Planar Geometry Research Articles

Circling a Regular Pentagon with Straightedge and Compass in Euclidean Geometry

“Circling The Regular Pentagon” is possibly used to describe a new challenge problem that comes from the exact solution for another new challenge problem “Circling the SQUARE”, published in January 2024 in the International Journal of Mathematics Trends and Technology (IJMTT). "Circling The SQUARE" means constructing a circle that has the exact area of the given square”. Therefore, the “Circling the Regular Pentagon” problem challenge has not existed in the mathematics field until it is solved and published nowadays. This paper, then, provides an exact solution to construct a circle that has the same area as a given regular pentagon. The solution does not use the number  and suits the exact constraint of Euclidean Geometry by straightedge & compass.

Structural and computational analysis of a nickel(II) complex with S-benzyldithiocarbazate Schiff base ligand: Synthesis, X-ray crystallography, MEP, HOMO-LUMO, Hirshfeld surface analysis, and molecular docking

A new compound, benzyl N-[1-(thiophen-2-yl)propylidene]hydrazinecarbodithioate (LH), and its nickel(II) complex (NiL2), were synthesized and structurally identified. Single crystal X-ray diffraction analyses determined that both the LH ligand and the NiL2 complex crystallized in the monoclinic system, with space group P21/c and Z = 4. The LH ligand exhibited lattice parameters of a = 9.661(2)Å, b = 8.995(2)Å, c = 18.939(4)Å, β = 102.677(4)°, V = 1605.7(7)Å3, whereas, NiL2 exhibited a = 17.805(7)Å, b = 10.548(4)Å, c = 18.100(7)Å, β = 108.279(6)°, V = 3228(2)Å3. The NiL2 complex was found to possess a slightly distorted square planar geometry with the LH deprotonated ligand acting as a monoanionic bidentate by coordinating through the azomethine nitrogen and thiolate sulfur atoms. It was observed that the HOMO-LUMO energy gap of the NiL2 complex (0.3654 eV) was lower than that of the LH ligand (2.4425 eV), indicating that NiL2 possesses a higher reactivity than the LH ligand. Intermolecular hydrogen bonding interactions were examined by Hirshfeld surface and 2D fingerprint analysis, which showed predominant H…H interactions for both the LH ligand and the NiL2 complex. A molecular docking simulation study with the dopamine D4 receptor (DRD4) and farnesyltransferase (FTase) revealed that NiL2 was able to interact with 13 residues in FTase, as compared to the farnesyl-Cys drug interacted with 11 residues.

Hydrogen embrittlement of Zircaloy-4 fabricated by ultrasonic additive manufacturing

Ultrasonic additive manufacturing (UAM) was successfully applied to the zirconium material system to create a planar geometry. Following fabrication, SS-J3 type tensile specimens of the UAM and wrought Zircaloy-4 with a nominal gage section of 5 × 1.2 × 0.75 mm were machined for hydriding studies and mechanical testing. The SS-J3 type tensile specimens were gas-charged with hydrogen using a custom system that precisely controls hydrogen gas flow rate, hydrogen partial pressure, and temperature. To avoid altering the UAM material, the maximum process temperature was limited to 550 °C. Using different initial hydrogen gas pressures and flow rates, various hydrogen contents (70–1755 wppm) were achieved for Zircaloy-4 specimens. Tensile testing shows that, regardless of hydrogen content, all UAM specimens measured yield strengths in the range of 557 ± 16 MPa and ultimate tensile strengths of 660 ± 4 MPa. However, total elongation clearly decreased as a function of increasing hydrogen content. At the lowest hydrogen content, the total plastic elongation measured 21.5 %, and this value decreased to less than 2 % when the hydrogen content was increased to 1000 wppm. Cross-sectional optical microscopy images revealed that the hydride distributions are randomly oriented. For the low hydrogen content specimen, these randomly distributed hydrides are isolated from each other. A sandwiched structure, consisting of high-density and low-density layers, was also observed for the specimens with hydrogen content between 200 and 400 wppm. As the hydrogen content increases, the hydrides diffuse into the low-density hydride layers to form a network across the whole specimen. Although the tensile properties of UAM and wrought Zircaloy-4 exhibit the same behavior with increasing hydrogen content, the distinctions in grain orientations led to differences in the hydride orientations at low and intermediate hydrogen concentrations.

