Slack Equation Research Articles

A theoretical investigation on the structure stability, electronic structures, optical properties, and transport properties of Zintl compounds CsZn4P3 and CsZn4As3

The present study investigates the structure stability, electronic structures, and optical properties of Zintl compounds CsZn4P3 and CsZn4As3 by first-principles calculations. An assessment of the phonon dispersion curves and elastic constants indicate both dynamic and mechanical stability for the both compounds. By employing the HSE06 hybrid functional, both compounds display direct bandgap with widths of 0.93 eV for CsZn4P3 and 0.66 eV for CsZn4As3. Furthermore, an analysis of the dielectric constant, refractive index, extinction coefficient, and energy loss as functions of photon energy is conducted to study the optical properties. Based on the semi-classical Boltzmann transport theory and the Slack's equation, a large Seebeck coefficient and minimum lattice thermal conductivity are obtained for CsZn4As3, which result in a figure of merit value of 0.79 at 700 K. These findings underscore the potential of CsZn4As3 as a promising candidate for future research and development in the realm of thermoelectric materials.

Elastic and thermal properties of selected 211 MAX phases: A DFT study

MAX phases are a class of ternary materials that have continued to play a key role in the field of materials science due to their unique properties that bridge the gap between metals and ceramics. However, there still remains a need to understand, particularly when the materials are used in harsh environments of say, elevated temperatures. This study sought to address some of these challenges, by determining the elastic and thermal properties of seven MAX-phase materials, namely, Ti2AlC, Ti2AlN, Ti2GaC, Ti2GaN, Ti2PbC, Ti2CdC and Ti2SnC using the density functional theory codes. The calculated values of the melting temperatures ranged from 1100 to 1700 K and hence indicate that the considered 211 MAX phase series are good candidates for high-temperature applications. Thermoelastic profiles obtained, under varying temperature and pressure conditions, show volume reduction with increasing pressure, for Ti2AlC and Ti2CdC, considered as being representative of the series under study. The temperature-dependent lattice thermal conductivity, κph, of the seven MAX phases, was also estimated from Slack's equation, revealing that Ti2AlN has the highest value while Ti2PbC has the lowest value at all the temperatures investigated.

Lattice thermal conductivity of ZnO: experimental and theoretical studies.

ZnO has been explored using different approaches such as doping, nanostructuring, 2D confinement, and introduction of interface effects for improving thermoelectric performance and lowering thermal conductivity. Herein, the lattice thermal conductivity (κL) of ZnO determined from Raman thermometry, the 3ω-method and simulation using three-phonon scattering is presented. The average Debye temperature (θD) of A1(TO) and E2(high) modes estimated utilizing bond-order-length-strength correlation with local bond averaging effects is ∼422 K. The average κL of ZnO calculated using the theoretical coefficient of Slack's equation, θD, and Grüneisen parameter (γ) in Slack's equation is 2.75 W m-1 K-1, which is significantly lower than the κL ∼ 50.9 W m-1 K-1 simulated by considering the Perdew-Burke-Ernzerhof functional with Hubbard and non-analytical corrections. The coefficient of Slack's equation is determined from the κL ∼ 31 W m-1 K-1 of ZnO measured using the 3ω-method for the accurate estimation of the κL. The experimental coefficient of Slack's equation with Raman thermometry data yields κL ∼ 50.6 W m-1 K-1, which is higher than the value obtained using the 3ω-method but consistent with the theoretical value. Thus, three-phonon anharmonicity describes the κL of ZnO associated with Raman scattering. The method adopted to calculate κL will help for the in-depth analysis of phonon dynamics and the design of wurtzite-ZnO-related power electronics.

