using a miniaturized electron gun, e.g. a carbon nanotube cathode. An electron-pumped device looks like a vacuum tube or valve [START_REF] Herbert | Principles of Electron Tubes[END_REF] , which can be scaled down to millimeter size. Such a configuration avoids the problems associated with the carrier injection (dopant ionization energy, asymmetric carrier mobility of electrons and holes, realization of ohmic contacts). In the case of pumping with an electron beam, the average energy required to generate an electron-hole pair in a given semiconductor is close to 3 times its band gap energy [START_REF] Leamy | [END_REF] . Therefore, the electron injection efficiency is wavelength independent (EIE » 1/3), and it should be possible to fabricate lamps operating close to the AlN band gap (e.g. 210 nm) if the IQE and LEE of the emitting structure is high enough. . In the spectral range generally used for disinfection, implantation of such a technology would depend basically on the cost per milliwatt of optical power, which could become comparable to LEDs thanks to the higher device output power (easily scalable with the semiconductor surface) and simpler material growth and processing.

Electron-pumped emission from GaN and AlGaN quantum wells has been reported [7][8][9][10][11][12] , but such lamps displayed low wall plug efficiency mostly due to the poor light extraction efficiency. Furthermore, stimulated emission around 350 nm has been demonstrated in electron-pumped GaN/AlGaN ridge structures under pulsed pumping [12][13][14] . For the implementation of such electron-pumped UV sources, the semiconductor geometry and conductivity must be adapted to maximize the energy conversion, light extraction and charge evacuation. There are several motivations to consider nanowires (NWs) for this application.

First, their as-grown shape spontaneously favors light extraction [15] , without additional processing requirements. In addition, the relatively easy implementation of three dimensionally confined objects (dots in a wire) results in high values of internal quantum efficiency (IQE) at room temperature [16] . Last but not the least, their direct growth self-assembled on silicon substrates reduces the cost of the system. One of the main challenges for the growth of NW ensembles for electron-pumped UV sources is the homogeneity of the dot thickness and composition along an active region that is hundreds-of-nm long. It is known that the luminescence from GaN/AlGaN and AlGaN/AlN superlattices (SLs) on GaN NWs presents strong spectral dispersion, with broad and sometimes multi-line spectral profiles [17,18] . Such dispersion is due the variation of the dot height and diameter along the growth axis [19][20][21] , the different strain relaxation in the dots along the growth axis [19,20,22] , the presence of monolayer thickness fluctuations in the dots [19] , and the alloy inhomogeneities and inter-diffusion phenomena at the hetero-interfaces, which depend on the strain state of the structure [18] . Additional perturbations can be introduced by coalesced areas and lateral GaN inclusions [23] . It is hence highly important to develop precisely controlled growth processes that allow the synthesis of homogeneous AlGaN/AlN quantum dot SLs in a NW geometry.

In this work, we describe the growth and characterization of N-polar AlxGa1-xN/AlN (x=0, 0.1) quantum dot SLs grown as extensions on GaN NWs and designed for their application in electron-pumped UV sources. The NWs are grown using plasma-assisted MBE on n-type Si(111) wafers using a low temperature AlN nucleation layer. Growth conditions are tuned to obtain a high density of non-coalesced GaN NWs. On top of them, the AlxGa1-xN/AlN active region is 400 nm long (88 periods of quantum dots). Such structures present single-line spectral emission with IQE that can reach 63%.

Results and discussion

Self-assembled N-polar GaN NWs were synthesized using plasma-assisted molecular beam epitaxy (PAMBE) on n-type Si(111) substrates using a low-temperature AlN nucleation layer, as described elsewhere [24,25] . To improve the height uniformity [26] , the growth began with a long (~900 nm) NW base with a diameter of 30-50 nm, grown under N-rich conditions (Ga/N flux ratio: FGa/FN = 0.25) at a substrate temperature TS = 810°C and at a growth rate vG = 290 nm/h. The choice of growth conditions is based on the study described in figure 1.

From these bird's eye and top-view scanning electron microscopy (SEM) images, we observe that increasing the substrate temperature from 770°C to 810°C results in a reduction of the NW density, at the price of a slight decrease of the growth rate due to the enhancement of GaN decomposition. However, at 825°C the decomposition rate is too high, and the NW height becomes highly inhomogeneous.

The growth of the GaN base was followed by the deposition of a 400-nm-long AlxGa1-xN/AlN (x = 0, 0.10) SL, i.e. 88 periods of quantum dots. The AlxGa1-xN dots were grown using the same Ga flux as for the NW base, and adding an Al flux FAl = x/vG, where x is the targeted Al mole fraction. The AlN sections were grown at the stoichiometry (FAl/FN = 1). The whole SL was synthesized without any growth interruption. To adjust the spectral response, we varied the dot/barrier thickness ratio and the Al content in the dots, maintaining the number of periods (88) and the period length (4.5 nm) constant. The values of Al mole fraction in the dots given in this manuscript are nominal values. From the position of the (0002) reflection of the quantum dot SL in x-ray diffraction studies we estimate that the error bar in the composition is in the range of ± 2%. The Al mole fraction in the AlxGa1-xN dots was kept intentionally low, around x = 0.1, to prevent deformations of the nanowire morphology and reduce the effects of alloy inhomogeneities, observed for x ≥ 0.3 [18] . The nominal characteristics of the samples presented here are listed in table 1.

Figures 2(a other, but highly packed. Top view images point to a NW density of 6-8×10 9 cm -3 . The NW diameter is in the range of 30-50 nm at the base, increasing to 50-70 nm in the top region due to the enhancement of lateral growth when depositing AlN [27,28] .

The as-grown NW ensemble was characterized using XRD in order to validate the periodicity of the structure, with the results illustrated in Figure 2(c). These measurements provide average information over a surface of several millimeters squared, and are hence sensitive to the dispersion of tilts and twists in the NW ensemble. Several satellites of the (0002) reflection of the SL are resolved, which confirms the thickness uniformity along the growth axis and the reproducibility of the period length from sample to sample (period = 4.3±0.2 nm).

It should be noticed that the (0002) reflection of GaN originated by the NW base is not located at the angular position of relaxed GaN (2q = 34.57°), but at slightly larger angles.

To get highly resolved information on the NW structure, the NWs were dispersed on holey carbon membranes for scanning transmission electron microscopy (STEM) observations. In both samples, we observe a decrease of the thickness of the AlN shell along the growth axis, which is more important in the sample with higher Al content in the SL. This is consistent with the fact that the AlN shell forms as a result of the low mobility and of the Al atoms impinging on the NW sidewalls. From STEM analysis of various NWs, the thickness of the shell is estimated to be only 1-2% of the axial AlN thickness. The presence of the AlN shell imposes a uniaxial compressive stress to the GaN base, which justifies the shift of the GaN (0002) XRD reflection as observed in Figure 2(c).

Optical characterization by cathodoluminescence (CL) at room temperature, with an electron acceleration voltage VA = 5 kV, is presented in Figure 4(a). The emission blue shifts when we decrease the dot height and increase their Al content. A spectral shift from 340 nm to 260 nm is demonstrated while maintaining approximately the same spectral linewidth. The acceleration voltage (VA) determines the penetration depth of the electrons in the structure, which depends mostly on the material density [START_REF] Leamy | [END_REF] . The thickness of the SL was initially decided by assuming that the sample behaves similar to a planar structure with the same geometry along the growth axis. However, in a NW ensemble, the average material density is smaller, and there is a risk of electron channeling when the direction of impinging electrons is perpendicular to the silicon substrate. To assess the relevance of these phenomena, we have studied the variation of the CL spectra as a function of VA, as shown in Figure 4(b). If the emission spectra are normalized to their maximum, the emission from the GaN base becomes visible only for VA > 5 kV. This implies that the NW ensemble is compact enough to prevent important deviations of their behavior with respect to planar structures.

Finally, we have assessed the IQE of the samples by analyzing the variation of their emission, either photoluminescence (PL) or CL, as a function of temperature. As an example, The IQE is often estimated as the ratio of the integrated luminescence intensity at room temperature and at low temperature: IQE » I(300 K) / I(0 K). However, it should be kept in mind that this expression assumes that I(0 K) is not affected by non-radiative processes.

Overestimation of the IQE can happen if non-radiative recombination paths are active at low temperature, but also if the pumping intensity is high enough to saturate non-radiative recombination paths at high temperature. The measured IQE is also higher under pulsed excitation, due to the difference in the behavior of radiative and non-radiative recombination with respect to time. To get a reliable measurement, our PL experiments were performed under continuous wave excitation, using very low power densities (10 µW pumping power focused on a spot with a diameter around 100 µm). The fact that the luminescence intensity is almost constant up to 50 K is an indication that non-radiative recombination is negligible at low temperature (» 6 K). Therefore, we can assume that the expression IQE » I(300 K) / I(» 6 K) is valid in our case. Following this reasoning, the IQE values of the samples under study are summarized in table 1. In particular, from figure 5(b) we can extract an IQE » 44% for sample S4, emitting at 286 nm. This value increases up to IQE » 63% for S1, emitting at 340 nm, and it decreases to » 22% for S5, emitting at 258 nm.