Mixed ligand complexes of Pt(II) Tertiary phosphines and 1-(benzo[d]thiazol-2-yl)-3-phenylthiourea; Crystal structure and DFT of [Pt(C14H9N3S2)(dppe)].CH2Cl2

This paper describes a series of Pt(II) thiourea complexes (1–4). The complexes were synthesized by a reaction between [PtCl2(diphos)] and the ligand 1-(benzo[d]thiazol-2-yl)-3-phenylthioure in the presence of triethylamine, with all reactants present in equimolar amounts. The produced complexes exhibited square planar geometry, which was validated by FT-IR and 1HNMR . Single crystal XRD study revealed that the coordination geometry was distorted square planar in [Pt(Btzt)(dpppe)]. Hirshfeld surface analysis was used to delve deeper into the intermolecular interactions which showed that the combined contribution of H⋯H and H⋯C contacts in the stabilization of the supramolecular assembly was 81.3%. A void analysis was carried out which showed that there was no large cavity in the supramolecular assembly. Theoretical studies were also conducted on [Pt(Btzt)(dpppe)]. The DFT analysis was carried out using the B3LYP Hybrid method in combination with Def2-SVP. The theoretical findings indicated just 3% differences between the theoretical and experimental Figures. Furthermore, the computed HOMO-LUMO gap was 3.4 eV, which was close to the anatase photocatalyst's band gap. DFT predicted that all complexes are non-magnetic with μB=0. There are high electron donation and back-donation between heteroatoms and transition metals in the complexes.

A Double Legendre Polynomial Order N Benchmark Solution for the 1D Monoenergetic Neutron Transport Equation in Plane Geometry

As more and more numerical and analytical solutions to the linear neutron transport equation become available, verification of the numerical results becomes increasingly important. This presentation concerns the development of another benchmark for the linear neutron transport equation in a benchmark series, each employing a different method of solution. In 1D, there are numerous ways of analytically solving the monoenergetic transport equation, such as the Wiener–Hopf method, based on the analyticity of the solution, the method of singular eigenfunctions, inversion of the Laplace and Fourier transform solution, and analytical discrete ordinates in the limit, which is arguably one of the most straightforward, to name a few. Another potential method is the PN (Legendre polynomial order N) method, where one expands the solution in terms of full-range orthogonal Legendre polynomials, and with orthogonality and series truncation, the moments form an open set of first-order ODEs. Because of the half-range boundary conditions for incoming particles, however, full-range Legendre expansions are inaccurate near material discontinuities. For this reason, a double PN (DPN) expansion in half-range Legendre polynomials is more appropriate, where one separately expands incoming and exiting flux distributions to preserve the discontinuity at material interfaces. Here, we propose and demonstrate a new method of solution for the DPN equations for an isotropically scattering medium. In comparison to a well-established fully analytical response matrix/discrete ordinate solution (RM/DOM) benchmark using an entirely different method of solution for a non-absorbing 1 mfp thick slab with both isotropic and beam sources, the DPN algorithm achieves nearly 8- and 7-place precision, respectively.

Regional Functionalization Molecular Design Strategy: A Key to Enhancing the Efficiency of Multi-Resonance OLEDs.

