Recent progress in enhancing built-in electric fields of perovskite solar cells via junction engineering

• Tong Xiao and
• Ke Xu

Beilstein J. Nanotechnol. 2026, 17, 602–621, doi:10.3762/bjnano.17.42

• gradient was induced across the film thickness. This induced a controllable potential gradient across the film thickness, increased the substrate work function by about 0.3 eV, and created a vertical p–n homojunction. Time-resolved photoluminescence (TRPL) measurements revealed that the carrier lifetime of

Probing tribological evolution in atomically thin MoS₂ at different scales

• Xingzhong Zeng and
• Miao Zhang

Beilstein J. Nanotechnol. 2026, 17, 586–597, doi:10.3762/bjnano.17.40

• tribology toward the sub-nanoscale. Experimental Sample preparation Single-layer and few-layer MoS2 were obtained from bulk MoS2 by mechanical exfoliation [19]. N-Doped Si covered with dry oxidation-generated 300 nm thick SiO2 was used as the substrate. Prior to exfoliation, substrates were cleaned

• as the actual tip displacement between adjacent sub-sites, which is critical for characterizing load- and layer thickness-dependent sub-nanoscale stick–slip motion. Results and Discussion Due to the relatively large fluctuation of atomically thin MoS2 caused by the rough substrate, the thickness of

• MoS2 on the rough substrate is determined by combining the cross-sectional height profile and the Raman spectra. The height profiles in topographic image are carefully located in the fluctuation direction of rough substrate. Thus, the thicknesses of MoS2 on the rough substrate along the black and red

Impacts of annealing on structural and photophysical properties of zinc phthalocyanine adsorbed on graphene

• Gautier Creutzer,
• Quentin Fernez,
• Nataliya Kalashnyk,
• Zohreh Safarzadeh,
• Lydia Sosa Vargas,
• Céline Fiorini-Debuisschert,
• Nicolas Fabre and
• Fabrice Charra

Beilstein J. Nanotechnol. 2026, 17, 576–585, doi:10.3762/bjnano.17.39

• treatments, as often shown for example with α and β phases of zinc phthalocyanines (ZnPc), with consequences on molecule orientations relative to the substrate and absorbance efficiency [21] or dynamics of charge migration and charge transfer to substrate [22]. Even inside a given phase, minute structural

• particular, ZnPc has been shown to present such 2D phase transitions on various substrates such as TiO2 [32], Au(111) [33][34], InSb [35], or ZnS [36]. These studies have emphasized the diversity of mechanisms involved in the relative stability of 2D phases, including intermolecular versus molecule–substrate

• Zn atom pointing outward (i.e., away from graphene) thus allowing the Pc-conjugated structure to approach closer to the graphene substrate. Returning to the absorption measurements, we infer that this planar-square to shuttlecock transition is accompanied by a shift of the Q-band, a small proportion

Electrochemical determination of ciprofloxacin using a MIL-101/reduced graphene oxide-modified electrode

• Nguyen Quang Man,
• Nguyen Ngoc Nghia,
• Nguyen Vinh Phu,
• Vo Thi Khanh Ly,
• Le Lam Son,
• Pham Khac Lieu,
• Le Thi Hong Phong,
• Nguyen Dinh Luyen and
• Dinh Quang Khieu

Beilstein J. Nanotechnol. 2026, 17, 541–554, doi:10.3762/bjnano.17.35

• morphology that offers a large surface area and a conductive substrate for material deposition. The pristine MIL-101 (Figure 4b) consists of irregular polyhedral particles that tend to aggregate, forming relatively compact clusters. In contrast, the MIL-101/rGO composite (Figure 4c) reveals that MIL-101

Defects and defect-mediated engineering of two-dimensional materials: challenges and open questions

• Arkady V. Krasheninnikov,
• Matthias Batzill,
• Anouar-Akacha Delenda,
• Marija Drndić,
• Chris Ewels,
• Katharina J. Franke,
• Mahdi Ghorbani-Asl,
• Alexander Holleitner,
• Ado Jorio,
• Ute Kaiser,
• Daria Kieczka,
• Hannu-Pekka Komsa,
• Jani Kotakoski,
• Manuel Längle,
• David Lamprecht,
• Yun Liu,
• Steven G. Louie,
• Janina Maultzsch,
• Thomas Michely,
• Katherine Milton,
• Anna Niggas,
• Hanako Okuno,
• Joshua A. Robinson,
• Marika Schleberger,
• Bruno Schuler,
• Alexander Shluger,
• Kazu Suenaga,
• Kristian S. Thygesen,
• Richard A. Wilhelm,
• E. Harriet Åhlgren and
• Carla Bittencourt