It is important to keep in mind that the IQE values reported above describe the intrinsic material properties of the structures, i.e. they refer to the ratio between radiative recombination and total recombination when the perturbation introduced by the excitation source is minimum (continuous wave excitation, power density around 0.00013 kW/cm 2 ). The most common values reported as IQE in the literature for AlGaN LEDs are measured under pulsed excitation [29][30][31][32][33][34][35][36][37][38] , using power densities in the range of 5-1000 kW/cm 2 [29][30][31][32]34,35] . This leads to photogenerated carrier densities higher than the doping level of the original structure, to emulate the carrier density under operation conditions. In such a situation, nonradiative recombination paths are partially saturated, so that the obtained IQE is often significantly higher than the lowinjection value, and depends on the excitation power density [29][30][31][32]35,39] .

In the case of high injection, the calculation of the IQE must take into account the drop of PL efficiency at low temperature due to the high-power excitation [35] :

𝐼𝑄𝐸(𝑃) = !(#$$ &,() !($ &,() × !($ &,() !($ &,*+, -./012-+.) (1)
where 𝐼(𝑇, 𝑃) is the integrated PL intensity as a function of temperature and excitation power (P). We have measured the PL of the NWs as a function of the excitation power using a pulsed Nd-YAG laser (266 nm, 2 ns pulses with a repetition rate of 8 kHz). Measurements were performed at 6 K and at 300 K, and the room-temperature IQE was calculated using equation 1. The results for samples S1 and S2, presented in the inset of figure 5(b), show that the NW samples are stable enough to stand pulsed operation at high power densities up to around 50 kW/cm 2 without degradation of the IQE. Note that in an electron-pumped UV lamp, using an acceleration voltage of 5 kV and an injection current of 1 mA to irradiate a spot with a diameter of 1 mm, the excitation density would be below 1 kW/cm 2

Conclusion

In summary, we have demonstrated the synthesis and characterization of AlxGa1-xN/AlN (x = 0, 0.1) quantum dot SLs on GaN NWs designed for application in electron-pumped UV sources.

The AlxGa1-xN/AlN active region is 400 nm long (88 periods of quantum dots), long enough to collect the electron-hole pairs generated by an electron beam with an acceleration voltage £ 5 kV. The NWs are grown using plasma-assisted MBE on n-type Si(111) wafers using a low temperature AlN nucleation layer. To adjust the spectral response, we have varied the dot/barrier thickness ratio and the Al content in the dots. By growing the AlxGa1-xN quantum dots under N-rich conditions and the AlN barriers at the stoichiometry, we obtain SLs with a regular periodicity along the whole active region. In all the cases, the room temperature emission is dominated by a single spectral line that can be tuned from 340 nm (IQE > 60%), to 286 nm (IQE = 44 %) and 258 nm (IQE = 22%).

In this manuscript, we have focused on the assessment of these nanostructures in terms of internal quantum efficiency and stability under electron pumping. The estimation of the wall plug efficiency should consider the complete system, including carrier injection losses and light extraction efficiency. To reduce carrier injection losses, cold cathodes with efficiencies close to unity are nowadays available [40,41] , and the design of the semiconductor structure should follow the advices discussed in this paper (particularly the superlattice total length and high nanowire density), so that the free carriers generated by impact ionization recombine within the active region of the device. Regarding light extraction, the nanowire geometry should result in improved performance in comparison with planar layers [15] , but the effect of the substrate and the GaN stem should be quantified. If relevant, the potential advantage of detaching the nanowire ensemble from the silicon substrate should be explored [42] .

Experimental Section

X-ray diffraction (XRD) measurements where performed in a Rigaku SmartLab x-ray diffractometer using a 4 bounce Ge(220) monochromator and a long plate collimator of 0.228° for the secondary optics.

The morphology of the as-grown NW ensemble was studied by field-emission scanning electron microscopy (SEM) using a Zeiss Ultra 55 microscope. Table 1. Nominal characteristics of the samples under study: AlN barrier thickness (tB), GaN dot height (tQD), Al mole fraction in the dots (x). Peak emission wavelength at room temperature (lRT), full width at half maximum of the emission at room temperature (FWHMRT) and IQE.

Sample tB (nm) tQD (nm)

x lRT (nm) FWHMRT (nm) IQE (%) S1

  Figures 2(a) and (b) present typical (a) cross-section and (b) top-view SEM images of an

Figure 3 (

 3 Figure 3(a-b) presents high-angle annular dark field (HAADF) STEM images of a GaN/AlN

Figure 5 (

 5 Figure 5(a) displays the evolution of the PL spectrum as a function of temperature for a sample

Figure 5 (

 5 b) depicts the integrated luminescence intensity from S4 as a function of the inverse temperature measured both by PL and CL, showing that there is no significant difference between the two measuring techniques. The variation of the integrated PL intensity from S1 with temperature is also presented.

Figure 1 .

 1 Figure 1. SEM images (bird's eye on the left and top view on the right side) of GaN NW samples grown at various substrates temperatures: (a) 770°C, (b) 790°C, (c) 810°C, and 825 °C. The NW height is indicated as a legend in the bird's eye figure. All the samples grew for 3 hours with a flux ratio FGa/FN = 0.25.

Figure 2 .

 2 Figure 2. (a) Cross-section and (b) top-view SEM images of a sample containing GaN NWs with a superlattice (SL) consisting of 88 periods of Al0.1Ga0.9N/AlN (S4). (c) XRD q-2q scans of the (0002) reflection of GaN NWs containing an 88-period AlxGa1-xN/AlN quantum dot SL. The nominal Al concentration is color-coded and the height of the quantum dots (QD) is indicated on the right side of the figure. The diffractograms are vertically shifted for clarity.

Figure 3 .

 3 Figure 3. (a) HAADF-STEM image of dispersed NWs containing 88 periods of GaN/AlN (1.5 nm/3.0 nm). The bright contrast is GaN and the darker contrast represents AlN. Note that in HAADF-STEM the contrast of the image provides chemical information. (b) Highmagnification HAADF-STEM image of a NW containing 88 periods of GaN/AlN (S1). (c) High-magnification HAADF-STEM image of a NW containing 88 periods of Al0.1Ga0.9N/AlN (S5).

Figure 4 .

 4 Figure 4. Room-temperature CL measurements of NWs containing an 88-period AlGaN/AlN quantum dot SL. (a) Measurements performed at an acceleration voltage VA = 5 kV of samples with the following SL structures: S1: 1.5 nm GaN / 3.0 nm AlN, S3: 0.75 nm GaN / 3.75 nm AlN, S4: 0.75 nm Al0.1Ga0.9N / 3.75 nm AlN, S5: 0.65 nm Al0.1Ga0.9N / 3.85 nm AlN. (b) Normalized CL spectra of the S4 structure as a function of the accelerating voltage. The emission from the GaN stem of the NWs is only resolved for VA > 5 kV.

Figure 5 .

 5 Figure 5. (a) Variation of the PL spectra as a function of temperature (sample S3: 0.75 nm Al0.1Ga0.9N / 3.75 nm AlN). (b) Variation of the integrated PL intensity of S1 and S4, and the integrated CL intensity of S4 as a function of the inverse temperature, normalized to their value at low temperature. The solid lines are fits to 1/[1 + A exp(-Ea/(kT))], where Ea is a thermal activation energy, kT is the thermal energy, and A is a fitting constant. For S1, Ea = 38±5 meV and A = 2.3±0.5. For S4, Ea = 48±4 meV and A = 7±2. Inset: Variation of the IQE as a function of the excitation power density, measured with a pulsed laser.

Acknowledgements

This work is supported by the French National Research Agency (ANR) via the UVLASE program (ANR-18-CE24-0014), and by the Auvergne-Rhône-Alpes region (grant PEAPLE). This project has also received funding from the European Research Council under the European Union's H2020 Research and Innovation programme via the e-See project (grant #758385). We also acknowledge technical support from F. Jourdan, Y. Curé and Y. Genuist. We benefited from the access to the technological platform NanoCarac of CEA-Minatech Grenoble in collaboration with the IRIG-LEMMA group.

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Detailed structural studies were conducted using high-resolution transmission electron microscopy (HR-TEM) and high-angle annular dark-field scanning transmission electron microscopy (HAADF-STEM) performed on a probe-corrected FEI Titan Themis microscope operated at 200 kV.

