Planar Growth Research Articles

Correlation between the Surface Undulation and Luminescence Characteristics in Semi-Polar 112¯2 InGaN/GaN Multi-Quantum Wells

Surface undulation was formed while growing InGaN/GaN multi-quantum wells on a semi-polar m-plane (1–100) sapphire substrate. Two distinct facets, parallel to 112¯2 and 011¯1, were formed in the embedded multi-quantum wells (MQWs). The structural and luminescence characteristics of the two facets were investigated using transmission electron microscopy equipped with cathodoluminescence. Those well-defined quantum wells, parallel and slanted to the growth plane, showed distinct differences in indium incorporation from both the X-ray yield and the contrast difference in annular darkfield images. Quantitative measurements of concentration in 011¯1 MQWs show an approximately 4 at% higher indium incorporation compared to the corresponding 112¯2 when the MQWs were formed under the same growth condition.

Distinct Optical and Structural (Nanoyarn and Nanomat-like Structure) Characteristics of Zinc Oxide Nanofilm Derived by Using Salvia officinalis Leaves Extract Made without and with PEO Polymer

This paper reports the optical properties of zinc oxide nanofilm fabricated by using organic natural products from Salvia officinalis leaves (SOL) extract and discusses the effect of the nanocrystal (NC) structure (nanoyarn and nanomat-like structure) on nanofilm optical properties. The surface-active layer of the nanofilm of ZnO nanoparticles (ZnO NPs) was passivated with natural organic SOL leaves hydrothermally, then accumulated on zinc oxide nanorods (ZnO NRs). The nanofilms were fabricated (with and without PEO) on glass substrate (at 85 °C for 16 h) via chemical solution deposition (CSD). The samples were characterized by UV-vis, PL, FESEM, XRD, and TEM measurements. TEM micrographs confirmed the nucleation of ZnO NPs around 4 nm and the size distribution at 1.2 nm of ZnO QDs as an influence of the quantum confinement effect (QCE). The nanofilms fabricated with SOL surfactant (which works as a capping agent for ZnO NPs) represent distinct optoelectronic properties when compared to bulk ZnO. FESEM images of the nanofilms revealed nanoyarn and nanomat-like structures resembling morphologies. The XRD patterns of the samples exhibited the existence of ZnO nanocrystallites (ZnO NCs) with (100), (002), and (101) growth planes. The nanofilms fabricated represented a distinct optical property through absorption and broad emission, as the optical energy band gap reduced as the nanofilms annealed (at 120 ℃). Based on the obtained results, it was established that phytochemicals extracted from organic natural SOL leaves have a distinct influence on zoic oxide nanofilm fabrication, which may be useful for visible light spectrum trapping. The nanofilms can be used in photovoltaic solar cell applications.

Prediction of Ethanol-Mediated Growth Morphology of Ammonium Dinitramide/Pyrazine-1,4-Dioxide Cocrystal at Different Temperatures.

The crystal morphology of high energetic materials plays a crucial role in aspects of their safety performance such as impact sensitivity. In order to reveal the crystal morphology of ammonium dinitramide/pyrazine-1,4-dioxide (ADN/PDO) cocrystal at different temperatures, the modified attachment energy model (MAE) was used at 298, 303, 308, and 313 K to predict the morphology of the ADN/PDO cocrystal under vacuum and ethanol. The results showed that under vacuum conditions, five growth planes of the ADN/PDO cocrystal were given, which were (1 0 0), (0 1 1), (1 1 0), (1 1 -1), and (2 0 -2). Among them, the ratios of the (1 0 0) and (0 1 1) planes were 40.744% and 26.208%, respectively. In the (0 1 1) crystal plane, the value of S was 1.513. The (0 1 1) crystal plane was more conducive to the adsorption of ethanol molecules. The order of binding energy between the ADN/PDO cocrystal and ethanol solvent was (0 1 1) > (1 1 -1) > (2 0 -2) > (1 1 0) > (1 0 0). The radial distribution function analysis revealed that there were hydrogen bonds between the ethanol and the ADN cations, van der Waals interactions with the ADN anions. As the temperature increased, the aspect ratio of the ADN/PDO cocrystal was reduced, making the crystal more spherical, which helped to further reduce the sensitivity of this explosive.