A comparative investigation on the fundamental physical properties of UX2 (X = O, N, C, Si and S) solid nuclear fuel materials: A DFT + U study

This study investigates and compares the physical properties of various UX2 (X = O, N, C, Si and S) nuclear fuels using a combination of Density Functional Theory (DFT) and Hubbard U correction parameter (U) to accurately model the electronic structure and account for strong correlation effects among 5f orbital electrons. Spin-polarized DFT + U method is employed to optimize the structures of the considered compounds. The computational analysis reveals that UO2 and UN2 exhibit Mott-insulating properties, while the remaining UX2 compounds demonstrate metallic nature. To assess the mechanical aspects of these materials, the study evaluates their mechanical stability, Vickers hardness and machinability index through calculations of elastic constants. Notably, all the UX2 fuels are found to be mechanically stable and possess ductility. Both the two-dimensional (2D) and three-dimensional (3D) depictions of the elastic moduli for the analyzed solid fuels demonstrate the presence of elastic anisotropy. The famous Slack's equation is used to determine the lattice thermal conductivity of the UX2 fuels, providing valuable insights into their thermal properties. Among the investigated materials, UC2 exhibits the highest values of Debye temperature and lattice thermal conductivity, which amount to 412 K and 15.73 Wm−1 K−1 at 300 K, respectively. Furthermore, UN2 demonstrates the highest melting temperature. Among the compounds under investigation, UC2 and UN2 fuels exhibit the most minimal thermal expansion and the greatest heat capacity. Based on these findings, UC2 and UN2 emerge as promising alternative fuel materials to UO2 for potential use in nuclear power reactors. This research contributes to the ongoing efforts in identifying alternative fuel materials.

Dependence of thermal conductivity of graphene on the coefficient of linear expansion and temperature

Graphene attracts particular interest recently due to its extraordinary properties. Xu et al. have studied length-dependent thermal conductivity in suspended single-layer graphene. They observed that the thermal conductivity of graphene increases and remains logarithmic divergence with sample length. Yoon et al. observed that the coefficient of linear expansion (α) for graphene is always negative between at least up to 2500 K. In this paper, we study the dependence of thermal conductivity of graphene on the coefficient of linear expansion (CLE). We find that the thermal conductivity of graphene bears an exponential relationship with its CLE and is directly proportional to the sample length. Using the Slack equation, we have also found that the thermal conductivity of graphene is inversely proportional to the temperature and verified the relationship between the thermal conductivity of graphene and CLE. Due to this property graphene can be utilized to remove fish plates in railroads which can increase the efficiency of railway tracks.

Optoelectronic properties and lattice thermal conductivity of Cs2CuBiX6 (X = F, Cl, Br, I) double perovskites: Thermodynamic and ab initio approaches

In this work, a first-principles method called full potential linearized augmented plane wave (FP-LAPW) within the generalized gradient approach (GGA) is used to investigate potential double free lead halide perovskites applications. The structural and elastic parameters of double perovskites Cs2CuBiX6 (X = F, Cl, Br, I) are investigated, and the results show that these compounds are stable, have a ductile character, and are characterized by a noticeable elastic anisotropy. Lattice thermal conductivity is calculated via Slack's equation for all studied materials, reflecting a good potential for thermoelectric applications. The calculations of the electronic and optical properties are performed using the GGA combined with the modified Becke-Johnson (mBJ) potential with the incorporation of the spin-orbit coupling (SOC). This SO effect allowed us to have correct bandgap behaviour for the Bi states. The studied compounds have large dispersion in their bands and low carrier effective masses. Furthermore, acceptable quality properties such as long diffusion length, tuneable bandgap, high mobility, ambipolar charge transport, and high absorption reinforce these double perovskites and make them even more suitable for photovoltaic applications.

Investigation of half-metallic ferromagnetism and transport properties of pnictide oxides REZnAsO (RE=La, Ce, Nd, Pr, Gd, Eu, Sm, Dy) for spintronics

The electronic, half-metallic and thermoelectric characteristics of pnictide oxides REZnAsO (RE = La, Ce, Nd, Pr, Gd, Eu, Sm, Dy) have been examined using full-potential linearized augmented plane wave (FP-LAPW) technique. The functional GdZnAsO, NdZnAsO and PrZnAsO alloys are predicted as half-metallic ferromagnets within a 100% spin-polarization and a total magnetic moment of 14.00, 6.00 and 4.00 μB, respectively that is promising for spintronic applications. The thermoelectric properties and lattice thermal conductivity of REZnAsO alloys are obtained using the Boltzmann transport theory within the constant relaxation time and Slack equation, respectively. The figure-of-merit (ZT) values of GdZnAsO, NdZnAsO, LaZnAsO, CeZnAsO and DyZnAsO alloys are found to be 0.992, 0.996, 0.943, 0.932, 0.980, respectively. These are promising for future thermoelectric applications.

