Dopant Research Articles

Quantum-Confined Tunable Ferromagnetism on the Surface of a Van der Waals Antiferromagnet NaCrTe2.

The surface of three-dimensional materials provides an ideal and versatile platform to explore quantum-confined physics. Here, we systematically investigate the electronic structure of Na-intercalated CrTe2, a van der Waals antiferromagnet, using angle-resolved photoemission spectroscopy and ab initio calculations. The measured band structure deviates from the calculation of bulk NaCrTe2 but agrees with that of ferromagnetic monolayer CrTe2. Consistently, we observe unexpected exchange splitting of the band dispersions, persisting well above the Néel temperature of bulk NaCrTe2. We argue that NaCrTe2 features a quantum-confined 2D ferromagnetic state in the topmost surface layer due to strong ferromagnetic correlation in the CrTe2 layer. Moreover, the exchange splitting and the critical temperature can be controlled by surface doping of alkali-metal atoms, suggesting the feasibility of tuning the surface ferromagnetism. Our work not only presents a simple platform for exploring tunable 2D ferromagnetism but also provides important insights into the quantum-confined low-dimensional magnetic states.

Dual Modulation of Bulk Electronic Structure and Surficial Active Sites in Sea Urchin‐Like MoO2 Nanoreactors Promoting Electrocatalytic Hydrogen Evolution

AbstractElectrocatalytic hydrogen evolution reaction (HER) is a promising strategy for realizing carbon neutralization as well as for the production of green hydrogen. Molybdenum dioxide (MoO2), possesses acid corrosion resistance and near‐metal‐level conductivity, endowing its widespread application in acidic HER. However, due to spatial barriers at the edge of sites and weak H* adsorption, the HER activity of MoO2 is greatly limited. Herein, a sea urchin‐like Pt@N‐MoO2 nanoreactor is designed, in which the bulk electronic structure and surface‐active sites are modulated by N doping and Pt single atoms anchoring, respectively. DFT calculations indicate that Mo─N coupling changes the charge density of Mo atoms, enhances the adsorption of H*, and thus optimizes the Gibbs free energy. The appearance of Pt‐O/N sites compensates for the lack of active sites exposed by MoO2, while promoting the desorption of H2 from the catalyst surface and accelerating the HER process. This work provides an effective strategy for activating inert electrocatalysts to promote energy conversion via a dual modulation strategy of bulk and surface engineering.

Study on interface characteristics and electron spin manipulation of two-dimensional [formula omitted] (001) heterostructures

Among numerous strongly correlated electron materials, cobalt ions in perovskite oxide SrCoO3 (SCO) possess rich valence states, playing a crucial role in determining the functionality of the material. Meanwhile, thin film preparation techniques such as epitaxial growth and pulsed laser deposition have accelerated the research progress of two-dimensional SCO films. In this study, we primarily investigate the interface characteristics and electron spin of two-dimensional SrCoO3/LaAlO3 (SCO/LAO) (SCO/LAO) (001) heterostructures using density functional theory, revealing the influence of atomic layer thickness and doping on the heterostructure. Our research results indicate that SCO/LAO heterostructures exhibit metallic behavior and ferromagnetism, and doping with oxygen vacancies and Nb2+ ions leads to significant changes in electronic properties and magnetism of the heterostructure. These findings provide insights for the experimental synthesis of two-dimensional heterostructures and suggest that two-dimensional SCO/LAO (001) heterostructures possess rich interface properties with tremendous potential applications in electronic devices.

Effects of Group IVA Elements on the Electrical Response of a Ge2Se3-Based Optically Gated Transistor.

The optically gated transistor (OGT) has been previously demonstrated as a viable selector device for memristor devices, and may enable optical addressing within cross-point arrays. The OGT current-voltage response is similar to a MOSFET device, with light activating the gate instead of voltage. The OGT also provides a naturally built-in compliance current for a series resistive memory element, determined by the incident light intensity on the gate, thus keeping the integrated periphery circuitry size and complexity to a minimum for a memory array. The OGT gate comprises an amorphous Ge2Se3 material that can readily be doped with other elements to alter the transistor's electrical properties. In this work, we explore the operation of the OGT when the Ge2Se3 gate material is doped with the Group IVA elements C, Si, Sn, and Pb. The dopant atoms provide changes to the optical and electrical properties that allow key electrical properties such as the dark current, photocurrent, switching speed, and threshold voltage to be tuned.