Anisotropic Behavior of Optical Properties in Edge-Modified Phosphorene Quantum Dots

The evolution of optical properties in low-dimensional phosphorene quantum dots remains a largely unexplored area, with great potential for applications as optoelectronic devices. Herein, we present a comprehensive analysis of the optical properties of edge-passivated (H, NH2, Cl, OCN, CN) monolayer phosphorene quantum dots using density functional theory and time-dependent density functional theory calculations. An extensive characterization of absorptions, nonradiative relaxations, radiative emissions, radiative lifetimes, and the high-frequency Raman modes (Ag1, B2g, and Ag2) is undertaken, emphasizing the role of material directionality, quantum confinement, and edge passivation on the evolution of these properties. Our results indicate that optical absorptions and emissions are preferred from the armchair direction of phosphorene regardless of the type of edge functionalization or system size, rendering these systems as potential natural linear optical polarizers in the UV–vis region. Larger phosphorene quantum dots have smaller Stokes shifts and less geometric variation upon relaxation, leading to decreased radiative lifetimes. Additionally, we identify an uncharacteristic Raman response associated with dissimilar shifting trends of the Ag peaks that arise from the planar growth of phosphorene quantum dots in the armchair or the zigzag direction. The behavior is attributed to the competing effects of mass coupling and improved bond strength.

Influence of Source Composition on the Planar Growth of Nanowires during Catalytic Growth in a Quasi-Closed Volume

Influence of Source Composition on the Planar Growth of Nanowires during Catalytic Growth in a Quasi-Closed Volume

Defect Location in Laterally Aligned Tin Oxide Nanowires: The Role of Growth Direction, Interface Dimensionality, and Catalyst

Laterally aligned SnO2 nanowires, which exhibit planar growth along crystallographically defined directions of a sapphire substrate surface, are superior for bio- and gas-sensing applications. Little is known about their cross-sectional geometry and the defect distribution within the nanowire cross section, although this substantially determines the electronic properties of the nanowires. In this study, SnO2 nanowires were grown on r-plane sapphire substrates. Scanning electron microscopy (SEM) and atomic force microscopy (AFM) reveal the highly oriented growth of the SnO2 nanowires toward the substrate edges. High-resolution transmission electron microscopy (HRTEM) of the NW cross section and strain mapping allowed for analyzing the crystallographic alignment of the SnO2 NW to the sapphire substrate and the defect distribution within the SnO2 nanowire cross section. These techniques revealed a defect-free SnO2–Al2O3 interface and a high alignment of the SnO2 NW lattice toward the sapphire substrate along the NW width. The determined high defect density close to the nanowire surface will be discussed in comparison to freestanding SnO2 nanowires and SnO2 thin films on sapphire substrates considering the differences in the growth direction and the interface dimensionality.

A General Analytical Solution for Two-Dimensional Columnar Crystal Growth during Laser Beam Welding of Thin Steel Sheets

A technique for calculating the main solidification parameters for a two-dimensional columnar crystal growth during complete penetration laser beam welding of thin steel sheets was developed. Given that the weld pool interface is described by Lamé curves (superellipses) within the horizontal plane of growth, general analytical solutions were derived for the geometry of the crystal axis and the corresponding growth rate and cross-sectional area of the crystal. A dimensionless analysis was performed to provide insights on the dependence of the solidification parameters on the shape and dimensions of the rear part of the weld pool boundary. The derived solutions were applied for the case of complete penetration laser beam keyhole welding of 2 mm thick 316L austenitic chromium-nickel steel sheets. It was shown that the reconstruction of the weld pool boundary with Lamé curves provides higher accuracy and flexibility compared to results obtained with elliptical functions. The validity of the proposed technique and the derived analytical solutions was backed up by a comparison of the obtained solutions to known analytical solutions and experimentally determined shapes and sizes of the crystals on the top surface of the sheet. The dimensions of the calculated crystal axis correlated well with the experimentally obtained results.

Theoretical study on the influence of different solvents on the growth morphology of 3,4-bis(3-nitrofurazan-4-yl)furoxan (DNTF).