Beilstein J. Nanotechnol. 2026, 17, 454–488, doi:10.3762/bjnano.17.31

• from triplet to singlet during dissociation [62]. Although O2 incorporation into the WS2 surface involves electron transfer from the substrate, the reaction may proceed via non-adiabatic pathways, as previously discussed for O2 adsorption on metal surfaces [66][67]. This underscores the limitations of

• MoSe2 on a gold substrate to the Janus MoSeS structure [77]. Also, a facile low-temperature thiol chemistry route was suggested to remove sulfur vacancies in 2D MoS2 [78], which resulted in a significant drop in the number of the charged impurities and traps and increased the mobility. Qualitatively

• has been found on S vacancies in a monolayer of MoS2 on Au(111) [146]. However, in this case, the metallic Au substrate provided not only the electron bath for screening the magnetic moment, but also enabled the electron transfer to the vacancy due to the favorable alignment of the work functions [147

First-principles study on elastic properties of Cu, (Cu_1−x,Ni_x)₃Sn and interfacial mechanical properties of (Cu_1−x,Ni_x)₃Sn/Cu in the lead-free solder joint

• Guomin Hua

Beilstein J. Nanotechnol. 2026, 17, 428–439, doi:10.3762/bjnano.17.29

• the substrate [9]. In contrast to the barrier layer formed with lead-containing solder, the lead-free solder forms a compact interface between the intermetallics and the substrate without a barrier layer [9]. The compact interface could result in poor resistance to high temperatures and thermal shocks

• interfacial mechanical properties of lead-free solder joints. Considering the important role the interface between the intermetallics and the substrate plays in the strength and reliability of a solder joint, Gan et al. investigated the formation of Cu3Sn and Cu6Sn5 on a Cu substrate and determined the

• orientation relationship of ε-Cu3Sn/Cu interfaces as (001)ε//(111)Cu and [100]ε//[−110]Cu [13], that is, the interface was constructed by attaching the (001) facet of ε-Cu3Sn to the (111) facet of the Cu substrate and making the [100] axis of ε-Cu3Sn parallel to the [−110] axis of Cu substrate. Based on the

Polycatecholamine nanocoatings on stainless steel: the effect on attachment of human fibroblasts and platelets

• Paulina Trzaskowska,
• Ewa Rybak,
• Maciej Trzaskowski,
• Kamil Kopeć,
• Jakub Krzemiński,
• Rafał Podgórski,
• Hatice Genc,
• Mehtap Civelek and
• Iwona Cicha

Beilstein J. Nanotechnol. 2026, 17, 365–380, doi:10.3762/bjnano.17.25

• used atmospheric oxidation (because the amount of oxidant in the solution is not limited by its dissolution) and allows for precise control over coating uniformity and thickness at the nanoscale. Also, for the synthesis of PTYR coatings, there is a limitation in the substrate availability for the

• oxidation reaction due to the limited solubility of ʟ-tyrosine in water. Thus, we used a supersaturated ʟ-tyrosine solution, to ensure that the oxidized part of the substrate is replaced by a fresh portion dissolved from the suspension [12]. The resulting SS-PDA and SS-PTYR samples were sterilizable, non

Ferroelectric nanodot reservoir for neuromorphic computing

• Anna Razumnaya,
• Yuri Tikhonov,
• Dmitrii Naidenko,
• Léo Boron,
• Valerii Vinokur and
• Igor Lukyanchuk

Beilstein J. Nanotechnol. 2026, 17, 352–364, doi:10.3762/bjnano.17.24

• , enabling rich dynamic response to time-varying inputs, as required for real-time neuromorphic tasks. Notably, our approach leverages the electrostatic physics of nanoscale ferroelectrics to implement a scalable, low-power analog computing substrate. The described system relies solely on dielectric

Interconnection morphology effects on the radio frequency response of carbon nanotube sponges

• Manuela Scarselli,
• Javad Rezvani,
• Zeno Zuccari,
• Mattia Scagliotti and
• Simone Tocci