Herein, we propose a regional functionalization molecular design strategy that enables independent control of distinct pivotal parameters through different molecule segments. Three novel multiple resonances thermally activated delayed fluorescence (MR-TADF) emitters A-BN, DA-BN, and A-DBN, have been successfully synthesized by integrating highly rigid and three-dimensional adamantane-containing spirofluorene units into the MR framework. These molecules form two distinctive functional parts: part 1 comprises a boron-nitrogen (BN)-MR framework with adjacent benzene and fluorene units forming a central luminescent core characterized by an exceptionally rigid planar geometry, allowing for narrow FWHM values; part 2 includes peripheral mesitylene, benzene, and adamantyl groups, creating a unique three-dimensional "umbrella-like" conformation to mitigate intermolecular interactions and suppress exciton annihilation. The resulting A-BN, DA-BN, and A-DBN exhibit remarkably narrow FWHM values ranging from 18 to 14 nm and near-unity photoluminescence quantum yields. Particularly, OLEDs based on DA-BN and A-DBN demonstrate outstanding efficiencies of 35.0 % and 34.3 %, with FWHM values as low as 22 nm and 25 nm, respectively, effectively accomplishing the integration of high color purity and high device performance.

Regional Functionalization Molecular Design Strategy: A Key to Enhancing the Efficiency of Multi‐Resonance OLEDs

AbstractHerein, we propose a regional functionalization molecular design strategy that enables independent control of distinct pivotal parameters through different molecule segments. Three novel multiple resonances thermally activated delayed fluorescence (MR‐TADF) emitters A‐BN, DA‐BN, and A‐DBN, have been successfully synthesized by integrating highly rigid and three‐dimensional adamantane‐containing spirofluorene units into the MR framework. These molecules form two distinctive functional parts: part 1 comprises a boron‐nitrogen (BN)‐MR framework with adjacent benzene and fluorene units forming a central luminescent core characterized by an exceptionally rigid planar geometry, allowing for narrow FWHM values; part 2 includes peripheral mesitylene, benzene, and adamantyl groups, creating a unique three‐dimensional “umbrella‐like” conformation to mitigate intermolecular interactions and suppress exciton annihilation. The resulting A‐BN, DA‐BN, and A‐DBN exhibit remarkably narrow FWHM values ranging from 18 to 14 nm and near‐unity photoluminescence quantum yields. Particularly, OLEDs based on DA‐BN and A‐DBN demonstrate outstanding efficiencies of 35.0 % and 34.3 %, with FWHM values as low as 22 nm and 25 nm, respectively, effectively accomplishing the integration of high color purity and high device performance.

Proof Without Words: Estimating Generalized Pi

Summary Euclidean geometry is a special case of the p-metric whose unit circle satisfies | x | p + | y | p = 1 , and we may use its area to define a generalized version of pi. We wordlessly deduce some simple bounds on the values of generalized pi from inscribed and exscribed octagons.

Near-threshold fatigue crack propagation in a Ni-based single crystal superalloy affected by crystallographic anisotropy

Fatigue crack propagation in the near-threshold regime was investigated experimentally and numerically in a Ni-based single crystal superalloy, namely, CMSX-4. Fatigue crack propagation tests were conducted at room temperature using four types of specimens with different primary and secondary crystallographic orientations. The results reveal that the crystallographic orientations strongly affect the fatigue crack propagation behavior, including the crack paths, fatigue crack propagation rates, and fatigue threshold. Crystal plasticity finite element analysis was conducted to quantify the slip activity of an octahedral slip system in front of the crack tip, considering the actual 3D geometry of the crack plane and elastic–plastic anisotropy. The fatigue damage parameter, considering the slip activities of the individual octahedral slip systems, provides reasonable explanation for the fatigue crack propagation rates and crack paths in the near-threshold regime, regardless of the crystallographic orientation. Furthermore, these damage parameters are equivalent at the fatigue thresholds for all specimens with different crystallographic orientation.

Discontinuous Galerkin approximations of the heterodimer model for protein–protein interaction

Mathematical models of protein–protein dynamics, such as the heterodimer model, play a crucial role in understanding many physical phenomena, e.g., the progression of some neurodegenerative diseases. This model is a system of two semilinear parabolic partial differential equations describing the evolution and mutual interaction of biological species. This article presents and analyzes a high-order discretization method for the numerical approximation of the heterodimer model capable of handling complex geometries. In particular, the proposed numerical scheme couples a Discontinuous Galerkin method on polygonal/polyhedral grids for space discretization, with a θ-method for time integration. This work presents novelties and progress with respect to the mathematical literature, as stability and a-priori error analysis for the heterodimer model are carried out for the first time. Several numerical tests are performed, which demonstrate the theoretical convergence rates, and show good performances of the method in approximating traveling wave solutions as well as its flexibility in handling complex geometries. Finally, the proposed scheme is tested in a practical test case stemming from neuroscience applications, namely the simulation of the spread of α-synuclein in a realistic test case of Parkinson’s disease in a two-dimensional sagittal brain section geometry reconstructed from medical images.