3,4-bis(3-nitrofurazan-4-yl)furoxan(DNTF) is a new energetic compound with high energy and high density, and it is an important component of propellant and melt cast explosive. In order to study the effect of solvent on the growth morphology of DNTF, the growth plane of DNTF in vacuum is predicted by attachment energy(AE) model, and then the modified attachment energy of each growth plane in different solvents is calculated by molecular dynamics simulation. The morphology of crystal in solvent is predicted by modified attachment energy (MAE) model. The factors affecting crystal growth in solvent environment are analyzed by mass density distribution, radial distribution function and diffusion coefficient. The results show that the growth morphology of crystal in solvent is not only related to the adsorption strength of solvent to crystal plane, but affected by the attraction of crystal plane to solute. The hydrogen bond plays an important role in the adsorption strength between solvent and crystal plane. The polarity of solvent has a great influence on the crystal morphology, and the interaction between the solvent with stronger polarity and the crystal plane is stronger. The morphology of DNTF in n-butanol solvent is closer to spherical, which can effectively reduce the sensitivity of DNTF. The molecular dynamics simulation is carried out under the COMPASS force field of Materials Studio software. Gaussian software is used to calculate the electrostatic potential of DNTF at B3LYP-D3/6-311 + G (d, p) theoretical level.

A Wox3-patterning module organizes planar growth in grass leaves and ligules

Grass leaves develop from a ring of primordial initial cells within the periphery of the shoot apical meristem, a pool of organogenic stem cells that generates all of the organs of the plant shoot. At maturity, the grass leaf is a flattened, strap-like organ comprising a proximal supportive sheath surrounding the stem and a distal photosynthetic blade. The sheath and blade are partitioned by a hinge-like auricle and the ligule, a fringe of epidermally derived tissue that grows from the adaxial (top) leaf surface. Together, the ligule and auricle comprise morphological novelties that are specific to grass leaves. Understanding how the planar outgrowth of grass leaves and their adjoining ligules is genetically controlled can yield insight into their evolutionary origins. Here we use single-cell RNA-sequencing analyses to identify a ‘rim’ cell type present at the margins of maize leaf primordia. Cells in the leaf rim have a distinctive identity and share transcriptional signatures with proliferating ligule cells, suggesting that a shared developmental genetic programme patterns both leaves and ligules. Moreover, we show that rim function is regulated by genetically redundant Wuschel-like homeobox3 (WOX3) transcription factors. Higher-order mutations in maize Wox3 genes greatly reduce leaf width and disrupt ligule outgrowth and patterning. Together, these findings illustrate the generalizable use of a rim domain during planar growth of maize leaves and ligules, and suggest a parsimonious model for the homology of the grass ligule as a distal extension of the leaf sheath margin.

A Comparison Study the Effect of Doping by Ga2O3 and CeO2 On the Structural and Optical Properties of SnO2 Thin Films

This research deals with the effect of gallium oxide and cerium oxide as dopants on the structural and optical characteristics of tin oxide. Gallium and cerium oxide doped tin oxide was prepared with different doping concentrations (0, 0.03, 0.05 and 0.07) wt. pure and doped tin oxide thin films were prepared by the pulsed laser deposition technique. X-ray diffraction and UV-Visible spectrophotometer were employed to investigate both oxides doping effects. Results showed that all prepared samples have poly-crystalline structure with a preferred plane of crystal growth along (110), where the crystal size grew from 40.3 nm to 64.5 nm and to 43.5 nm for Ga2O3 and CeO2 doped tin oxide thin films, respectively. Transmittance decreased drastically by increasing the doping ratio of gallium oxide. In contrast, it increased by increasing the doping ratio of cerium oxide. The optical energy gap was found to change in nonsystematic sequence with the increase of Ga2O3 doping concentration, while it decreased monotonically by increasing the CeO2 doping concentration.

Plasma enhanced planar crystal growth of laser induced graphene

The effect of wettability of Polyimide (PI) surface on the crystallite size and electrical conductivity of Laser-Induced Graphene (LIG) was studied for the first time. The wettability of PI was increased by varying the scan speed of a localised Argon cold plasma pen. The surface energy of PI increases with wettability which reduces the activation energy of diffusion of graphene nuclei and improves the crystallinity of LIG. The effect of plasma exposure was studied by contact angle measurement using two liquid solvents such as water and Ethylene Glycol (EG). Raman spectra of LIG showed that the crystallite size increased by 21 % at lower CO2 laser power (0.77 W) on PI pre-treated at a lower plasma scan speed (200 mm/min) which improved the conductivity by 51.13%. This study demonstrated that increased wettability of PI surface assists in the planar growth of LIG crystallites.