Cathodoluminescence (CL) experiments were performed using a field-emission SEM FEI Inspect F50 equipped with a low-temperature Gatan stage to cool the sample down to 6 K, and with an IHR550 spectrometer.

Photoluminescence (PL) spectra were obtained by excitation with a frequency-doubled continuous-wave solid-state laser (λ = 244 nm), with an optical power of » 10 μW focused on a spot with a diameter of » 100 μm. Additionally, PL measurements under pulsed excitation were performed using a Nd-YAG laser (266 nm, 2 ns pulses with a repetition rate of 8 kHz) focused on a spot with a diameter of » 100 μm. In both cases, the sample was mounted on a cold-finger cryostat, and its emission was collected by a Jobin Yvon HR460 monochromator equipped with a UV-enhanced charge-coupled device (CCD) camera.

ToC entry:

This paper describes the fabrication of AlGaN/AlN quantum dot superlattices on GaN nanowires for application in electron-pumped UV sources. The active region is designed to collect the carriers generated by an electron beam with an acceleration voltage of 5 kV. The emission is tuned in the 340 to 258 nm range by varying the dot/barrier thickness ratio and the Al content in the dots.


Jorick Maes

Nicolo Castro

Kim De Nolf

Willem Walravens

Benjamin Abécassis
email: benjamin.abecassis@ens-lyon.fr

Zeger Hens
email: zeger.hens@ugent.be

Size and Concentration Determination of Colloidal Nanocrystals by Small-Angle X-ray Scattering

Keywords: Colloidal Nanocrystals, Small-Angle X-ray Scattering, Sizing Curves, Concentration

published or not. The documents may come

Introduction

The development of colloidal synthesis methods that result in the formation of semiconductor nanocrystals (NCs) with a narrow size dispersion and tunable sizes have turned these materials into model systems for nanoscience, with research interests covering diverse topics such as size quantization effects, [START_REF] Murray | Synthesis and characterization of nearly monodisperse CdE (E = sulfur, selenium, tellurium) semiconductor nanocrystallites[END_REF][START_REF] Alivisatos | Semiconductor Clusters, Nanocrystals, and Quantum Dots[END_REF][START_REF] Murray | Synthesis and Characterization of Monodisperse Nanocrystals and Close-Packed Nanocrystal Assemblies[END_REF] the chemistry of nanoscale surfaces, [START_REF] Nag | Metal-free Inorganic Ligands for Colloidal Nanocrystals: S 2-2 , HS -, Se 2-, HSe -, Te 2-, HTe -, TeS 2-3 , OH -, and NH -2 as Surface Ligands[END_REF][START_REF] Anderson | Ligand Exchange and the Stoichiometry of Metal Chalcogenide Nanocrystals: Spectroscopic Observation of Facile Metal-Carboxylate Displacement and Binding[END_REF][START_REF] Hens | A Solution NMR Toolbox for Characterizing the Surface Chemistry of Colloidal Nanocrystals[END_REF] and the selfassembly of nanoscale colloids. [START_REF] Murray | Colloidal synthesis of nanocrystals and nanocrystal superlattices[END_REF][START_REF] Evers | Low-Dimensional Semiconductor Superlattices Formed by Geometric Control over Nanocrystal Attachment[END_REF][START_REF] Shevchenko | Structural diversity in binary nanoparticle superlattices[END_REF] In addition, they have been widely studied as alternative opto-electronic materials [START_REF] Kovalenko | Prospects of Nanoscience with Nanocrystals[END_REF] that combine size-and surface-dependent optical and electronic properties [START_REF] Giansante | Darker-than-Black" PbS Quantum Dots: Enhancing Optical Absorption of Colloidal Semiconductor Nanocrystals via Short Conjugated Ligands[END_REF][START_REF] Brown | Energy Level Modification in Lead Sulfide Quantum Dot Thin Films through Ligand Exchange[END_REF] with a suitability for solution-based processing. [START_REF] Mcdonald | Solution-processed PbS quantum dot infrared photodetectors and photovoltaics[END_REF][START_REF] Kim | 25th Anniversary Article: Colloidal Quantum Dot Materials and Devices: A Quarter-Century of Advances[END_REF][START_REF] Yuan | Colloidal quantum dot solids for solution-processed solar cells[END_REF] A general need faced by both scientific research on NCs and the further development of NC technology is the accurate determination of the average size of NCs in a given batch. NC sizing is, for example, essential to quantify size-dependent effects, and compare experimental results with theoretical predictions. Moreover, looking at the synthesis of NCs, both practical synthesis protocols and studies into nucleation and growth mechanisms rely on knowledge of the NC size, either to determine NC concentrations or to map the variation of the NC size throughout the reaction. [START_REF] Owen | Precursor Conversion Kinetics and the Nucleation of Cadmium Selenide Nanocrystals[END_REF][START_REF] Abe | Reaction Chemistry/Nanocrystal Property Relations in the Hot Injection Synthesis, the Role of the Solute Solubility[END_REF] In addition, size and concentration are key quantities to know when applying NCs as biolabels, [START_REF] Bruchez | Semiconductor Nanocrystals as Fluorescent Biological Labels[END_REF] assessing NC toxicity, [START_REF] Alkilany | Toxicity and cellular uptake of gold nanoparticles: what we have learned so far[END_REF][START_REF] Elsaesser | Toxicology of nanoparticles[END_REF] or when simulating the functioning of NC-based opto-electronic devices. [START_REF] Xie | Colloidal Quantum Dots Enabling Coherent Light Sources for Integrated Silicon-Nitride Photonics[END_REF] At present, the average diameter of NCs in a given batch is mostly determined by means of transmission electron microscopy (TEM). Operated in the most common, bright field mode, contrast in the 2D images results from a difference in mass and thickness between the sample and the carbon-film support. Therefore, the crystal regions of NCs will appear dark and this enables sizes to be determined by automated software packages, such as Digital Micrograph and ImageJ. Herein, particles are localized and measured by a variety of thresholding algorithms. [START_REF] Pyrz | Particle Size Determination Using TEM: A Discussion of Image Acquisition and Analysis for the Novice Microscopist[END_REF] It should be noted that the interpretation of such images is often hampered by the simultaneous occurrence of the contrast-forming phenomena, [START_REF] Chattopadhyay | An Introduction to Nanoscience and Nanotechnology ; PHI Learning[END_REF] and that the particle-background boundary determination is prone to operator bias. In the particular case of semiconductor NCs or quantum dots, TEM imaging is typically used to construct calibration curves that relate the NC diameter to the energy of the band-edge transition. As this quantity is conveniently determined using absorbance spectroscopy, such so-called sizing curves make sizing straightforward, up to the point where it can be done in-situ.

Small-angle x-ray scattering (SAXS) is an alternative to determine NC diameters. As compared to TEM, SAXS has several advantages for the purpose of NC sizing. It is performed in solution, which makes that possibly detrimental drying artifacts, such as evaporation induced aggregation, can be avoided. Moreover, in a SAXS measurement, an ensemble of typically 10 6 particles is probed; a number that exceeds by 4 orders of magnitude the population adressed in a TEM analysis. Furthermore, analysis procedures induce little user bias. This point was recently made clear by a round-robin experiment, [START_REF] Pauw | Nanoparticle size distribution quantification: results of a small-angle X-ray scattering inter-laboratory comparison[END_REF] in which the same sample was measured by different teams and their results were compared. Across the different laboratories, measured values of the mean radius and the polydispersity of a gold nanoparticle sample had an error below 1%, a result that confirms SAXS to be a very reliable technique to determine the size distribution of colloidal NCs.

In the case of semiconductor NCs, SAXS has been used for a long time to measure diameters.