Plasmonic Al, Ga and in-doped zinc oxide nanoparticles as key components for the design of electrophoretic inks with MWIR and LWIR absorbance properties

Doped zinc oxide nanocrystals (NCs) are halfway through semiconductors and metals. They exhibit unique optoelectronic properties from a high surface density of free-charge-carriers, which are responsible for localized surface plasmon resonance (LSPR). Here, a one-pot approach is presented to synthesize doped aluminum, gallium or indium zinc oxide NCs, making them all stable in non-polar media. The effect of doping on the growth mechanisms and the final crystalline structure were studied as a function of the aliovalent doping atom used. Doping atoms were integrated by substitution of Zn atom into the crystalline mesh of ZnO with a wurtzite phase identified as the primary crystalline phase for all samples by X-ray Diffraction (XRD). Typical aluminiun doped ZnO NCs (AZO) or Gallium doped NCs (GZO) nanoflowers were identified as a single crystalline structure by High-Resolution-Transmission Electron Microscopy (HR-TEM). Indium doped ZnO NCs (IZO) and pristine ZnO appeared significantly different with spherical and heart-like shapes, respectively. The doping level tendency was identified through Energy Dispersive X-ray (EDX) and increased form Al to Ga and In doping atoms. All plasmonic doped NCs have broadband infrared absorption in the Middle-wave (MWIR) and Long-wave infrared (LWIR) wavelengths, making them interesting for thermal regulation or thermal camouflage. As part of the latter application, dispersions of doped ZnO NCs were formulated as electrophoretic inks and their IR-absorbing performances determined by using a homemade setup including an infrared camera. AZO, GZO, and IZO NCs based inks achieved temperature contrasts of 15 °C and 6 °C in the MWIR and LWIR wavelengths.

Investigation of mechanical behavior of porous carbon-based matrix by molecular dynamics simulation: Effects of Si doping

Understanding the mechanical properties of porous carbon-based materials can lead to advancements in various applications, including energy storage, filtration, and lightweight structural components. Also, investigating how silicon doping affects these materials can help optimize their mechanical properties, potentially improving strength, durability, and other performance metrics. This research investigated the effects of atomic doping (Si particle up to 10 %) on the mechanical properties of the porous carbon matrix using molecular dynamics methods. Young's modulus, ultimate strength, radial distribution function, interaction energy, mean square displacement and potential energy of designed samples were reported. MD outputs predict the Si doping process improved the mechanical performance of porous structures. Numerically, Young's modulus of the C-based porous matrix increased from 234.33 GPa to 363.82 GPa by 5 % Si inserted into a pristine porous sample. Also, the ultimate strength increases from 48.54 to 115.93 GPa with increasing Si doping from 1 % to 5 %. Silicon doping enhances the bonding strength and reduces defects in the carbon matrix, leading to improved stiffness and load-bearing capacity. This results in significant increases in mechanical performance. However, excess Si may disrupt the optimal bonding network, leading to weaker connections within the matrix. Also, considering the negative value of potential energy in different doping percentages, it can be concluded that the amount of doping added up to 10 % does not disturb the initial structure and stability of the system, and the structure still has structural stability. So, we expected our introduced atomic samples to be used in actual applications.

Boosting the Electrical Transfer by Molybdenum Doping for Robust and Flexible NiSe-Based Supercapacitor.

NiSe is a promising electrode material for enhancing the energy density of supercapacitors, but it faces challenges such as sensitivity to electrolyte anions, limited specific capacity, and unstable cycling. This study employs a strategy of metal atom doping to address these issues. Through a hydrothermal reaction, Mo-doped NiSe demonstrates significant improvement in electrochemical performance, exhibiting high capacity (799.90Cg-1), splendid rate performance, and excellent cyclic stability (90% capacity retention). The introduction of Mo induces charge redistribution in NiSe, leading to a reduction in the band gap. Theoretical calculation reveals that Mo doping can effectively enhance the electrical conductivity and the adsorption energy of NiSe. A flexible printed hybrid Mo-doped NiSe-based supercapacitor is fabricated, demonstrating superior electrochemical performance (367.04mFcm-2) and the ability to power timers, LEDs, and toy fans. This research not only deepens the understanding of the electrochemical properties of metal-doped NiSe but also highlights its application potential in high-performance supercapacitors.