Beilstein J. Nanotechnol. 2026, 17, 343–351, doi:10.3762/bjnano.17.23

• , Italy 10.3762/bjnano.17.23 Abstract In this work, the high-frequency response of a multiwalled carbon nanotube (MWCNT) film grown on a silicon substrate is compared with that of MWCNT sponges (CNSs). Different from the CNT film, CNSs are a self-standing material that can operate in the absence of a

• supporting substrate, showing high flexibility, light weight, and mechanical robustness. We tested our synthesized CNSs as active material for the production of antennas working in the radio frequency (RF) range to determine whether CNT sponges present, in addition to practical advantages over CNT films

• from the as-grown CNS sample and the film of MWCNTs grown on a Si substrate as comparison (Figure 1). We observed that both samples have a resonance peak; for the CNT film, it is S11 = −23.0 dB at 4.40 GHz, and for the CNS sample, it is S11 = −22.6 dB at 4.78 GHz. The result found for the MWCNT film is

Fast vortex dynamics and relaxation times in NbRe-based heterostructures

• Francesco De Chiara,
• Zahra Makhdoumi Kakhaki,
• Francesco Avitabile,
• Francesco Colangelo,
• Abhishek Kumar,
• Carmine Attanasio and
• Carla Cirillo

Beilstein J. Nanotechnol. 2026, 17, 292–302, doi:10.3762/bjnano.17.20

• capping layer. In addition, the reduced strip width of the NbRe/Py microbridge leads to a less efficient heat evacuation toward the substrate, thereby enhancing the role of Joule heating and amplifying the impact of thermal dissipation on vortex dynamics. For this reason, and since, according to the LO

• , some considerations can be done. First, in both samples, the NbRe layer is in contact with the substrate. Therefore, the dominant heat dissipation channel into the thermal bath is common to both systems and does not affect the relative comparison between Au- and Py-capped samples. Consequently

• , substrate-related thermal effects can be considered equivalent, and the observed differences can be attributed to the specific properties of the capping layers and their interaction with the superconducting film. In this respect, Au and Py are expected to affect the vortex dynamics in qualitatively

Advancing nanolithography: a comprehensive review of materials for local anodic oxidation with AFM

• Matteo Lorenzoni

Beilstein J. Nanotechnol. 2026, 17, 275–291, doi:10.3762/bjnano.17.19

• patterns through a biased conductive tip. The process relies on the formation of a nanoscale water meniscus [14] at the tip–sample interface, enabling electrochemical reactions that oxidize the substrate, creating protruding nanostructures mainly composed of oxide. Unlike other techniques requiring complex

• preparation steps, LAO enables direct chemical modification, with precise control of the tip–sample separation during the writing process. This is typically achieved in contact mode, where the tip apex is in direct contact with the substrate, ensuring capillary condensation and the formation of a stable water

• materials considered and illustrating a comparison between conventional and electrode-free LAO (EFLAO). 2 Principles of local anodic oxidation (LAO) As already said, LAO is a nanoscale fabrication technique that utilizes an AFM to generate precise oxidation patterns on a substrate. The process is based on

Multilayered hyperbolic Au/TiO₂ nanostructures for enhancing the nonlinear response around the epsilon-near-zero point

• Fernando Arturo Araiza-Sixtos,
• Mauricio Gomez-Robles,
• Rafael Salas-Montiel and
• Raúl Rangel-Rojo

Beilstein J. Nanotechnol. 2026, 17, 251–261, doi:10.3762/bjnano.17.17

• glass substrate, then a dip in the visible region, and finally an increase towards the IR region. We can observe a relatively high absorbance around 800 nm with transmittances down to 40%, which could lead to losses and undesired thermal effects. The complex permittivity of single layers of the

Gold nanoparticle-decorated reduced graphene oxide as a highly effective catalyst for the selective α,β-dehydrogenation of N-alkyl-4-piperidones

• Brenda Flore Kenyim,
• Mihir Tzalis,
• Marilyn Kaul,
• Robert Oestreich,
• Aysenur Limon,
• Chancellin Pecheu Nkepdep and
• Christoph Janiak