3

One of the first papers reporting quantum confinement of CuCl in glass used SAXS to determine the size of the crystallites. [START_REF] Golubkov | Growth kinetics of CuCl microcrystals in a glassy matrix[END_REF] Later, Mattoussi et al. [START_REF] Mattoussi | Characterization of CdSe nanocrystallite dispersions by small angle x-ray scattering[END_REF][START_REF] Mattoussi | Properties of CdSe nanocrystal dispersions in the dilute regime: Structure and interparticle interactions[END_REF] used SAXS to investigated the interaction between CdSe NCs. Vossmeyer et al. used (powder) SAXS to assign diameters to different batches of CdS NCs, arguing that for small sizes TEM images were insufficiently clear. [START_REF] Vossmeyer | CdS Nanoclusters: Synthesis, Characterization, Size Dependent Oscillator Strength, Temperature Shift of the Excitonic Transition Energy, and Reversible Absorbance Shift[END_REF] For the few batches these authors analyzed both by TEM and SAXS -involving sub 4 nm CdS NCs -they systematically report TEM-based diameters that exceed SAXS-based diameters by 10 to 20%. Whereas several studies continued to use SAXS for the in-situ observation and analysis of NC nucleation and growth, [START_REF] Abécassis | Real-Time in Situ Probing of High-Temperature Quantum Dots Solution Synthesis[END_REF] reports in which SAXS-based diameters are correlated with absorbance data to calibrate sizing curves are limited. A rare example is the study of Weidman and coworkers, who compared SAXS and TEM data obtained on PbS NCs. [START_REF] Weidman | Air-Stable PbS Nanocrystals via Precursor Stoichiometry Control[END_REF] Focusing on NC larger than 4.5 nm, they found that both approaches yield largely identical results. Grinolds et al. have also determined a sizing curve for 3 to 5.5 nm-sized PbS NCs using SAXS, and reported a significant deviation from previous findings. [START_REF] Grinolds | Quantum-Dot Size and Thin-Film Dielectric Constant: Precision Measurement and Disparity with Simple Models[END_REF] For a given band-gap, they found that diameters estimated from SAXS measurements exceeded values calculated from TEM-based sizing curves. In both studies, the q-range was limited (0.07 to 0.4 Å -1 and 0.17 to 0.46 Å -1 for Weidman et al. and Grinolds et al. respectively).

These contradictory results and limited q-range motivated us to perform an in depth study on nanocrystal sizing by SAXS that involves both Pb-and Cd-based sulfides and selenides.

For all these materials, colloidal synthesis methods cover a considerable range of diameters, and absorbance spectra exhibit a pronounced peak corresponding to band-edge transition. To make the method more accessible as an experimental technique for size and concentration determination, we first provide very detailed experimental methods and a step-by-step protocol for SAXS measurements, background correction and absolute intensity calibration. Next, focusing on a series of PbS QDs, we compare several methods to retrieve the diameter distribution from the SAXS patterns. These methods include direct fits of the SAXS curve using predefined distributions, next to a more recently developed Monte-Carlo approach that does not start from an analytical expression of the diameter distribution. We show that all methods yield similar results for samples having a narrow and monomodal size distribution. For all four materials studied, this analysis yields a diameter that varies smoothly as a function of the NC band gap measured by means of absorbance spectroscopy. In the case of PbS and PbSe NCs, we find that the SAXS-based calibration of the sizing curve presented here, and previously published TEM-based calibrations coincides for larger NCs -typically with a diameter above 5 nm. Significant deviations, however, are observed for smaller NCs. In the case of zinc-blende CdSe NCs, where TEM-based sizing curves are limited to relatively small diameters, we find that the SAXS-based calibration yields systematically larger diameters than the previously published TEM-based calibration. In the case of CdS NCs, no such comparison is possible since no consistent, TEM-based sizing curve is available in the literature. Finally, we took the analysis of SAXS patterns one step further by showing that also NC concentrations can be determined reliably through SAXS. We thus provide compelling evidence that SAXS can be used as a preferred, quantitative method for size and concentration determination of dispersed NCs.

Materials and Methods

Colloidal Nanocrystals

For this work, we synthesized four series of CdS, CdSe, PbS and PbSe NCs with different diameters, using established hot-injection approaches as outlined in detail in Section 1 of the Supporting Information. [START_REF] Murray | Colloidal synthesis of nanocrystals and nanocrystal superlattices[END_REF][START_REF] Abe | Tuning the Postfocused Size of Colloidal Nanocrystals by the Reaction Rate : From Theory to Application[END_REF][START_REF] Flamee | Fast, high yield, and high solid loading synthesis of metal selenide nanocrystals[END_REF][START_REF] Hendricks | A tunable library of substituted thiourea precursors to metal sulfide nanocrystals[END_REF] Figure 1 represents TEM images of representative samples from each series, showing that the batches used in this study are composed of quasi-spherical NCs with a narrow size dispersion. The latter conclusion is corroborated by the absorbance spectra of samples analysed by SAXS shown in Figure 2. These spectra feature a pronounced band-edge transition that gradually shifts to longer wavelengths with increasing nanocrystal diameter. To estimate the spectral position of these transitions, spectra were fitted to a set of Gaussian peaks on top of a polynomial that accounted for the increasing background (see Supporting Information, Section 2.2). In what follows, the mean of the Gaussian that fits the band-edge transition will be used to construct sizing curves that relate the band-gap transition energy to the NC diameter.

Small Angle X-Ray Scattering

For characterization of NC dispersions using SAXS, 0.01 mm thin-wall glass capillaries were filled with a dispersion of NCs in toluene. In the case of PbS and PbSe, this procedure was carried out inside a nitrogen-filled glovebox. The capillaries were sealed with a UV-curable glue. Part of the synchrotron SAXS experiments were performed on the SWING beamline at the SOLEIL synchrotron (Saint-Aubin, France) using an x-ray energy of 8 keV and a sample to detector distance of 1 m. Other experiments were performed on the ID2 beamline at the ESRF (Grenoble, France). In this case, the energy was 12 keV and the distance was comprised between 1 and 1.5 m. All the scattering patterns were radially averaged with beamline specific procedures to yield the intensity versus scattering vector q curve.

Procedures

Measuring SAXS patterns SAXS measurements on NC dispersions can be performed in two ways (see Figure 3). First, one can load each sample in a separate glass capillary and use a capillary holder with multiple positions to measure several samples at a time (Figure 3A). The main advantage of this approach is throughput. With exposure times less than 1 seconds at a synchrotron, it is possible to measure tens of samples in hours. On the other hand, as capillaries will always have different thicknesses and background signals, measurements on an absolute scale are difficult. The alternative is to use a flow-through setup, which enables the scattering pattern of the empty capillary, the solvent and the solution to be measured at exactly the same spot in the capillary. By doing so, one ensures that exactly the right background signals are subtracted and the thickness of the sample can be measured very accurately through a secondary standard.

Another advantage is that scattering patterns can be recorded on dispersions whilst flowing, an approach that limits possible radiation damage. Hence, from the point of view of SAXS data quality, flow setups are superior, yet they make for longer measurement times since the capillary must be carefully washed in between measuring two different dispersions.

The following describes a detailed protocol for measuring SAXS patterns that can be used to determine the size distribution and concentration of dispersed NCs. It is intended for researchers with a minimal hands-on experience of SAXS. First, we describe the preparation of capillaries. Next, it is outlined how to obtain a net NC SAXS pattern by radially averaging and correcting for the signal of the capillary and the solvent. Finally, we describe how a SAXS diagram can be put on an absolute scale using standards.

Preparation of Capillaries

Caution. These glass capillaries are very thin and hence extremely fragile. Handle them with care to avoid their fracturing.

1. Use x-ray boro-silicate or quartz capillaries dedicated to xx-ray-ray experiments.

2. Fill the top of the capillary with 100 µL of the dispersion.

3. Make the dispersion flow at the bottom of the capillary by holding the top of the capillary with two fingers and make a fast circular move with your arm straight from top to bottom.

Another option is to fill the capillary with a syringe equipped with a long and fine needle.

4. Close the top of the capillary to avoid evaporation of the solvent. We found that wrapping some Parafilm around the top of the glass capillary is enough to avoid significant evaporation for a few days. If there is a long time period between the capillary preparation and the measurement, it is better to seal the capillary with a flame or UV-curable glue. The latter can be performed under nitrogen atmosphere for air-sensitive materials.

The procedure to subtract the solvent and capillary signal from the raw SAXS patterns vary with the measurement protocol. It is straightforward when a flow cell is used and it is done alongside the normalization at the absolute scale (see the corresponding section). When capillaries are used, the procedure is more complicated because the capillary signal has to be measured separately and the thickness of the capillary is not known precisely.

Radially averaging of raw SAXS patterns

In the case of isotropic samples (i.e. samples which do not display macroscopic orientation which is usually the case for nanoparticle dispersion) the SAXS intensity does not depend on the azimuthal angle on the detector and no information is lost by averaging the data over this angle. This azimuthal integration groups points with similar azimuthal angles into bins, yielding an intensity versus scattering vector q curve which is henceforth called the SAXS pattern (see Figure 4). This treatment, along with numerous other corrections [START_REF] Pauw | Everything SAXS: small-angle scattering pattern collection and correction[END_REF] is usually performed by automated programs at the beam-line and we consider in the following that we deal with radially averaged I(q) curves.

Calculation of

I disp net = I disp raw -I cap (1)
Remark. This step is essential. The glass capillary yields a large SAXS signal at small q, where the scattered intensity increases quickly as q decreases. This is due to inho- mogeneities in the glass. This signal gets much larger than that of the nanocrystals and dominates the overall intensity for small q. Hence it is important to subtract it from the signal.

Troubleshooting. Subtraction might not be perfect at small q because each capillary will have a different signal.