Inspecting the Thermoelectric Response and Mechanical Stability of Novel Cobalt‐Based Heusler Alloys: A DFT Insight

The potential of Heusler materials to provide interesting aspects predominantly of vital physical properties along with typical variations has extended their prospects for thermoelectric applicability. Herein, highly precise and high‐throughput density functional theory (DFT) calculations along with Boltzmann transport simulation scheme are intensely applied to deliver the numerous physical properties of two Cobalt‐based Heusler materials. The geometric structural optimization of both these alloys suitably approves the cubic crystalline geometry at elevated temperatures. The electronic band profile validates the half‐metallic behavior. The fetched density of states and band occupation reflect the ferromagnetic half‐metallic character of the materials. The calculated elastic parameters reveal that the presented set of Heusler materials is mechanically stable holding ductile nature. These cobalt materials exhibit anisotropic performance for both longitudinal and transverse velocities, so these demonstrate directional‐dependent velocities. The utmost significant lattice part of thermal conductivity is accurately figured out by the help of Slack's equation. The current exploration demonstrates spin half metallicity as well as good value of transport parameters, which opens up a route typical for numerous research areas like thermoelectrics and spintronic applicability.

First-principles investigations of the electronic, magnetic and thermoelectric properties of VTiRhZ (Z= Al, Ga, In) Quaternary Heusler alloys

Calculations using density functional theory (DFT) were performed to investigate the structural, dynamical, mechanical, electronic, magnetic, and thermoelectric properties of VTiRhZ (Z = Al, Ga, In) alloys. The most stable structure of these alloys was found to be the type-I configuration. Using GGA-PBE functional, VTiRhGa, and VTiRhIn alloys are predicted as half-metallic ferromagnets with a 100% spin-polarization and a total magnetic moment of 3μB, which is promising for spintronic applications. The thermoelectric properties and lattice thermal conductivity of VTiRhZ alloys were obtained using the Boltzmann transport theory within the constant relaxation time and Slack equation, respectively. The figure-of-merit (ZT) values of VTiRhAl, VTiRhGa, and VTiRhIn alloys were found to be 0.96, 0.88 and 0.64, respectively, which are promising for future thermoelectric applications.

High thermoelectric performance in cubic inorganic halide perovskite material AgCdX3 (X = F and Cl) from first principles

The need for high-performance thermoelectric materials is a hot topic under research to harvest the waste heat generated in the environment. In the present work, we have studied the inorganic halide perovskite AgCdX3 (X = F and Cl) in the cubic phase with the Pm − 3 m space group. We have explored the structural, electronic, elastic, and thermoelectric properties for both materials using first -principles calculations based on density functional theory. The thermoelectric transport coefficients have been calculated from band structure results with the aid of the BoltzTraP2 code based on the semiclassical Boltzmann theory under the constant relaxation time approximation. The deformation potential theory has been implemented to estimate the relaxation time for charge carriers. The Slack equation is solved to evaluate the contribution of lattice thermal conductivity. The results show that the AgCdCl3 has a high value of the figure of merit (ZT) in comparison to that of AgCdF3 at any temperature irrespective of the type of charge carriers. Our findings will provide a guide for the experimentalists to develop high-efficient thermoelectric materials.