Beilstein J. Nanotechnol. 2026, 17, 218–238, doi:10.3762/bjnano.17.15

• reduced graphene oxide (AuNPs/rGO) were demonstrated to be a highly reactive catalyst for the selective α,β-oxidative dehydrogenation (ODH) of N-alkyl-4-piperidones, using N-methyl-, N-ethyl- and N-benzyl-4-piperidone. The substrate N-methyl-4-piperidone represents a pharmaceutically relevant system as

• stability of Au-SiW9/rGO. It not only achieved the highest TON (118) among the three catalysts but also maintained a steady increase in TON and only a slight decrease in TOF after two hours, reflecting its ability to continuously convert substrate over time with minimal deactivation. In contrast, Au-Cit/rGO

• comparable catalytic activity. The Au-SiW9/AC catalyst, originally synthesized by Xia et al. [13], was reproduced in this study with a key difference of gold loading. Xia et al. used 3.17 wt % Au, corresponding to 0.025 mol Au/mol substrate ratio in the oxidation reaction. In this study, we used only 0.42 wt

Micro- and nanoscale effects in biological and bioinspired materials and surfaces

• Thies H. Büscher,
• Rhainer Guillermo Ferreira,
• Manuela Rebora and
• Stanislav N. Gorb

Beilstein J. Nanotechnol. 2026, 17, 214–217, doi:10.3762/bjnano.17.14

• adhesion. They combined molecular dynamics for the simulation of the adhesive contact between the gecko spatula and the substrate with finite element modeling of the mechanical behavior of the adhesive seta to understand key aspects of gecko seta adhesion across scales. Besides these studies, focusing on

Time of flight secondary ion mass spectrometry imaging of contaminant species in chemical vapour deposited graphene on copper

• Barry Brennan,
• Vlad-Petru Veigang-Radulescu,
• Philipp Braeuninger-Weimer,
• Stephan Hofmann and
• Andrew J. Pollard

Beilstein J. Nanotechnol. 2026, 17, 200–213, doi:10.3762/bjnano.17.13

• substrate, depending on processing conditions employed and the chemical species present on the surface. This has implications for the gas permeation barrier properties of this material, graphene transfer mechanisms, as well as the effectiveness of using the oxidation of the copper foil as a rapid graphene

• diffusion of material present in the copper foil before growth [36][37], which could lead to variations in the properties of graphene once transferred from the Cu foil to an alternative substrate. We also explore the gas permeation properties of the CVD graphene on Cu [38] by examining the oxygen detected

• directly under the graphene after post-growth exposure to atmosphere. This could have implications for better understanding transfer mechanisms that rely on oxidation of the Cu substrate [39][40], defect characterisation [41], or the heat dissipation ability of graphene on Cu [42]. Methods CVD graphene

Structure-dependent thermochromism of PAZO thin films: theory and experiment

• Georgi Mateev,
• Dean Dimov,
• Nataliya Berberova-Buhova,
• Nikoleta Kircheva,
• Todor Dudev,
• Ludmila Nikolova,
• Elena Stoykova,
• Keehoon Hong,
• Dimana Nazarova,
• Silvia Angelova and
• Lian Nedelchev

Beilstein J. Nanotechnol. 2026, 17, 186–199, doi:10.3762/bjnano.17.12

• 1200–4000 nm on a glass substrate. Thicker films were prepared because many potential applications – such as the inscription of polarization-selective holographic optical elements – require greater film thickness for optimal performance. The samples were heated in an oven and the measurement of ∆n

• based on PAZO polymer. Preparation of thin-film samples All thin films used in this work were fabricated using spin-coating technique. Methanol was used as solvent for all samples, and a magnetic stirrer ensured complete dissolution. The solution was then poured onto the substrate (quartz or glass) and

• spun at 1000 rpm to achieve thin film thicknesses ranging from 200 to 4000 nm. For spectral measurements, quartz substrates were used due to their lack of absorbance in the PAZO absorption region; thin film thickness was about 200 nm. A glass substrate was used for the measurement of birefringence (∆n

Capabilities of the 3D-MLSI software tool in superconducting neuron design

• Irina E. Tarasova,
• Nikita S. Shuravin,
• Liubov N. Karelina,
• Fedor A. Razorenov,
• Evgeny N. Zhardetsky,
• Aleksandr S. Ionin,
• Mikhail M. Khapaev and
• Vitaly V. Bol’ginov