Collision of rare-gas atoms on helium nanodroplets: Theoretical evidence for an efficient coagulation of heavy rare-gas atoms.

The coagulation of rare-gas atoms (RG = Ne, Ar, Kr, Xe, and Rn) in helium nanodroplets (HNDs) composed of 1000 atoms is investigated by zero-point averaged dynamics where a He-He pseudopotential is used to make the droplet liquid with proper energies. This method reproduces the qualitative abundances of embedded Arn+1 structures obtained by Time-Dependent Density Functional Theory and Ring Polymer Molecular Dynamics for Ar + ArnHe1000 collisions at realistic projectile speeds and impact parameters. More generally, coagulation is found to be much more efficient for heavy rare-gases (Xe and Rn) than for light ones (Ne and Ar), a behavior mainly attributed to a slower energy dissipation of the projectile in the HND. When coagulation does not occur, the projectile maintains a speed of 10-30m s-1 within the HND, but its velocity vector is rarely oriented toward the dopant, and the projectile roams in a limited region of the droplet. The structure of embedded RGn+1 clusters does not systematically match their gas-phase global minimum structure, and more than 30% of RGn-RG unbound structures are due to one He atom located in between the projectile and a dopant atom.

Theoretical investigation of Ag and Au modified CSiN monolayer as a potential gas sensor for air decomposition components detection

Monitoring and detection of air decomposition components (ADCs) is an effective approach for diagnosing the faults in air switchgear devices. Herein, the adsorption behavior of three ADCs (CO, NO, NO2) on pristine CSiN and Ag/Au-doped CSiN (Ag-CSiN and Au-CSiN) monolayers was investigated using first-principles calculations. The electronic properties of different adsorption systems were systematically analyzed. Moreover, the gas-sensing performance of these systems was evaluated by the analyses of conductivity, work function, and recovery time, which aims to explore the potential sensitive materials for monitoring the ADCs. The results reveal that the pristine CSiN exhibits weak adsorption towards CO and NOx. However, the doping of Ag and Au atoms significantly enhances the adsorption effects of CSiN monolayer, resulting in a transformation from physisorption to chemisorption, with the adsorption energy values ranging from − 0.73 eV to − 1.45 eV. The related adsorption mechanism is further elucidated through the analyses of density of states, band structures, charge density difference, and electron localization function. Furthermore, both Ag-CSiN and Au-CSiN exhibit excellent sensitivity towards CO and NO due to the moderate adsorption strength and significant changes in conductivity and work function. Specifically, Ag-CSiN exhibits a sensing response of 5.59 × 103 towards NO at 348 K, with a recovery time of 1.45 s. While Au-CSiN shows sensing responses of 498 and 6.00 × 105 towards CO and NO at 298 K, respectively, with recovery times of 2.20 s and 7.06 s. Therefore, Au-CSiN can be utilized a promising and recyclable gas sensor for detecting CO and NO at room temperature, and its sensing performance is unaffected by moisture in the air. This study offers a theoretical foundation for the design of novel gas sensors that can effectively diagnose the faults in air switchgear equipment.

Polyacrylonitrile-based 3D N-rich activated porous carbon synergized with Co-doped MoS2 for promoted electrocatalytic hydrogen evolution

MoS2 is a promising electrocatalyst for the hydrogen evolution reaction (HER), but it is limited by its own agglomeration, low conductivity, and surface inertness. In this study, we report a well-organized 3D hybrid structure of N-rich active porous carbon (CN) synergized with Co-doped molybdenum disulfide (Co-MoS2@CN) for promoted electrocatalytic hydrogen evolution. MoS2 nanosheets were grown vertically on CN and the pore structure was improved, which exposed abundant edge active sites. The N-rich graphite structure and abundant pore structure of CN enhanced the electron transfer ability and stability of the 3D hybrid structure. Meanwhile, doping of Co atoms optimized the catalytic activity of the in-plane sulfur sites and induced a partial transition from 2H-MoS2 to 1 T-MoS2. The resulting Co-MoS2@CN 3D hybrid structure exhibited excellent HER electrocatalytic performance, had a current density of 10 mA cm−2 in a 0.5 M H2SO4 aqueous solution by an overpotential of 137 mV, and showed excellent stability in a 24 h HER test. DFT calculations confirmed that the heterogeneous interface constructed by CN and Co-MoS2 effectively optimized Gibbs free energy (ΔGH*) of the MoS2 hydrogen adsorption site and enhanced the interfacial electron transfer capacity, which ensured rapid HER kinetics during the electrocatalytic process.