The lattice thermal conductivity in monolayers group-VA: from elements to binary compounds

The lattice thermal conductivity () of the monolayers of partial group-VA elements and binary compounds are systemically investigated by the first-principles calculations and phonon Boltzmann transport equation (PBTE), including aW-antimonene, α-arsenene, black phosphorus, α-SbAs, α-SbP and α-AsP. The values decrease with the increasing of atomic mass for these materials with similar geometry and valence structures. It is ascribed to phonon branches softening, low phonon group velocity, and large Grüneisen parameters. Due to the neutralization of phonon group velocity and phonon lifetime, of binary compounds is between their corresponding elements. As the atomic radius and mass increase, the bond strength and the phonon group velocity decreases. Furthermore, the dimensionless parameter , which comes from the Slack equation and only has the dependence of Grüneisen parameter, grows up with the atomic mass rising, which indicates that a larger anharmonicity is present in the heavier V-V monolayers. For SbAs and SbP compounds, the thermal conductivity anisotropy mainly results from the anisotropy of elastic coefficients along armchair and zigzag directions. Our results highlight the impact of atomic arrangement on the thermal conductivity of group VA binary compounds. This work paves a way to modulate the thermal conductivity of 2D VA elements by incorporation atoms with suitable mass and may guide to improve thermoelectrical performance via the alloying method.

First principles study of thermodynamic properties of Cd-substituted Zn Chalcogenides by employing quasi harmonic Debye model

First Principles study of thermodynamic properties of Cd-substituted Zn Chalcogenides have been performed by using quasi harmonic Debye Model. A non linear variation in bulk modulus of is found in Chalcogen’s group (X = O, S, Se, Te). Slack equation is used to calculate the thermal conductivity of materials and it is found that Cd-rich Zn Chalcogenides have less thermal conductivity. The anharmonicity and thermal expansion coefficient of increase linearly from X = O to X = Se but then interestingly a crossover is appeared in their values at X = Te. Several other thermodynamic parameters like Debye temperature, molar heat capacities, internal energy and entropy are also determined and compared to available theoretical and experimental results.

First-principles insight on elastic, electronic, and thermoelectric transport properties of BAgX (X = Ti, Zr, Hf)

The geometrical structures of BAgX (X = Ti, Zr, Hf) are relaxed and their dynamical stability is confirmed by phonon dispersion curves, based on the first-principles density functional theory. Band energy structures, effective masses, relaxation time and carrier mobility of these structures are also obtained. The elastic constants and the deformation potential constants of each structure are calculated with the deformation potential theory. The lattice thermal conductivity is calculated by employing the Slack equation and elastic matrix. The electronic thermal conductivity, Seebeck coefficient, and power factors are obtained by solving the Boltzmann equation. Finally, the dimensionless figure of merit (ZT) is evaluated. The results demonstrate that a maximum ZT value of 0.8–1.2 could be achieved for the n-type BAgX, and the largest one is 1.2 for the n-type BAgZr at the carrier concentration value of 1.17 × 1020 with the temperature of 500 K, implying that the n-type BAgZr is a promising candidate for thermoelectric materials.

Magneto-electronic, mechanical, thermoelectric and thermodynamic properties of ductile perovskite Ba2SmNbO6

We have extensively explored crystal properties based on the structural optimization inclusive of elasto-mechanical, thermoelectric and thermodynamics of double layered perovskite Ba22+Sm3+Nb5+O62−. The generalised gradient approximation (GGA) within Perdew-Burke-Ernzerhof (PBE) functional along with Hubbard correction (U) were integrated to figure out exact electronic structure. Magnetic inetarctions suggest ferromagnetic phase as stable phase as also confirmed by thermodynamic calculations. Spin dependent band structures along with density of states envisages its perfect half-metallic character with a direct band gap of 3.32 eV under GGA + U approximation. The elastic properties has been evaluated to descript its mechanical strength, which convey its ductile nature. Boltzmann theory is employed to check the thermoelectric response, where Seebeck coefficient of 211.4 μVK−1 and power-factor of 1.12 × 1011 W/mK2s is seen. Furhter, the lattice thermal conductivity via Slack's equation is found to be 1.04 W/mK at room temperature which show a decaying trend towards high temperatures. The high temperature and pressure variation of thermodynamic properties were calculated using the quasi-harmonic Debye approximation to descript its stability.