Beilstein J. Nanotechnol. 2026, 17, 122–138, doi:10.3762/bjnano.17.8

• is not an impassable barrier on the way to implementation of superconducting neurons. Increasing the neuron integration density in practical devices can be achieved by expanding the screen to cover the entire substrate area. In this case, the actual size of each neuron will be determined by the outer

• parameter here is the London penetration depth [5], which was taken as 85 nm according to manufacturer data. Details of 3D-MLSI numerical technique can be found in [49][50]. An input data file, specified in text format, must contain the geometry of the device in the plane of the substrate, the parameters of

Development and in vitro evaluation of liposomes and immunoliposomes containing 5-fluorouracil and R-phycoerythrin as a potential phototheranostic system for colorectal cancer

• Raissa Rodrigues Camelo,
• Vivianne Cortez Sombra Vandesmet,
• Octavio Vital Baccallini,
• José de Brito Vieira Neto,
• Thais da Silva Moreira,
• Luzia Kalyne Almeida Moreira Leal,
• Claudia Pessoa,
• Daniel Giuliano Cerri,
• Maria Vitória Lopes Badra Bentley,
• Josimar O. Eloy,
• Ivanildo José da Silva Júnior and
• Raquel Petrilli

Beilstein J. Nanotechnol. 2026, 17, 97–121, doi:10.3762/bjnano.17.7

• and immunoliposomes were first stabilized by adding 5% glutaraldehyde for 2 h [25]. After fixation, the formulations were diluted at a ratio of 1:750 in deionized distilled water (ddH2O). A 3.0 µL aliquot of the diluted sample was deposited onto a freshly cleaved mica substrate, followed by vacuum

• dehydration process and adhesion forces onto the substrate [57][58]. On the other hand, the z-axis and 3D projections for HSPC-IM-R-PE (Figure 3E and Figure 3H, respectively) demonstrated a more irregular nanoparticle surface, with visibly rougher and more heterogeneous topography, a result consistent with

Functional surface engineering for cultural heritage protection: the role of superhydrophobic and superoleophobic coatings – a comprehensive review

• Giuseppe Cesare Lama,
• Marino Lavorgna,
• Letizia Verdolotti,
• Federica Recupido,
• Giovanna Giuliana Buonocore and
• Bharat Bhushan

Beilstein J. Nanotechnol. 2026, 17, 63–96, doi:10.3762/bjnano.17.6

• , pollutants, and oily substances. These functional surfaces, often based on coatings consisting of polymeric, ceramic, and composite materials, can provide durable, non-invasive protection tailored to specific substrate weaknesses and exposure environments (indoor and outdoor). Objective of this review

• article is to critically examine the most recent studies and materials innovations relevant to cultural heritage site preservation. First the assessment of substrate vulnerabilities and environmental threats is presented, followed by a detailed analysis of coating types and compositions. It concludes with

• protective materials; nanocomposite; substrate vulnerability; superhydrophobic coatings; sustainability; Introduction The conservation of cultural heritage is not only an endeavor to preserve the physical integrity of artifacts but also to maintain the cultural identity of societies [1]. With the growing

Subdigital integumentary microstructure in Cyrtodactylus (Squamata: Gekkota): do those lineages with incipiently expressed toepads exclusively exhibit adhesive setae?

• Philipp Ginal,
• Yannick Ecker,
• Timothy Higham,
• L. Lee Grismer,
• Benjamin Wipfler,
• Dennis Rödder,
• Anthony Russell and
• Jendrian Riedel

Beilstein J. Nanotechnol. 2026, 17, 38–56, doi:10.3762/bjnano.17.4

• habitats, in contrast, can vary considerably in roughness and stability of the substrate (e.g., sandstone outcrops are generally less stable and rougher than granite boulders) [31][33][34]. However, some plant surfaces can be even rougher than saxicoline ones [30]. The questions of how a fully functional

•  1), yet shows adhesive competency [7][13][50]. Once this occurs, the filament–substrate interaction can be seen as the “trigger” that promotes further elaboration of spines and prongs into setae (enhancing adhesive interactions) and the modification of the scales to support more filaments (promoting

• among microstructure types Spines, prongs, and setae most likely evolved from spinules in association with the adoption of a more scansorial lifestyle [3][22], with accompanying selective demands for increasingly stronger adherence to the substrate. Spines, which resemble the simplest microstructure