• When the signal of the empty cap is larger than the signal in which the NC solution is present, the aformentioned subtraction will lead to overcorrection. In that case, the only solution left is to crop the SAXS pattern and exclude the points where negative intensities are obtained. In this case, the q-range is reduced, only scattering intensities at larger q can be used, where the signal of the NC is significantly larger than that of the capillary.

• If the signal of the capillary is under-subtracted, an increasing scattering intensity at small q remains. This can be falsely attributed to an aggregation of nanoparticles, which would also result in an increase of intensity at small q.

3. Subtract capillary from solvent:

I sol net = I sol raw -I cap (2)
4. Subtract solvent from the net dispersion to obtain the net NC SAXS pattern I N C net and divide by the capillary thickness provided by the capillary manufacturer e:

I N C net = I disp net /e d -I sol net /e s (3)
Remark. Since different capillaries are used, their thicknesses can vary by 20%, whereas it must be known with a 1% accurary to get a neat subtraction at high q. Hence, our using of the symbols e d and e s to indicate thicknesses in Eq 3. To achieve a good subtraction under such conditions, we introduce a parameter x, plot the difference

I disp net -x × I sol net ,
and adjust x until the intensities of both signal are equal at high q. At large scattering vector the intensity of the nanoparticle is supposed to decrease like q -4 to become much smaller than the signal of the solvent which dominates the overall scattering signal. When we find the right x parameter, the thus obtained intensity is used as

I N C
net for further analysis. We usually find x values comprised between 0.8 and 1.2, a range that is consistent with thickness variation of around 20%.

Protocol for Obtaining SAXS Patterns on an Absolute Scale

To estimate the NC concentration from a SAXS pattern, the measured scattered intensity has to be normalized at an absolute scale. This needs further steps as described below.

Apart from intensity normalization by means of a primary standard, which may or may not be a routine procedure at synchrotron beamlines, it also requires accurate knowledge of the sample thickness. For this, calibration measurements using a primary and a secondary standard with the same cell are needed, which makes that absolute measurements are only possible with flow-cells.

1. Normalize the SAXS intensity using a standard such as glassy carbon. [START_REF] Zhang | Glassy Carbon as an Absolute Intensity Calibration Standard for Small-Angle Scattering[END_REF] Record a SAXS pattern of the standard (I st ) and tune a constant parameter c such that c × I st yields exactly the same intensity as provided with the standard.

2. Using the flow cell, record the SAXS patterns (in that order) of ( 1) the empty capillary (I cap ), and (2) pure water (I w raw ).

3. Obtain the net SAXS pattern of water as:

I w net = I w raw -I cap 4.
Find the thickness e of the water layer from the ratio of I w net and the absolute scattering intensity of water (1.65 10 -3 mm -1 ) at ambient temperature and pressure. [START_REF] Dreiss | On the absolute calibration of bench-top smallangle X-ray scattering instruments: a comparison of different standard methods[END_REF] 5. Record the SAXS pattern of the solvent (I sol raw ) and the NC dispersion (I disp raw ).

6. Determine the absolute SAXS intensity for the NC dispersion and the solvent:

I disp abs = c × I disp

Diameter determination by SAXS General Approach

For size determination, NC dispersions loaded in separate capillaries were analyzed using synchrotron x-ray Small Angle Scattering. The 2D SAXS images were radially averaged using beamline specific procedures. Afterwards, the capillary and solvent signal are subtracted according using the protocol discussed above. The resulting net NC SAXS patterns were used for the size determination.

Figure 6 shows the SAXS pattern for a series of PbS NCs, which will be used as the reference system in this section. Qualitatively, the SAXS curves display a clear plateau at small scattering vectors (q). This indicates that the dispersion analyzed is stable and does not contain aggregated NCs, a result in line with the TEM observations shown in Figure 1. This ensures that we can neglect any structure factor in the SAXS analysis and consider a model of independent scatterers. With increasing q, the scattering intensity shows an abrupt decrease, followed by several intensity oscillations. This indicates that the samples contain highly monodisperse nanocrystal ensembles.

For the quantitative analysis of the SAXS pattern, we start from the assumption that such samples of PbS NCs consist of spheres of homogeneous electron density, dispersed in a solvent. The SAXS intensity on an absolute scale I is then given by: 38,39

I(q) = n(∆ρ) 2 ∞ 0 V (R) 2 D(R)P (q, R)dR, (4)
Here, n is the NC concentration, ∆ρ is the difference of scattering length density between the NCs and the solvent (the contrast), V (R) is the volume of a sphere of radius R, D(R) is the radius distribution and P (q, R) is the form factor of a sphere. Equation 4 makes clear that the shape of the SAXS curve only depends on the NC radius distribution D(R), whereas the overall intensity is dictated by the NC concentration n. Hence, descriptors such as the average diameter and the size dispersion can be obtained from an intensity versus q signal in arbitrary units (a.u.), whereas a precise measure of the NC concentration requires absolute intensities.

In essence, two different methods can be used to retrieve the radius distribution from the SAXS pattern. First of all, one could fit the experimental spectrum to an intensity pattern simulated using Eq 4 by means of a parametrized analytical distribution function. Alternatively, the mean diameter and the size dispersion can be estimated by means of Monte-Carlo (MC) simulations, [START_REF] Bressler | McSAS : software for the retrieval of model parameter distributions from scattering patterns[END_REF] which extract a volume-weighted size distribution from the SAXS pattern. In what follows, we compare both approaches by means of PbS NC SAXS patterns shown in Figure 6.

Diameter Analysis Using Presupposed Analytical Distributions

To assess the stability of the diameter estimation using a presupposed distribution function,

we evaluated SAXS patterns obtained on PbS QDs using a Gaussian, a Schulz and a Log-Normal distribution. These respective distributions were parameterized as indicated in Table 1, while the expressions for the NC size disperion can be found in Section 3 of the Supporting Information.

Table 1: Expressions of Gaussian, Log-normal and Schulz distribution functions.

Distribution Expression for D(R)

Gaussian As indicated by the example shown in Figure 7, the best fits of the SAXS pattern using either of these distributions superpose, and can not be distinguished. Furthermore, all the fits are excellent, except for the sample with a band gap of 0.99 eV, which is the pattern marked with * in Figure 6. In line with the coinciding fits, we found that the estimates of the mean diameter and the size dispersion do not depend on the actual analytical expression of the distribution. As can be seen from Table 2, the mean diameter is the same within 0.02 nm and the differences on the size dispersion are also very small. This indicates that these values are robust and do not depend on the actual radius distribution that was put forward. A detailed fitting protocol using the software package 'SASView' can found in Section 3.1 of the Supplementary Information.

1 √ 2πσ e -(R-R 0 ) 2 2σ 2 Schulz R R 0 Z (Z + 1) (Z+1) e -(Z+1) R/R 0 Γ(Z+1) Lognormal 1 Rσ √ 2π exp -(ln R-µ) 2 2σ 2 R = radius; R 0 = mean radius; σ = standard deviation; Γ(z) = ∞ o t z-1 e -t dt
Table 2: Results from different fitting procedures for the PbS NC series. For each fitting method, the mean diameter (d) and the polydispersity (poly.) is indicated. The mathematical expression of the different size distributions is described in Table 1. NC radius (nm)

Gaussian

A) B)
Figure 8: Result of the Monte-Carlo fitting of the outlier PbS sample. Note the bi-modal distribution with a maximum at 1.5 nm and a second maximum at 2.1 nm.

examination of the SAXS pattern and the associated fits based on presupposed distributions, shows that all these monomodal distributions underestimate the SAXS intensity at high q.

When the Monte-Carlo approach is used instead (Figure 8), the retrieved size distribution appears as bi-modal with a population of small particles that have a mean radius of around 1.5 nm appearing alongside the main population with a mean radius 2.1 nm. Possibly, this population of smaller nanoparticles corresponds to unbound lead oleate molecules which are still present due to insufficient purification. Note that the Monte-Carlo approach yields a correct fit of the SAXS intensity at high q (Figure 8A).

Error estimation

The comparison of descriptor estimation in the case of the PbS series shows that all the methods studied yield essentially the same mean diameter and size dispersion. Apart from the error on these estimates that comes with a given fitting method, we identify two other sources of uncertainty in our measurements. The first one is the overall error on the SAXS measurement of a given sample. This includes the measurement itself, and all the data treatment procedures. These errors have been shown to impact the final mean radius and polydispersity by less than 1%. [START_REF] Pauw | Everything SAXS: small-angle scattering pattern collection and correction[END_REF] The other uncertainty is that of the choice of fitting procedure (least square fit or Monte-Carlo). Our data show that this error is even smaller and is well below 1%. Hence, the dominating uncertainty of our measurement lies in the SAXS modeling choice. When the fit for a monomodal size distribution is excellent, which is the case for large nanoparticles where multiple oscillations are visible, this error is a few percent at a maximum. Only for smaller particles, such as the first of the PbS series, the uncertainty is higher, yet it remains within the 10% range. Most likely, this increased error is caused by the solvent scattering masking the oscillations of the scattered intensity by the NCs at high q.