High‐Quality CsAg2I3Microwires Grown by Spatial Confinement Method for Self‐Powered UV Photodetectors

AbstractRecently, lead halide perovskites arouse intensive attentions in photoelectric fields owing to their low trap state density, high carrier mobility, large extinction coefficient, and low‐temperature solution processing technique. However, their future commercial development is severely hampered by the lead toxicity and instability. Although lead‐free metal‐halide perovskites recently have been extensively studied in optoelectronics, their low‐dimensional counterparts can further expand their novel optical and electronic properties for potential applications. In this work, the planar growth of one‐dimensional (1D) CsAg2I3 single‐crystal microwires (MWs) on diverse substrates by space‐confined method is reported. The growth kinetic process of CsAg2I3 single‐crystal MWs is elucidated by in situ observation. Furthermore, the CsAg2I3 single‐crystal MWs exhibit potential applications in ultraviolet light photodetectors as a photoactive layer. The fabricated heterojunction photodetector exhibits pronounced photoelectric performances working in self‐powered mode, including a specific detectivity of 2.09 × 108 Jones, a high responsivity of 26.5 mA W−1, and fast response speed of 258.43/289.32 ms under light illumination of 265 nm, due to the high crystallinity and effective photogenerated carrier transfer. This study provides a vital insight into the rational design of low‐dimensional perovskites for versatile optoelectronic applications.

Crystal morphology prediction of CL-20 and 1,4-DNI co-crystal at different temperatures.

The morphologies of hexanitrohexaazaisowurtzitane (CL-20) and 1,4-dinitroimidazole (1,4-DNI) co-crystal under vacuum or solvent at different temperatures were predicted. The CL-20/1,4-DNI co-crystal has six important growth crystal planes: (002), (011), (101), (11‒1), (110), (111). The areas of (002), (101), and (011) planes account for a relatively large proportion, which are important crystal planes that affect the crystal morphology. The crystal habits at different temperatures were simulated. The simulation results showed that the crystal plane attachment energy of CL-20 and 1,4-DNI co-crystal increases with the increase of temperature, indicating that the increase of temperature is conducive to the growth of crystal planes. The aspect ratio decreases with the increase of temperature and the morphology of co-crystal becomes more spherical at a higher temperature. The theoretical predictions are in good agreement with the experiment. The simulation results can provide guidance for the crystallization of CL-20/1,4-DNI to obtain a nearly spherical crystal morphology. The CL-20/1,4-DNI unit cell structure was geometrically optimized by the COMPASS force field. The AE model was used to predict the morphology of CL-20/1,4-DNI under vacuum, resulting in the most morphologically important growth planes. Ethyl acetate was selected as the solvent. The interaction energy between the solvent and the crystal plane, and the attachment energies in solvent at 298 K, 320 K, 340 K, 360 K, and 380 K were predicted. The NVT ensemble is used in the molecular dynamics calculation process. The simulation step is 1 fs and the total simulation time is 500 ps. The Andersen thermostat is selected as the temperature control method. In the potential energy calculation, the atom-based and Ewald methods were selected to calculate the van der Waals force and the electrostatic interaction force, respectively.

An Inorganic-Rich SEI Layer by the Catalyzed Reduction of LiNO3 Enabled by Surface-Abundant Hydrogen Bonding for Stable Lithium Metal Batteries.

Lithium (Li) metal anodes (LMAs) are promising anode candidates for realizing high-energy-density batteries. However, the formation of unstable solid electrolyte interphase (SEI) layers on Li metal is harmful for stable Li cycling; hence, enhancing the physical/chemical properties of SEI layers is important for stabilizing LMAs. Herein, thiourea (TU, CH4 N2 S) is introduced as a new catalyzing agent for LiNO3 reduction to form robust inorganic-rich SEI layers containing abundant Li3 N. Due to the unique molecular structure of TU, the TU molecules adsorb on the Cu electrode by forming CuS bond and simultaneously form hydrogen bonding with other hydrogen bonds accepting species such as NO3 - and TFSI- through its NH bonds, leading to their catalyzed reduction and hence the formation of inorganic-rich SEI layer with abundant Li3 N, LiF, and Li2 S/Li2 S2 . Particularly, this TU-modified SEI layer shows a lower film resistance and better uniformity compared to the electrochemically and naturally formed SEI layers, enabling planar Li growth without any other material treatments and hence improving the cyclic stability in Li/Cu half-cells and Li@Cu/LiFePO4 full-cells.