Quantitative estimation of nanoparticle/substrate adhesion by atomic force microscopy

• Aydan Çiçek,
• Markus Kratzer,
• Christian Teichert and
• Christian Mitterer

Beilstein J. Nanotechnol. 2026, 17, 1–14, doi:10.3762/bjnano.17.1

• particles between 6 and 12 nm. In addition, an applied positive substrate bias voltage led to more energetic landing conditions and thus to increased adhesion forces. This study underscores the suitability of atomic force microscopy in characterizing adhesion on the nanoscale and offers insights into future

• strategies for tailoring nanoparticle/substrate interactions. Keywords: adhesion; atomic force microscopy; magnetron sputtering; nanomanipulation; nanoparticles; Introduction Nanoparticles (NPs) are at the forefront of basic research and technological innovation, captivating researchers and engineers from

• [5] and physicochemical properties [6]. Among these properties, particle adhesion (which is determined by the interaction between the NP and the substrate) and the interface formed between NPs and substrate [7][8] play a decisive role. Particularly when the contact area between NPs and the substrate

Terahertz-range on-chip local oscillator based on Josephson junction arrays for superconducting quantum-limited receivers

• Fedor V. Khan,
• Lyudmila V. Filippenko,
• Andrey B. Ermakov,
• Mikhail Yu. Fominsky and
• Valery P. Koshelets

Beilstein J. Nanotechnol. 2025, 16, 2296–2305, doi:10.3762/bjnano.16.158

• those in the sample arrays. The S21 parameter for the coplanar line is shown by the cyan dotted line in Figure 6a and Figure 6b. The values are indicated on the right axis. The decrease is caused by the leak of the JJ array radiation into the substrate. Moreover, the parameters of the films that form

Chiral plasmonic nanostructures fabricated with circularly polarized light

• Tian Qiao and
• Ming Lee Tang

Beilstein J. Nanotechnol. 2025, 16, 2245–2264, doi:10.3762/bjnano.16.154

• chemical reactions by modulating the phase of plasmonic modes. Several parameters must be carefully considered when preparing chiral PNSs with CPL. The first consideration is whether to immobilize the achiral starting PNSs on a substrate. The reactions can be easily scaled up if PNSs are dispersed in a

• to PNSs immobilized on a substrate under unidirectional CPL [75][76][77][78]. Therefore, in most cases, PNSs are immobilized on a substrate. The second parameter to consider is the energy of the excitation CPL, as it affects the EM distribution and defines the energetics of hot carriers to modulate

• electron sink, and the hot holes oxidize Pb2+ in solution to form PbO2. The TiO2 substrate is very efficient in removing the unwanted hot electrons to suppress carrier recombination. Ag+ in the reaction mixture eventually consumes these hot electrons. Previous studies have shown that the oxidation of Pb(II

Optical bio/chemical sensors for vitamin B₁₂ analysis in food and pharmaceuticals: state of the art, challenges, and future outlooks

• Seyed Mohammad Taghi Gharibzahedi and
• Zeynep Altintas

Beilstein J. Nanotechnol. 2025, 16, 2207–2244, doi:10.3762/bjnano.16.153

• provided the recognition event, the chemiluminescent substrate (CDP-Star) acted as the indicator, and the sensing signal was the measured photon emission. In this competitive assay, VB12 antibodies were fixed onto a nitrocellulose membrane, then exposed to both VB12 and its enzyme-bound form for

Electromagnetic study of a split-ring resonator metamaterial with cold-electron bolometers

• Ekaterina A. Matrozova,
• Alexander V. Chiginev,
• Leonid S. Revin and
• Andrey L. Pankratov

Beilstein J. Nanotechnol. 2025, 16, 2199–2206, doi:10.3762/bjnano.16.152

• solver of CST MWS in 3D mode. The simulated receiving structure is placed on a 500 μm thick silicon substrate. A 4 mm-diameter silicon hyperhemispherical lens is placed on the rear side of the substrate to efficiently couple the incident radiation into the planar structure. The external signal is

• parameters of the structure. Specifically, increasing the substrate thickness to 700 μm enables an increase in absorbed power up to 0.3 a.u. across an ultrabroad frequency range of 200–1200 GHz (Figure 5e). Such a thick substrate can be realized by using commercially available substrates with greater