SAXS-Calibrated Sizing Curves

SAXS Analysis of PbSe, CdSe and CdS Dispersions

A major result of the analysis of the SAXS patterns recorded on PbS dispersions is that very similar NC descriptors are obtained, regardless of the fitting methodology. We therefore extended the SAXS study to PbSe, CdSe and CdS NCs, where we restricted ourselves to a data analysis based on Gaussian distributions. All experimental patterns, and Gaussian fits are shown in Figure 9. The thus obtained estimates for the NC diameter and the size dispersion have been summarized in the Supporting Information (Table S2).

A) CdS B) CdSe C) PbSe

SAXS-Based Sizing Curves for PbS, PbSe, CdS and CdSe Nanocrystals

Having an estimated NC diameter d N C from SAXS and the optical band gap E g from absorbance measurements, we can construct calibration curves that link both quantities. Figure 10 displays the resulting sizing curves, where each marker represents an experimental (d, E g ) data point and each trend line is an interpolation determined by fitting the set of data points to the empirical expression:

E g = E g,bulk + 1 a + b.d N C + c.d 2 N C (5)
Here, E g,bulk stands for the bulk band gap of the material and a,b and c are fitting parameters.

These parameters are summarized in Table 3 for all 4 materials, together with their bulk band gap and the energy range in which the sizing curve is valid. All experimental data points are in good agreement with the final sizing curve, clearly demonstrating the high reliability of SAXS to determine NC diameters. For practical considerations, a straightforward relation between d N C and the wavelength of the band-edge transition is often useful. Such relations are given in the Supporting Information (Section 5).

Concentration determination by SAXS

Next to the NC diameter, the NC concentration is a second key descriptor of NC dispersions.

Currently, such concentrations are determined by combining estimates of the NC diameter and the total volume of semiconductor material in a dispersion. The former is often calculated from [START_REF] Moreels | Size-dependent optical properties of colloidal PbS quantum dots[END_REF][START_REF] Capek | Optical Properties of Zincblende Cadmium Selenide Quantum Dots[END_REF][START_REF] Moreels | Composition and Size-Dependent Extinction Coefficient of Colloidal PbSe Quantum Dots[END_REF] while the dotted line denotes the bulk band gap. follow directly from the intensity limit when q → 0. As outlined in the protocol section, such absolute measurements involve a judicious calibration procedure that relies on an accurate knowledge of the sample thickness. To assess the accuracy of this approach, we measured SAXS patterns for a series of PbS NCs with a flow cell and determined the NC concentration from the thus obtained absolute scattering intensity. In addition, the same dispersions were analyzed using absorbance spectroscopy, and the NC molar concentration was determined from their absorbance at 400 nm according to literature. [START_REF] Moreels | Size-dependent optical properties of colloidal PbS quantum dots[END_REF] Figure 11 shows the ratio of the NC concentration determined by absorbance spectroscopy and SAXS on a sample-by-sample basis. For all samples, these ratios fall randomly within a band of ±10% around the expected value of 1. A striking result since two completely different approaches were used to determine the same quantity. This indicates that SAXS is a viable alternative method for concentration determination and therefore can be useful in future fundamental studies concerning nucleation and growth mechanisms of NC syntheses.

Discussion

The overview of the SAXS-based sizing curves given in Figure 10 enables us to compare SAXS and TEM as methods to calibrate nanocrystal sizing curves. As compared to TEMbased curves, [START_REF] Moreels | Size-dependent optical properties of colloidal PbS quantum dots[END_REF][START_REF] Capek | Optical Properties of Zincblende Cadmium Selenide Quantum Dots[END_REF][START_REF] Moreels | Composition and Size-Dependent Extinction Coefficient of Colloidal PbSe Quantum Dots[END_REF] we find that in the case of PbS and PbSe NCs larger than ∼ 5 nm, both methods yield coinciding sizing curves. For smaller diameters, however, we find that SAXS calibration always assigns a larger diameter to a given band-edge energy then TEM calibration. This trend points towards systematic, size-dependent deviations between both techniques. Most likely, this differences are due to TEM-based calibrations underestiming the actual NC diameter due to a poor electron scattering contrast between small NCs and the underlying amorphous carbon-film support. This will lead to a low pixel intensity difference at the particlesupport boundary that jeopardizes factual particle edge determination for very small NCs.

Moreover, the fractional contribution of a single pixel will become substantial, leading to large errors if misinterpreted. [START_REF] Pyrz | Particle Size Determination Using TEM: A Discussion of Image Acquisition and Analysis for the Novice Microscopist[END_REF] If indeed limited contrast induces errors in TEM-based sizing, the problem such be more pronounced for materials giving weaker contrast, i.e., materials composed of small atomic number elements. This effect is clearly noticeable in the case of CdSe NCs, where we find a considerable larger underestimation of the NC diameter by TEM.

A full quantification of TEM-induced errors is not possible since instrumental parameters like magnification, Cs-correction and contrast transfer function can all have a strong influence on the edge determination. Moreover, the user-dependent character of the data analysis further decreases the reliability of TEM for sizing, especially for small and light-element NCs. To counterbalance such effects, imaging is better done in an aberration corrected microscope in higher magnification conditions, and, if possible, using high contrast techniques such as high angle annular dark field (HAADF) scanning TEM (STEM). In this latter case, one would be able to enhance the particle-support contrast ratio, and have a clear and sharp definition of the quantum dot contour, achieving even atomic column definition. Nevertheless, sampling of an inadequate number of particles, [START_REF] Murphy | Best Practices for the Reporting of Colloidal Inorganic Nanomaterials[END_REF] and the inevitable increase in electron flux causing beam damage, can still induce errors. SAXS, on the other hand, does not have such limitation, which makes it most likely the more reliable technique for size calibration over the complete size range analyzed here.

Conclusions

We have proposed protocols for the sizing of dispersed nanocrystals by means of small angel x-ray scattering. Using dispersions of PbS nanocrystals of different sizes as a reference material, we show that in the case of monodisperse nanocrystal batches, data analysis by means of a parameterized analytical radius distribution or a Monte Carlo approach that does not use a presupposed expression for the radius distribution yield coinciding results. We extended the study to PbSe, CdSe and CdS nanocrystal dispersions, and present so-called sizing curves for these materials in which the band-edge energy is related to the nanocrystal diameter by means of SAXS calibration. We show that in the case of sub 5 nm nanocrystals, a similar calibration using TEM imaging systematically yields smaller diameters for a given band-edge energy, a difference we assign to the difficulty to delineate the nanocrystal boundary in case of low-contrast TEM images. Finally, we show that SAXS patterns calibrated to an absolute scale can be used to estimate nanocrystal concentrations, providing values that differ by less than 10 % deviation from concentrations obtained using absorbance spectroscopy. We conclude that SAXS is a most reliable and versatile technique for the quantitative analysis of the diameter and concentration of dispersed nanocrystals, that may become a mainstream analysis method in nanocrystal science thanks to the analysis protocols and reference data provided here.

Figure 1 :

 1 Figure 1: TEM micrographs of well-separated CdS(e) and PbS(e) NCs to illustrate the quasispherical shape of NCs.

Figure 2 :

 2 Figure 2: Typical (NIR)-UV-Vis absorbance spectra of (A) CdS, (B) CdSe, (C) PbS and (D) PbSe NCs, an offset is given for clarity.

Figure 3 :

 3 Figure 3: Illustration of the two different SAXS measurement procedures: A) using a series of capillaries and a capillary holder B) using a flow cell. This particular model was purchased from XENOCS. C) Beamline setup during SAXS experiments. The x-ray beam comes from the left, hits the sample and the scattered signal is detected by a camera.

  Net NC SAXS Patterns when capillaries are used 1. Start from the SAXS patterns of the nanocrystal dispersion in a capillary I disp raw , the solvent in a capillary I sol raw and an empty capillary I cap . 2. Obtain the net SAXS pattern I disp net of the dispersion by subtracting the patterns of the dispersion and the capillary.

Figure 4 :

 4 Figure 4: A) Illustration of the working principal of SAXS. The focused incident beam is scattered by the capillary and yields a pattern on the detector. B) As an example: simulated 2D pattern for spherical nanoparticles with a diameter of 14 nm and a polydispersity of 4.3 % and C) the corresponding 1D pattern I(q) after radial averaging.