Urea adsorption and detection using silicon nanowires doped with B, Al, C, Ge, N, and P: A DFT investigation

Urea can serve as a biomarker for the detection of various illnesses, including renal and hepatic failure. Consequently, the development of novel devices and materials capable of adsorbing and identifying urea is a crucial objective for the scientific community. This study theoretically investigates the adsorption and detection capabilities of doped silicon nanowires (SiNWs) for urea using Density Functional Theory (DFT). Doping involves substituting a silicon atom on the surface with a dopant atom; B, Al, C, Ge, N, and P were employed for this purpose. This study presents an innovative method for enhancing urea adsorption and detection by doping SiNWs with group XIII elements, specifically aluminum and boron atoms. The results indicate that this doping significantly improves urea adsorption on SiNWs compared to undoped SiNWs. Notable changes in the bandgaps and work functions of the doped nanowires following urea adsorption suggest their potential use as diagnostic tools for uremia.

Efficient Holes Abstraction by Precisely Decorating Ruthenium Single Atoms and RuOx Clusters on ZnIn2S4 for Photocatalytic Pure Water Splitting.

Developing efficient photocatalysts for two-electron water splitting with simultaneous H2O2 and H2 generation shows great promise for practical application. Currently, the efficiency of two-electron water splitting is still restricted by the low utilization of photogenerated charges, especially holes, of which the transfer rate is much slower than that of electrons. Herein, Ru single atoms and RuOx clusters are co-decorated on ZnIn2S4 (RuOx/Ru-ZIS) to employ as multifunctional sites for efficient photocatalytic pure water splitting. Doping of Ru single atoms in the ZIS basal plane enhances holes abstraction from bulk ZIS by regulating the electronic structure, and RuOx clusters offer a strong interfacial electric field to remarkably promote the out-of-plane migration of holes from ZIS. Moreover, Ru single atoms and RuOx clusters also serve as active sites for boosting surface water oxidation. As a result, an excellent H2 and H2O2 evolution rates of 581.9 µmolg-1h-1 and 464.4 µmolg-1h-1 is achieved over RuOx/Ru-ZIS under visible light irradiation, respectively, with an apparent quantum efficiency (AQE) of 4.36% at 400nm. This work paves a new way to increase charge utilization by manipulating photocatalyst using single atom and clusters.

Non-trivial evolution of the Dirac cone in chromium doped Dirac semimetal Cd3As2

The magnetic doping of Dirac semimetals breaks their time-reversal symmetry thus giving rise to anomalous transport phenomena promising for applications. While the Dirac cone (DC) manifestations were observed in the electron transport of (Cd1−xCrx)3As2 alloys up to x = 0.06, their mechanism remains challenging. To address this challenge, we performed DFT calculations of the (Cd0.96Cr0.04)3As2 alloy with the non-magnetic (NM), ferromagnetic (FM), and antiferromagnetic (AFM) spin orders. It was found that the NM state is unstable, while the FM state has the lowest energy. The band structure analysis revealed that Cr 3d states are distributed in energy not uniformly, but with Cr-free windows, in which the Dirac spectrum can survive. The only exception is the Dirac point vicinity, where a narrow gap is formed. Outside the window, the DC band strongly hybridizes with the Cr 3d states and disappears. Near EF, such a Cr-free window with the DC band exists only in the FM↓ state and is absent in the NM, FM↑, and AFM states. The Fermi surface of FM (Cd1−xCrx)3As2 has two parts: the DC↓ sheet with the velocity vF ≈ 1∙106 m/s and several sheets with low vF, originating from Cr 3d↑ states. As an example of transport properties, the dc conductivity of FM (Cd1−xCrx)3As2 was estimated. We found that at T→0 K the DC↓ electrons have a very large transport lifetime and therefore dominate in the conductivity. In this dominance, the role of the Cr-free window is double: it ensures the DC↓ surviving and greatly reduces an admixture of Cr 3d↓ orbitals to DC↓ states, so suppressing the scattering of DC↓ electrons by doped Cr atoms. This mechanism looks rather general and may be applied to the design of magnetic topological alloys.