Inscribing Spheres in Topologically Embedded Simplices

In plane geometry, it is well-known that one can inscribe a circle in any triangle. A less trivial yet natural question could be asked about inscribing circles in topologically embedded triangles. This paper presents a proof for the following general theorem: Given any boundary of an n-simplex topologically embedded in Euclidean k-space, with n ≤ k , there exists a ( k − 1 ) -sphere whose interior is disjoint from the embedded boundary of the simplex and which touches each ( n − 1 ) -dimensional face on the boundary of the simplex at least once. We say this sphere is inscribed in the embedding. This paper also shows that ellipsoids, cubes, and octahedrons can be inscribed in any topologically embedded boundary of the n-simplex. The uniqueness of inscribed spheres is also discussed.

Facile Synthesis and Global Aromaticity of Aza‐Superbenzene and Aza‐Supernaphthalene at Different Oxidation States

AbstractAll‐benzenoid polycyclic aromatic hydrocarbons or macrocycles usually display localized aromaticity. On the other hand, incorporation of quinoidal units into the skeleton could lead to effective electron delocalization and global (anti)aromaticity. In this work, fully π‐conjugated macrocycle 1 and bismacrocycle 2 containing both para‐quinodimethane and triphenylamine units are efficiently synthesized mainly through intermolecular Friedel–Crafts alkylation reaction. They can be considered as a tetraazasuperbenzene and a hexaazasupernaphthalene, respectively, due to their similar geometry and electronic structures to the benzene and naphthalene. X‐ray crystallographic analyses reveal a largely planar geometry for both 1 and 2 and variable‐temperature NMR measurements disclose slow dynamic processes owing to restricted ring flipping of the phenyl rings. 1 and 2 can be easily oxidized into higher‐oxidation‐state species. NMR and theoretical calculations indicate that 12+ and 14+ show global anti‐aromaticity and aromaticity, respectively, with a dominant 32π and 30π conjugation pathway, while for the bismacrocycle 2, its dication 22+, tetracation 24+ and hexacation 26+ exhibit global aromaticity, antiaromaticity, and aromaticity with a 54π, 52π and 50π conjugation pathway along the outermost backbone, respectively.

Facile Synthesis and Global Aromaticity of Aza-Superbenzene and Aza-Supernaphthalene at Different Oxidation States.

All-benzenoid polycyclic aromatic hydrocarbons or macrocycles usually display localized aromaticity. On the other hand, incorporation of quinoidal units into the skeleton could lead to effective electron delocalization and global (anti)aromaticity. In this work, fully π-conjugated macrocycle 1 and bismacrocycle 2 containing both para-quinodimethane and triphenylamine units are efficiently synthesized mainly through intermolecular Friedel-Crafts alkylation reaction. They can be considered as a tetraazasuperbenzene and a hexaazasupernaphthalene, respectively, due to their similar geometry and electronic structures to the benzene and naphthalene. X-ray crystallographic analyses reveal a largely planar geometry for both 1 and 2 and variable-temperature NMR measurements disclose slow dynamic processes owing to restricted ring flipping of the phenyl rings. 1 and 2 can be easily oxidized into higher-oxidation-state species. NMR and theoretical calculations indicate that 12+ and 14+ show global anti-aromaticity and aromaticity, respectively, with a dominant 32π and 30π conjugation pathway, while for the bismacrocycle 2, its dication 22+, tetracation 24+ and hexacation 26+ exhibit global aromaticity, antiaromaticity, and aromaticity with a 54π, 52π and 50π conjugation pathway along the outermost backbone, respectively.