7 .Figure 5 :

 75 Figure 5: Typical SAXS patterns of a NC dispersion in a glass capillary (green), solvent in capillary (red) properly normalized at an absolute scale and the subtraction yielding the NC SAXS pattern to further analyze (blue).

Figure 6 :

 6 Figure 6: Experimental SAXS patterns for 1.55-0.59 eV PbS NCs (open circles), fitted by means of a Gaussian mono-modal size distribution (line).

Figure 7 :

 7 Figure 7: Fitting of a PbS NC SAXS pattern with a model of polydisperse spheres for different mathematical form of the distribution: gaussian, log-normal or Schulz. The three fits are excellent and indistinguishable.

Figure 9 :

 9 Figure 9: Experimental SAXS patterns for (A) 3.3-2.6 eV CdS, (B) 2.6-1.9 eV CdSe, (C) 1.2-0.6 eV PbSe NCs (open circles) with Gaussian size distribution (line).

Figure 10 :

 10 Figure 10: Relation between the band gap and the diameter for (A) CdS, (B) CdSe, (C) PbS and (D) PbSe NCs. To the the set of SAXS data (open circles), a sizing curve is fitted (colored line). The black lines represent a TEM-based sizing curves from literature, 41-43 while the dotted line denotes the bulk band gap.

Table 3 :

 3 Empirical fitting parameters of SAXS-based sizing curves (Equation5) with bulk band gap and energy range.

		CdS	CdSe	PbS	PbSe
	a (eV -1 )	-1.123	0.756	0.0261	-0.891
	b (nm -1 • eV -1 )	0.768	-0.249	0.235	0.621
	c (nm -2 • eV -1 )	0.0522	0.139	-	-0.0141
	E g,bulk (eV)	2.42	1.74	0.41	0.278
	E g,max -E g,min (eV) 3.29-2.62 2.64-1.93 1.55-0.59 1.19-0.58

Acknowledgement

We are grateful to J. Perez and T. Bizien for assistance during beamtime and to the SOLEIL staff for smoothly running the facility. We thank ESRF for provision of the beamtime and S. Prevost for help during the measurements. We appreciate the scientific discussions concerning imaging of nanocrystals by transmission electron microscopes with Jordi Arbiol. Z.H. acknowledges Ghent University (GOA 01G01513) for funding. Benjamin, do you want to include some fundings?

Monte Carlo Analysis of Scattering Patterns

Next, we applied the Monte-Carlo approach as developed by Bressler et al. to fit the SAXS patterns of the PbS NCs series. In this method, no particular shape of the radius distribution is assumed. Even so, we found that all the size distribution retrieved by this method for our samples are monomodal, a result that is consistent with the excellent fits found with presupposed distributions. The mean diameters retrieved by MC are slightly higher, probably because a volume weighted size distribution is obtained instead of a number weighted size distribution.

An interesting case to compare both approaches is the SAXS pattern recorded on the 0.99 eV sample of the PbS series, which emerged as an outlier in Figure 6 (marked with * ). A closer the energy of the band-edge transition by means of a sizing curve, whereas the latter can be obtained from the absorbance of the dispersion at energies well above the band-edge transition. For many materials, the absorption coefficient µ i at these higher energies is constant, and agrees with a theoretical value that can be calculated based on bulk optical properties. [START_REF] Hens | Light Absorption by Colloidal Semiconductor Quantum Dots[END_REF] Combining both elements yields a molar absorption coefficient as (N A : Avogadro's number):

Having at hand, the molar concentration of NCs c N C is readily obtained from the absorbance Equation 4 suggests that SAXS provides an alternative to determine the NC concentration as the scattered x-ray intensity is directly proportional to the NC concentration. Hence, when special care is taken to normalize the intensity on an absolute scale, the NC concentration should

Supporting Information Available

The Supplementary Information contains detailled NC synthesis methods, an example of absorption spectra fitting (Igor Pro) and SAXS patterns (SASView), datapoints of sizing curves and alternative sizing curves, relating the NC diameter to the wavelength of the first optical transition.


Vinh Le-Van

Gérard Meunier

Olivier Chadebec

Jean-Michel Guichon

A Magnetic Vector Potential Volume Integral Formulation for Nonlinear Magnetostatic Problems

Keywords: Edge elements, integral equation, magnetic vector potential, nonlinear magnetostatic, team Workshop problem 13, volume integral method

Volume integral method (VIM) has been known as an interesting alternative to the finite-element method for electromagnetic field computation. Since only the active regions have to be meshed, VIM is very efficient for modeling of electromagnetic devices containing a predominated air volume. It is also adapted for multistatic studies with motion or optimization strategies. The aim of this paper is to propose a magnetic vector potential volume integral formulation in order to deal with the 3-D nonlinear magnetostatic problems. The main advantage of this formulation is that the convergence of nonlinear material solution can be easily reached after a few iterations without any relaxation. Moreover, the use of coarse meshes can lead to accurate results. Computed results for the TEAM Workshop problem 13 and for an actuator are reported.

treated straightforwardly, and the nonlinear problems convergence was easily reached without any relaxation technique. Nevertheless, the B-facet formulation requires independent loops search in order to restore the flux solenoidality. This task can be time consuming and decreases the convergence rate of linear systems solutions.

In this paper, an original magnetostatic volume integral formulation based on the interpolation of the magnetic vector potential on edge elements is proposed. This novel approach, called A-edge formulation, does not require the independent loops search algorithms. The formulation being auto-gauged, no gauge condition has to be added and the convergence rate of the linear solution is then improved. In others words, this new formulation has all the benefits of the B-facet one but without its drawbacks.

II. MAGNETIC VECTOR POTENTIAL FORMULATION

Let us consider a 3-D magnetostatic problem composed by ferromagnetic regions and coils with DC current density J. We note the boundary between and the free space.

I. INTRODUCTION

V

A. Basis Equations

From Maxwell's equations, the governing equations for magnetostatics can be given in terms of magnetic flux density B, magnetic field strength H, and

J divB = 0 (1) curl H = J. (2)
These equations are associated with constitutive laws

H = ν(B)B (3) M = (ν 0 -ν(B))B ( 4
)
where M is the magnetization; ν 0 and ν(B) are the reluctivity of air and of ferromagnetic materials, respectively. The decomposition of the magnetic field strength into the sum of the source magnetic field H 0 and reduced magnetic OLUME integral method (VIM) is known today as an interesting alternative to the classical finite-element method (FEM) for solving nonlinear magnetostatic problems. The main advantage of VIM versus FEM is that neither free space mesh nor boundary condition is required, only active regions have to be meshed. Associated with fast algorithms such as the fast multipole method (FMM) or hybrid cross approximation, the fully populated matrices system obtained by VIM can be compressed and treated rapidly.

To this day, several magnetostatic volume integral formulations have already been introduced [START_REF] Harrington | Field Computation by Moment Methods[END_REF]- [START_REF] Le-Van | A volume integral formulation based on facet elements for nonlinear magnetostatic problems[END_REF]. The magnetic moment method (MMM) [1]- [START_REF] Morandi | A modified formulation of the volume integral equations method for 3-D magnetostatics[END_REF], which is the oldest and also the most popular one, is based on uniform magnetization on each element. This method is straightforward to implement and well adapted for simple geometry devices. However, it leads to numerical instability with the presence of nonlinear magnetic materials. In addition, the MMM is also strongly limited by the size of the obtained matrix (three unknowns per element). H-edge, and ϕ-nodal formulations were proposed in [START_REF] Kettunen | A volume integral formulation for nonlinear magnetostatics and eddy currents using edge elements[END_REF]- [START_REF] Le-Van | The adaptive cross approximation technique for a volume integral equation method applied to nonlinear magnetostatic problems[END_REF]. They seem to be more flexible and robust than the MMM for the treatment of complex devices. In general, both formulations present a quite poor convergence rate for nonlinear solutions due to the use of a B(H) curve. Moreover, the use of these formulations in order to model the multiply connected regions requires cutting techniques. Most recently, the B-facet formulation based on interpolation of magnetic flux on facet elements associated with an equivalent circuit approach has been proposed in [START_REF] Le-Van | A volume integral formulation based on facet elements for nonlinear magnetostatic problems[END_REF]. This formulation ensures a good convergence rate due to the use of a H(B) curve. Its results present an excellent accuracy even with a very coarse mesh. Multiply connected regions were field H r leads to write [START_REF] Le-Van | A volume integral formulation based on facet elements for nonlinear magnetostatic problems[END_REF] ν(B)B + grad ϕ r = H 0 [START_REF] Canova | Integral solution of nonlinear magnetostatic field problems[END_REF] where ϕ r is the reduced magnetic scalar potential, determined by [START_REF] Durand | Magnétostatique[END_REF]

ϕ r = 1 4π M • grad 1 r d . ( 6
)
By introducing the magnetic vector potential A that guarantees B = curlA, (5) can be written as

ν(B)curl A + gradϕ r = H 0 . ( 7
)
From ( 4) and ( 6), the reduced scalar potential can be expressed as

ϕ r = 1 4π (ν 0 -ν(B))curl A • grad 1 r d . (8)

B. Edge Elements Interpolation

The magnetic vector potential A can be approximated by the first-order edge elements interpolation as

A = N e j =1
w j A j [START_REF] Durand | Magnétostatique[END_REF] where w j is the shape function of edge element j , A j the tangent component of vector potential on this edge, and N e is the number of edge elements in the ferromagnetic regions mesh. For an edge element i belonging to the facet f (Fig. 1), its first-order shape function satisfies [START_REF] Bossavit | Whitney forms: A class of finite elements for threedimensional computations in electromagnetism[END_REF]

curl w i • n f = curl w in = 1 S f ( 10
)
where n f and S f are the normal vector and the surface of the facet f , respectively.