In-situ construction of ionic liquid salt-derived graphene-like dual-heteroatoms doping engineering as efficient metal-free electrocatalyst for oxygen reduction reaction

Metal-free heteroatom-doped carbon catalysts can effectively eliminate the degradation and pollution problems caused by metal dissolution. However, single heteroatom doping often restricts the regulation of geometric and electronic structures for oxygen reduction reaction (ORR) electrocatalysts. Herein, a dual atomic doping engineering to optimize the electronic structure and local coordination environment of ionic liquid salt-derived graphene-like host material as an efficient metal-free electrocatalyst for ORR. The resultant PNG catalyst with a typical nanosheets morphology with a high specific surface area and unique hierarchical porous structure. Benefiting from the uniform doping of P, N heteroatoms, graphene-like structure, optimized electronic structure and coordination environment, thus-prepared PNG-2 exhibits outstanding ORR activity, which the onset potential and half-potential are negatively shifted by around 70 mV and 46 mV, respectively, as compared to the commercial Pt/C (20 wt.%) for ORR. Furthermore, the PNG-2 exhibits the higher diffusion limiting current density, superior long-term stability as well as excellent resistance to methanol oxidation than Pt/C. Experimental studies and density functional theory (DFT) calculations reveal that the enhanced chemisorption of oxygen species (*OOH, *O and *OH) onto the P and N sites of the PNG catalysts can effectively accelerate the charge transfer kinetics and promote a favorable ORR performance.

A novel approach to detecting doping agents in food using electrochemical sensor based on zinc oxide/graphene oxide nanocomposites

A novel approach to detecting doping agents in food using electrochemical sensor based on zinc oxide/graphene oxide nanocomposites

Interface engineering assisted La1-xSrxCoO3/FeOOH heterostructure as a high-performance electrocatalyst for oxygen evolution reaction

Perovskite oxides play a crucial role as catalysts in the oxygen evolution reaction (OER) owing to their distinctive structure and diverse physical and chemical characteristics. Currently, enhancing the electronic structure and specific surface area of the perovskite surface through strategic regulation poses a significant challenge in the advancement of perovskite catalyst development. Here, we propose a novel method for enhancing the OER activity of LaCoO3 (LCO) perovskite through atomic doping and interface engineering. The La0.6Sr0.4CoO3–FeOOH (Fe-LSCO) composite exhibits a notable enhancement in electrocatalytic performance, manifesting a low overpotential of 296.3 mV for the oxygen evolution reaction (OER) at a current density of 10 mA cm−2, while maintaining a high level of activity for a duration of 120 h. The enhanced activity of Fe-LSCO can be attributed to oxygen vacancy defects and the increase of the active site caused by A-site-doping and amorphous nanosheets of FeOOH, which synergistic effect reduces the energy barrier for water dissociation. The highly disordered surface and high surface energy of amorphous FeOOH render it highly active. This study presents a viable interface engineering approach for manipulating the reconstruction of the active phase in high-performance perovskite-based electrocatalytic materials.

Atomic-level tailoring of single-atom tungsten catalysts for optimized electrochemical nitrate-to-ammonia conversion

Nitrate contamination of water resources poses significant health and environmental risks, necessitating efficient denitrification methods that produce ammonia as a desirable product. The electrocatalytic nitrate reduction reaction (NO3RR) powered by renewable energy offers a promising solution, however, developing highly active and selective catalysts remains challenging. Single-atom catalysts (SACs) have shown impressive performance, but the crucial role of their coordination environment, especially the next-nearest neighbor dopant atoms, in modulating catalytic activity for NO3RR is underexplored. This study aims to optimize the NO3RR performance of tungsten (W) single atoms anchored on graphene by precisely engineering their coordination environment through first and next-nearest neighbor dopants. The stability, reaction paths, activity, and selectivity of 43 different nitrogen and boron doping configurations were systematically studied using density functional theory. The results reveal W@C3, with W coordinated to three carbon atoms, exhibits outstanding NO3RR activity with a low limiting potential of −0.36 V. Intriguingly, introducing next-nearest neighbor B and N dopants further enhances the performance, with W@C3-BN achieving a lower limiting potential of −0.26 V. This exceptional activity originates from optimal nitrate adsorption strengths facilitated by orbital hybridization and charge modulation effects induced by the dopants. Furthermore, high energy barriers for NO2 and NO formation on W@C3 and W@C3-BN ensure their selectivity towards NO3RR products. These findings provide crucial atomic-level insights into rational design strategies for high-performance single-atom NO3RR catalysts via coordination environment engineering.