Classes of graphs embeddable in order-dependent surfaces

Given a function \(g=g(n)\) we let \(\mathcal{E}^g\) be the class of all graphs \(G\) such that if \(G\) has order \(n\) (that is, has \(n\) vertices) then it is embeddable in some surface of Euler genus at most \(g(n)\), and let \(\widetilde{\mathcal E}^g\) be the corresponding class of unlabelled graphs. We give estimates of the sizes of these classes. For example we show that if \(g(n)=o(n/\log^3n)\) then the class \(\mathcal{E}^{g}\) has growth constant \(\gamma_{\mathcal{P}}\), the (labelled) planar graph growth constant; and when \(g(n) = O(n)\) we estimate the number of n-vertex graphs in \(\mathcal{E}^{g}\) and \(\widetilde{\mathcal E}^g\) up to a factor exponential in \(n\). From these estimates we see that, if \(\mathcal{E}^g\) has growth constant \(\gamma_{\mathcal{P}}\) then we must have \(g(n)=o(n/\log n)\), and the generating functions for \(\mathcal{E}^g\) and \(\widetilde{\mathcal E}^g\) have strictly positive radius of convergence if and only if \(g(n)=O(n/\log n)\). Such results also hold when we consider orientable and non-orientable surfaces separately. We also investigate related classes of graphs where we insist that, as well as the graph itself, each subgraph is appropriately embeddable (according to its number of vertices); and classes of graphs where we insist that each minor is appropriately embeddable. In a companion paper, these results are used to investigate random \(n\)-vertex graphs sampled uniformly from \(\mathcal{E}^g\) or from similar classes.Mathematics Subject Classifications: 05C10, 05C30Keywords: Embeddable graphs, order-dependent surfaces, approximate counting, labelled graphs

The effect of ammonia addition on soot nanostructure and composition in ethylene laminar flames

The formation of soot in ammonia (NH3) combustion with a hydrocarbon such as ethylene (C2H4) is investigated using the analysis of particles’ nanostructure and surface chemical composition to identify the mechanisms by which NH3 suppresses soot growth. Young and mature soot particles were extracted from laminar diffusion co-flow flames of NH3C2H4 blends up to 50% NH3. The soot nanostructure is examined using lattice fringe analysis of high-resolution transmission electron microscopy (HRTEM) images, Raman Spectroscopy, and Electron Energy Loss Spectroscopy (EELS). The chemical composition of the soot surface is analyzed using X-ray Photoelectron Spectroscopy (XPS). With increasing NH3 addition, reduced planar growth of the soot graphitic layers is observed through the analysis of the soot nanostructure which indicates the suppression of carbon addition at the edge sites of the layer planes. It is also found that the carbon sp2/sp3 bonding ratio increases which shows a nanostructure with less carbon sp3 bonding at defect sites. By exploring the chemical composition, the nitrogen content of the soot surface increases with NH3 addition which is caused by the bonding of nitrogenated species with carbon at defect and edge sites of the carbon layer, reducing the potential for carbon addition, and thus soot growth. The results also show an increased risk of formation of nitrogenated polyaromatic hydrocarbons (N-PAHs) on the soot surface by NH3 addition. The study presents a novel investigation of the impact of NH3 on the soot surface chemistry and particles nanostructure which explains the role of NH3 in suppressing soot growth and reveals the potential for new toxic characteristics for the soot emissions formed in NH3-hydrocarbon cofired systems.

Epitaxial Lateral Overgrowth of GaAsP for III‐V/Si‐Based Photovoltaics

GaAsP/Si with high crystalline quality fabricated by cost‐effective heteroepitaxial technology is a promising pathway for realizing low‐cost Si‐based tandem solar cell with efficiency higher than 30%. Herein, hydride vapor‐phase epitaxy is used to perform selective area growth of GaAsP with high lateral coverage, referred to as epitaxial lateral overgrowth (ELOG). The ELOG is performed on GaAs‐based substrates as a prestudy, followed by GaAs/Si and GaAsP/Si seed wafers employing chemical mechanical polishing to fabricate full 2″ GaAsP/Si templates. These are subsequently used to grow and process GaAsP/Si pn‐junction structures for electrical characterization. The ELOG GaAsP is studied by spatially resolved photoluminescence (PL) mapping and high‐resolution X‐ray diffraction measurements. PL analysis of the GaAsP/GaAs ELOG samples reveals an enhanced P‐incorporation during lateral growth of GaAsP. This is also observed for the GaAsP/Si ELOG templates along with evidence of improved material quality, clearly distinguishing the laterally grown GaAsP from the planar growth directly above the Si substrate. Leakage pathways causing reduced electrical performance of the ELOG GaAsP/Si pn‐junction structures are identified.