The Impact of GeoGebra in Analytic Algebra: A Bibliometric Review

GeoGebra is a software that is often used in mathematics learning. GeoGebra is an electronic program that contains a set of tools that equip students with mathematical skills. This research aims to look at research trends related to algebra learning that integrates GeoGebra. This research is literature review research, there are 95 publications collected from the Scopus database which are then analyzed using the bibliometric analysis method assisted by the Vos viewer application. Data taken from the Scopus database was refined in 4 stages, namely identification, filtering, eligibility, and inclusion, resulting in 95 publications. The research results show that Spain and Austria are the most influential countries and have high cooperation with other countries in this field. The focus of research related to GeoGebra on algebra material is, 1) geometry and students; 2) GeoGebra and algebra; 3) elementary geometry and computational theory; 4) dynamic geometry and computer algebra. The new themes in this research are symbolic computation, quantifier elimination and real quantifier elimination. This research not only enriches the scientific literature in the fields of GeoGebra and algebra but also provides practical guidance for educators and researchers to improve mathematics teaching and learning methods in the future.

High-Precision Calibration Method and Error Analysis of Infrared Binocular Target Ranging Systems

Infrared binocular cameras, leveraging their distinct thermal imaging capabilities, are well-suited for visual measurement and 3D reconstruction in challenging environments. The precision of camera calibration is essential for leveraging the full potential of these infrared cameras. To overcome the limitations of traditional calibration techniques, a novel method for calibrating infrared binocular cameras is introduced. By creating a virtual target plane that closely mimics the geometry of the real target plane, the method refines the feature point coordinates, leading to enhanced precision in infrared camera calibration. The virtual target plane is obtained by inverse projecting the centers of the imaging ellipses, which are estimated at sub-pixel edge, into three-dimensional space, and then optimized using the RANSAC least squares method. Subsequently, the imaging ellipses are inversely projected onto the virtual target plane, where its centers are identified. The corresponding world coordinates of the feature points are then refined through a linear optimization process. These coordinates are reprojected onto the imaging plane, yielding optimized pixel feature points. The calibration procedure is iteratively performed to determine the ultimate set of calibration parameters. The method has been validated through experiments, demonstrating an average reprojection error of less than 0.02 pixels and a significant 24.5% improvement in calibration accuracy over traditional methods. Furthermore, a comprehensive analysis has been conducted to identify the primary sources of calibration error. Ultimately, this achieves an error rate of less than 5% in infrared stereo ranging within a 55-m range.

Quantitative Signal Analysis of Sum-Frequency Scattering Experiments from Aerosol Surfaces.

Droplet interfaces are instrumental in processes of biology, engineering, production, and environmental systems. The chemical and physical properties of heterogeneous interfaces are known to be different from those of their underlying bulk phases, and different again when considering the curved surface of submicron aerosol droplets. The recently developed technique of vibrational sum-frequency scattering (VSFS) spectroscopy from airborne particles has emerged as an interface-specific method for the in situ analysis of this unique system. While the technique has shown promise in debut works, a quantitative analysis of the VSFS system has not yet been performed. Here we provide a comprehensive analysis of a VSFS spectrometer with reference to the well-documented planar analog. We decompose the VSFS signal into coherent and incoherent as well as resonant and nonresonant components as a function of incident pulse delay time. We then quantify and compare resonant and nonresonant VSFS and VSFG experimental data using the same laser and detection systems. Using the air/water interface as a guide, we show that the resonant and nonresonant contributions to the SF responses are comparable for the two systems by extracting second-order susceptibilities and hyperpolarizabilities, and using them to estimate single-particle susceptibilities. A quantitative analysis of the signal detection systems for the scattering and planar geometries is made, and conversion efficiencies for VSFG, VSFS, and other nonlinear scattering experiments are compared. Lastly, the possibility of a low-repetition (1 kHz) VSFS spectrometer is considered, determining that it may be possible with modern laser technology but is inevitably less efficient than a high-repetition (100 kHz) system. Though this multistep analysis we obtain a better understanding of the components of the VSFS signal from aerosol particles, further validate the feasibility of the experiments, and provide insight to those wishing to conduct similar experiments and how they may be improved.