C. Integral Formulations

The projection of (7) into ferromagnetic domain by the Galerkin method with the test functions curlw i leads to

curl w i • ν(B)curlA d + curl w i • grad ϕ red d = curlw i • H 0 d . ( 11
)
The two terms on the left hand can be developed as follows: curl

curl w i • ν(B)curl A d = N e j =1 curl w i • ν(B)curl w j d A j (12)
w i • gradϕ red d = div(curl w i • ϕ r )d = curlw in ϕ r d = curl w in 1 4π (ν 0 -ν(B))curlA • grad 1 r d d . ( 13
)
Since edge elements of first-order are used, the magnetic field B is constant in each tetrahedral element and nearly constant in each hexahedral element. Therefore, the reluctivity is constant in each volume element and ( 13) is transformed as

curl w i • gradϕ red d = curl w in ⎛ ⎝ 1 4π N e j =1 ⎛ ⎝ N f f =1 δν f f curl w j n r d f ⎞ ⎠ A j ⎞ ⎠ d ( 14
)
where N f is the number of facets in the mesh and δν f is the jump of reluctivity between two adjacent elements sharing the facet f or between the reluctivity an element belonging to the boundary and with the reluctivity of the air. With ( 12) and ( 14), [START_REF] Ren | Influence of the RHS on the convergence behaviour of the curl-curl equation[END_REF] can be written as

([R] + [L]){A} = {U 0 }. (15)
The matrices R, L, and the vector U 0 are determined by

R i j = curlw i • ν(B)curlw j d L i j = 1 4π curlw in ⎛ ⎝ N f f =1 δν f curlw j n f 1 r d f ⎞ ⎠ d U 0i = curlw i • H 0 d . ( 16
)
The dimension of sparse matrix R is N e × N e and one of the fully populated matrix L being N e × N e with N e is the number of edge elements on the boundary . Their dimension is shown in Fig. 2. In general, N e N e . Since (15) is autogauged, using an iterative solver such as GMRES (Generalized Minimal Residual Method) can solve it straightforwardly without any gauge condition requirement [START_REF] Ren | Influence of the RHS on the convergence behaviour of the curl-curl equation[END_REF]. If the Newton-Raphson (NR) method is used in order to deal with nonlinear system, the integral matrix L should be computed only once before the resolution and then be updated at each iteration with a new δν. Note that the singularity of surface integral of Green's function can be analytically computed [START_REF] Rubeck | Analytical calculation of magnet systems: Magnetic field created by charged triangles and polyhedra[END_REF].

III. APPLICATIONS AND NUMERICAL RESULTS

A. T.E.A.M. Workshop Problem 13

Benchmark problem 13 defined in the T.E.A.M. Workshop is a 3-D nonlinear magnetostatic model. A coil excited by DC current is set between two steel channels, and a steel plate is inserted between the channels. The ampere turns are 1000 and 3000 AT which is sufficient to saturate the steel. The aim of this problem is to calculate the magnetic fields at various positions [START_REF] Nakata | Summary of results for benchmark problem 13 (3-D nonlinear magnetostatic model)[END_REF], [START_REF] Nakata | Summary of results for team workshop problem 13 (3-D nonlinear magnetostatic model)[END_REF]. Its description is summarized in Fig. 3.

Three meshes composed by 1235, 4855 and 20 738 tetrahedral elements (Fig. 4) were considered. The distribution of magnetic flux density in steel plates are shown in Fig. 5 corresponding with the mesh of 20 737 elements.

The average magnetic flux density in steel plates (sections A-B-C-D-E-F) and magnetic flux density in the air (segment a-b) are shown in Figs. 6 and7, respectively. For both ampere turns, the numerical results are very close to the measurement. A coarse mesh of 1235 elements was able to give reasonable results. Since the results obtained with the mesh of 4855 elements and with the one of 20 737 elements were superposed, it can be concluded that the convergence of solution was reached using the mesh with 4855 elements.

The discrepancies between the numerical results and the measurement at 3000 AT for the distribution of absolute value of flux density in air are larger than those at 1000 AT. However, the same difference was reported in [START_REF] Nakata | Summary of results for benchmark problem 13 (3-D nonlinear magnetostatic model)[END_REF].

The NR method was used for the resolution of nonlinear system with an absolute criterion 10 -7 . The linear system is solved by GMRES method using the incomplete LU decomposition of R as the preconditioner. Computational cost is reported in Table I.

We notice that the use of B-facet formulation [START_REF] Le-Van | A volume integral formulation based on facet elements for nonlinear magnetostatic problems[END_REF] leads to the same results as the A-edge one. However, the A-edge formulation is preferred in terms of linear system resolution. In effect, for this formulation, GMRES solver converged after 220 iterations without any preconditioning technique and computational time was 103 s. In the same conditions, it took 580 iterations and 240 s for B-facet formulation. This difference can be explained by the fact that A formulation resolves the original linear equation system whereas the B formulation has to deal with a linear equations system depending the choice of independent loops [START_REF] Le-Van | A volume integral formulation based on facet elements for nonlinear magnetostatic problems[END_REF], [START_REF] Meunier | A-T volume integral formulation for solving electromagnetic problems in the frequency domain[END_REF].

B. Global Force Computation of an Actuator

The proposed formulation was applied to model an actuator excited by a dc current coil [Fig. 8(a)]. The nonlinear material is approximated by an arctan law as reported in [START_REF] Le-Van | A volume integral formulation based on facet elements for nonlinear magnetostatic problems[END_REF] with J s = 1 T and μ r = 1000. The aim of this example is to compute the global force acting on the mobile part of the actuator.

The distribution of the magnetic flux density in the material is shown in Fig. 8(b). The global force was computed from the solution of the proposed formulation because of the equivalent magnetic charge method [START_REF] Rakotoarison | Formal sensitivity computation of magnetic moment method[END_REF]. The convergence of the global force is given in Fig. 9. Compared with the results obtained by FEM with FLUX software, the convergence rate obtained with VIM is faster than FEM. With mesh of 100 tetrahedral elements, the relative error of the computed force given by VIM is ∼5%, the computation time being 3 s. To reach the same results, FEM has to use a mesh composed by more than 10 000 elements, the computation time being more than 1 min.

IV. CONCLUSION

An original integral volume formulation for solving nonlinear magnetostatic problems has been presented. This formulation is based on the interpolation of the magnetic vector potential on edge elements. The obtained equation system is autogauged and can be solved by an iterative solver without the imposition of a gauge condition. The singularity of Green's function can be analytically computed. The convergence of nonlinear solution is easily reached without any relaxation technique associated.

In general, the proposed formulation gives accurate results with a small computational cost. Even when coarse meshes were used, the obtained results remain acceptable. The formulation can be straightforwardly associated with fast algorithms in order to solving complex problems.

Fig. 1 .

 1 Fig. 1. Edge elements.

Fig. 2 .

 2 Fig. 2. Dimension of matrices R and L.

Fig. 3 .

 3 Fig. 3. Description of the problem 13. (a) Device geometry. (b) B-H curve of steel. (c) Specified positions for the magnetic field computation.

Fig. 4 .

 4 Fig. 4. Meshes of steel plates of the problem 13. (a) 1235 elements. (b) 4855 elements. (c) 20 737 elements.

Fig. 5 .

 5 Fig. 5. Magnetic flux density in steel plates. (a) 1000 AT. (b) 3000 AT.

Fig. 6 .

 6 Fig. 6. Spatial distribution of average magnetic flux density in steel plates.

Fig. 7 .

 7 Fig. 7. Spatial distribution of magnetic flux density in air.

Fig. 8 .Fig. 9 .

 89 Fig. 8. Actuator. (a) Geometry view. (b) Magnetic flux density distribution.

TABLE I COMPUTATIONAL

 I COST OF TEAM PROBLEM 13 SOLUTION