Ion diffusion retarded by diverging chemical susceptibility

For first-order phase transitions, the second derivatives of Gibbs free energy (specific heat and compressibility) diverge at the transition point, resulting in an effect known as super-elasticity along the pressure axis, or super-thermicity along the temperature axis. Here we report a chemical analogy of these singularity effects along the atomic doping axis, where the second derivative of Gibbs free energy (chemical susceptibility) diverges at the transition point, leading to an anomalously high energy barrier for dopant diffusion in co-existing phases, an effect we coin as super-susceptibility. The effect is realized in hydrogen diffusion in vanadium dioxide (VO2) with a metal-insulator transition (MIT). We show that hydrogen faces three times higher energy barrier and over one order of magnitude lower diffusivity when it diffuses across a metal-insulator domain wall in VO2. The additional energy barrier is attributed to a volumetric energy penalty that the diffusers need to pay for the reduction of latent heat. The super-susceptibility and resultant retarded atomic diffusion are expected to exist universally in all phase transformations where the transformation temperature is coupled to chemical composition, and inspires new ways to engineer dopant diffusion in phase-coexisting material systems.

Screening and comparison of 18 transition metal atoms doped to modulate the electrocatalytic synthesis of ammonia with multi-S vacancies MoS2: A study of density functional theory

Electrocatalytic nitrogen reduction reaction (NRR) represents a promising approach for the sustainable production of ammonia (NH3). However, current electrocatalysts exhibit limitations in terms of active sites, high reaction potential, and poor reaction selectivity. Consequently, there is a pressing need for the development of NRR electrocatalysts with high activity and selectivity under mild conditions. Among them, defect engineering and transition metal doping are two effective methods to improve the electronic structure of catalysts. In this paper, density functional theory was used to study the activity of electrocatalytic NRR on Molybdenum disulfide (MoS2) surfaces with 1–4 S vacancy numbers (VSx–MoS2, x = 1–4) and transition metal (TM)-doped VS3–MoS2 (TM@VS3–MoS2). Both VSx–MoS2 showed better structural stability, with S vacancy defects altering the electronic structure of the surface. As more Mo atoms were exposed on the surface of the structure, the adsorption of N2 gradually enhanced and the reaction was more inclined to undergo NRR. Further investigation of 18 TM-doped VS3–MoS2 revealed that the doping of Sc, Ti, V, Y, Zr and Nb atoms effectively inhibits the hydrogen precipitation reaction and reduced the energy of NH3 desorption, which contributed to the continuation of the reaction. Therefore, the improvement of MoS2 electrocatalytic NRR performance can be achieved by constructing S vacancies and TM doping to provide a better basis and reference for experimental exploration.

A triple-benefit strategy of microstructure, electronic property and active site modulation on ZnIn2S4 photocatalyst by Sn atom doping

To overcome the high cost and complex preparation of cocatalysts in photocatalytic H₂ production, this study pioneers a triple-benefit strategy in visible-light-absorbing semiconductors through microstructure optimization, electronic property modulation, and active site modification in two-dimensional (2D) hexagonal ZnIn₂S₄ (ZIS) via Sn atom doping. Facilely synthesized through a one-step hydrothermal method without surfactants, 4–6 nm thick Sn-doped ZIS nanosheets exhibit reduced charge recombination by preventing excessive self-assembly and aggregation. Density Functional Theory (DFT) and X-ray Absorption Fine Structure (XAFS) confirm Sn’s substitution for Zn on (001) surface, shifting the Fermi level into the conduction band to facilitate electron migration and charge separation. This adjustment also modulates the electronic properties of adjacent S atoms, triggering the inert basal plane for an enhanced H₂ evolution reaction kinetics. Consequently, Sn-ZIS achieves a H₂ evolution rate of 62.18 μmol h−1 under visible light, significantly outperforming pure ZIS and Pt@ZIS by 6.7 and 3.5 times, respectively.