Relocation of the 2021 MW 7.4 Maduo, Qinghai, China earthquake sequence and implications for seismogenic structure

An MW 7.4 earthquake struck Maduo County, Qinghai, China on 22 May 2021. To better understand the seismogenic structure of this region, we collect local earthquake arrival time data at the MAD station in the China Earthquake Networks Center observational bulletins during 1 June to 20 September 2021, and manually pick P and S wave arrival times from high-quality seismograms recorded at 34 recently deployed MaduoArray portable seismic stations. Using these arrival times, we relocate the Maduo earthquake sequence using earthquake association, absolute location and relative location methods. Our results show that the 2021 Maduo earthquake sequence occurred along the Kunlun Mountain Pass-Jiangcuo fault zone in the NWW-SEE direction and the aftershocks are located on both sides of the mainshock, showing characteristics of bilateral rupture. Vertical cross-sections of the aftershock distribution illustrate a nearly vertical shape of the seismogenic fault that tilts toward the northeast and southwest in different sections, reflecting a complex geometry of the fault plane. There is a horsetail bifurcation phenomenon at the eastern end of the fault zone. A sparse area of aftershocks appears at about 30 km east of the mainshock epicenter, which may be associated with a uniform fault friction and sufficient release of rupture energy caused by super-shear rupture of the mainshock. Taking into account many geophysical results including seismic tomography and magnetotelluric soundings, we speculate that the occurrence of the Maduo earthquake could be affected by crustal fluids in the fault zone. The fluids may ascend from the lower crustal flow beneath northeastern Tibet.

Supramolecular Assembly of Nitrogen-Containing Curved π-Conjugated Molecules to Polar Crystal

The chemistry of curved π-conjugated molecules has been extensively studied because of the development of their synthetic approach. Among the numerous studies, we have been engaged in nitrogen-containing curved π-conjugated molecules. These molecules have extended optical properties, redox-active behaviors, and electron-donating ability due to the electron-rich nature of nitrogen atoms. Recently, we have focused on the supramolecular assembling behavior of these compounds. For example, we have reported buckycatcher consisted of two azabuckybowls, providing two-photon absorbing host-guest complexes with fullerenes. In this presentation, we would like to show the packing behavior using π-extended azahelicenes and azabuckybowls. Our group has developed π-extended aza[5]helicene derivatives, showing brilliant emissive features. Recently, we succeeded in the synthesis of unsymmetrically substituted azahelicenes. We found that their stackings in the crystal were dependent on the substituent. X-ray diffraction analysis for all products showed π-stacked conformation with each other in the crystal (Figure 1b). In particular, azahelicenes with triisopropylsilyl groupswere found to align their dipole moments to form a one-dimensional columnar structure in the crystal. Polar crystals are expected to have applications in ferroelectric, piezoelectric, and nonlinear optical response materials. This is the first example of a polar crystal with helicenes, paving the way for novel piezoelectric organic materials.The second topic is related to azabuckybowl. In the crystal, azabuckybowl forms a face-to-face dimer by concave-convex interaction. In this study, we prepared two types of azabuckybowl perylene bisimide (PBI) dyads connected with alkyne linkages. PBIs have also been reported to show columnar stacking in solution. We anticipated that the segregate stacking should be achieved by recognizing the difference in structures between the bowl and planar geometry. The UV/vis absorption spectra revealed the presence of intramolecular charge-transfer interaction. In addition, the concentration-dependent absorption spectra and 1H NMR spectra indicated PBI and azabuckybowl units stacked with each other independently in solution (Figure 1a). Here, we would like to discuss about the effect of the effect of molecular shape on the supramolecular assembly with the data obtained using these compounds. Figure 1