Core-Shell Nanorods as Ultraviolet Light-Emitting Diodes.

Existing barriers to efficient deep ultraviolet (UV) light-emitting diodes (LEDs) may be reduced or overcome by moving away from conventional planar growth and toward three-dimensional nanostructuring. Nanorods have the potential for enhanced doping, reduced dislocation densities, improved light extraction efficiency, and quantum wells free from the quantum-confined Stark effect. Here, we demonstrate a hybrid top-down/bottom-up approach to creating highly uniform AlGaN core-shell nanorods on sapphire repeatable on wafer scales. Our GaN-free design avoids self-absorption of the quantum well emission while preserving electrical functionality. The effective junctions formed by doping of both the n-type cores and p-type caps were studied using nanoprobing experiments, where we find low turn-on voltages, strongly rectifying behaviors and significant electron-beam-induced currents. Time-resolved cathodoluminescence measurements find short carrier liftetimes consistent with reduced polarization fields. Our results show nanostructuring to be a promising route to deep-UV-emitting LEDs, achievable using commercially compatible methods.

Investigating defects in InGaN based optoelectronics: from material and device perspective

III-nitride optoelectronics have revolutionized solid-state lighting technology. However, non-radiative defects play a major bottleneck in determining the performance of InGaN-based optoelectronics devices. It becomes especially challenging when high indium is required to be incorporated to obtain emission at higher wavelength (>500 nm). In this research article, we are going to discuss our investigation on the origin of defects in InGaN-based optoelectronics devices from the material and device perspective and characterize them through various techniques. This article broadly consists of two parts. In the first part, we investigate defects in InGaN based optoelectronics from a material point of view. Here, we discuss the challenges in the growth of InGaN planar (2-dimensional) and nanowires (1-dimensional) with high indium (≥20%) incorporation using the plasma-assisted molecular beam epitaxy (PA-MBE) technique. Photoluminescence spectroscopy (PL) has been performed to characterize these grown samples to assess their optical quality. Atomic force microscopy (AFM) has been employed to characterize the surface morphology of grown InGaN layers. High-resolution transmission electron microscopy (HRTEM) and scanning electron microcopy (SEM) are also used to characterize InGaN planar and nanowire samples grown under various process conditions. In the second part, we investigate the role of defects on InGaN optoelectronics from a device point of view. Here, we discuss the fabrication of InGaN multi-quantum well-based light emitting diodes (LEDs). Temperature-dependent current versus voltage measurements are carried out to investigate the role of defects on carrier dynamics under forward and reverse bias conditions. Frequency-dependent capacitance versus voltage (CV) and conductance versus voltage (GV) techniques are employed extensively to characterize defects in fabricated InGaN LEDs.

Effective Density Derived from Laboratory Measurements of the Vapor Growth Rates of Small Ice Crystals at −65° to −40°C

Abstract An electrodynamic levitation thermal-gradient diffusion chamber was used to grow 268 individual, small ice particles (initial radii of 8–26 μm) from the vapor, at temperatures ranging from −65° to −40°C, and supersaturations up to liquid saturation. Growth limited by attachment kinetics was frequently measured at low supersaturation, as shown in prior work. At high supersaturation, enhanced growth was measured, likely due to the development of branches and hollowed facets. The effects of branching and hollowing on particle growth are often treated with an effective density ρeff. We fit the measured time series with two different models to estimate size-dependent ρeff values: the first model decreases ρeff to an asymptotic deposition density ρdep, and the second models ρeff by a power law with exponent P. Both methods produce similar results, though the fits with ρdep typically have lower relative errors. The fit results do not correspond well with models of isometric or planar single-crystalline growth. While single-crystalline columnar crystals correspond to some of the highest growth rates, a newly constructed geometric model of budding rosette crystals produces the best match with the growth data. The relative frequency of occurrence of ρdep and P values show a clear dependence on ice supersaturation normalized to liquid saturation. We use these relative frequencies of ρdep and P to derive two supersaturation-dependent mass–size relationships suitable for cloud modeling applications.