In-Situ Raman Spectroscopy of Li+ and Na+ Storage in Anodic Nanotubes

Anodizing is a high-voltage electrochemical conversion process that forms barrier-type oxide layers or self-organized nanoporous/nanotubular structures. So far, the Al2O3-like nanopores and TiO2-like nanotubes could be successfully synthesized on many valve metals and alloys.Recently, it become possible to form nanotubular structures on iron and consequently control the ratio of hematite (Fe2O3) to magnetite (Fe3O4) by subsequent thermal treatment.1 Electrochemically active nanotubes, such as self-organized TiO2 are of high interest in the battery field due to a unique one-dimensional (1D) geometry offering high volume expansion tolerance and applications without binders and conductive additives. Herein, we report an in - situ Raman spectroscopy study under current control for a better fundamental understanding of Li+/Na+ storage in nanotubes and correlate the structural fingerprints with the electrochemical data on differential capacity plots of d(Q–Q 0) dE –1.2 Real-time measurements revealed that the nanotubes had undergone two major phase transformations with increasing lithium content, disclosing the sequential steps of a lithium intercalation type of storage. In contrast, sodium-ion insertion induced no significant crystal structure modification but instead a slight crystallinity rupture, signifying a dominant nondiffusion-limited capacitive type of storage. The insight into the charge storage in a 1D material in relation to other nanostructures such as nanoparticles3 will be discussed.1. L. Fadillah, D. Kowalski, M. Vincent, C. Zhu, S. Kitano, Y. Aoki and H. Habazaki, ACS Applied Materials & Interfaces , 2023, 15, 52563-52570.2. M. Vincent and D. Kowalski, ACS Applied Nano Materials , 2023, 6, 6528-6537.3. M. Vincent, S. S. Kumar and D. Kowalski, Electrochimica Acta , 2023, 469, 143161. Figure 1

CMP: Four Decades of Innovation and a Promising Future

The invention of CMP by IBM in 1983 represents an inflection point in the evolution of semiconductors. It was a bold and risky idea to mechanically apply abrasives to the surface of a wafer, but this new process for restoring a wafer to a planar state was the key to continued areal scaling and increased device complexity. Initially applied to a simple application for filling trenches with insulator, the potential of this technique was evident. Through understanding of material properties and subsequent manipulation of slurry chemistry, CMP was applied to enable the more challenging and technically critical processes for Shallow Trench Isolation and Tungsten contacts. The late 1990’s saw the introduction of copper interconnects. IBM was first to market this technology which continues to support conductors down to lines widths on the order of 10 nanometers! The CMP process required for copper interconnects demanded a thorough understanding of the materials present and potential surface phenomena for exploitation. For this need, a new and complex mechanism for copper planarization was invented. Equally challenging was the need for a subsequent non-selective barrier removal step. For this a unique, self-balancing approach was also invented, borrowing some elements from previous STI technology. Device architecture has progressed from planar geometry to FINFETs to Gate All Around, or Nanosheet technology. This evolution has required increasingly more CMP process steps with a concurrent demand for increased process control. Gate height control for Nanosheeets is currently paramount for performance, affected by many CMP steps in the process flow. A typical requirement is less than 1nm added variability per CMP step! Looking forward, CMP continues to be tapped to enable technology evolution. Polish steps are becoming part of the patterning process to enable pitch splitting for upcoming nodes. A VTFET integration approach can enable further area scaling but presents integration challenges; CMP is being asked to enable “third color” schemes for the protection of NFET vs PFET devices which is necessary at various points in the in the VTFET process flow. Heterogeneous Integration is demanding nanometer control of topography across large contact pads and large lateral distances. Finally, extreme scaling is creating fundamental challenges for metal polish at the contact and interconnect levels. CMP tool architecture currently does not provide for control of ambient conditions at the wafer surface during wet transport and post CMP cleaning. Subsequently, unwanted metal dissolution is observed. New tool designs will be needed to control wafer surface conditions, and particularly oxygen exposure through all steps within the CMP tool. This talk is dedicated to the special session in honor of Professor S.V. Babu