A scattering rate model for accelerated evaluation of lattice thermal conductivity bypassing anharmonic force constants

Predicting the lattice thermal conductivity from the atomic structure is important to many scientific and engineering applications. However, the state-of-the-art method based on first-principles calculations of the three-phonon scattering process is bound with high computational cost, while semiempirical models such as the Slack equation are less accurate. In this work, we examined the theoretical background of the commonly used computational models for thermal conductivity evaluation and proposed an improved quasiharmonic model based on an early approximation for three-phonon scattering strength. This model has significantly reduced computational cost as compared to the full anharmonic lattice dynamics calculations but retains a fairly good quantitative accuracy comparing to many semiempirical models. It also allows one to include normal processes in phonon-phonon scattering and obtain the phonon relaxation times.

A Projection Neural Network for Circular Cone Programming

A projection neural network method for circular cone programming is proposed. In the KKT condition for the circular cone programming, the complementary slack equation is transformed into an equivalent projection equation. The energy function is constructed by the distance function and the dynamic differential equation is given by the descent direction of the energy function. Since the projection on the circular cone is simple and costs less computation time, the proposed neural network requires less state variables and leads to low complexity. We prove that the proposed neural network is stable in the sense of Lyapunov and globally convergent. The simulation experiments show our method is efficient and effective.

Assessing the thermoelectric properties of ScRhTe half-heusler compound

We investigate structural, elastic, electronic and thermoelectric properties of ScRhTe with frame work of density functional theory and Boltzmann equations. The physical properties such as elastic constants, bulk modulus, shear modulus, Young's modulus and Poisson's ratio of ScRhTe compound are calculated. The elastic properties of ScRhTe reveal that this compound is mechanically stable. Further, the phonon calculation of ScRhTe indicates that it also dynamically stable. The band structure as well as DOS spectra indicates that ScRhTe compound has a semiconductor nature. The calculated energy band gap of ScRhTe compound is 0.8eV.The thermoelectric properties such as Seebeck coefficient, electrical conductivity scaled by relaxation time, power factor scaled by relaxation time and electronic thermal conductivity has been investigated with variation of carrier concentration and temperature. The highest power factor obtained at 1200 K is 11.9 × 1011V2SK−2ms−1 at optimum carrier concentration n∼1.3 × 1021cm−3 for n-type ScRhTe. The maximum dimensionless figure of merit for n-type ScRhTe is 0.63. Using the slack equation, the lattice thermal conductivity of ScRhTe is also estimated. At room temperature, ScRhTe attains low thermal conductivity as compared to well know ZrNiPb half heusler compound. The lower lattice thermal conductivity of ScRhTe suggest that this compound may be act as good thermoelectric material. Our results suggested that power factor can't improve further when electron carrier concentration reaches to n∼1.3 × 1021cm−3 and hole carrier concentration reaches to n∼7.5 × 1020cm−3. This is the first quantitative theoretical estimation of these properties.

Ab-initio calculations of elastic constants and thermodynamic properties of LuAuPb and YAuPb half-heusler compounds

In this work, density functional theory plane-wave pseudo potential method with generalized gradient approximation is used to investigate the elastic, mechanical, phonon and thermodynamic properties of LuAuPb and YAuPb compounds. The physical properties such as elastic constants, bulk modulus, shear modulus, Young's modulus, Poisson's ratio and shear anisotropy factor of both the compounds are calculated. The elastic properties of these compounds reveal that these compounds are mechanically stable. Further, the phonon dispersion curves prove that these compounds are also dynamically stable. The study of shear anisotropic factor gives purely anisotropic nature of these compounds. Pugh's and Frantsevich's ratio shows that both these compounds exhibit ductile nature. The sound velocities and Debye temperature of these compounds are also estimated from elastic constants. Using the slack equation, the lattice thermal conductivity of these compounds is also estimated. At room temperature, LuAuPb attains low thermal conductivity as compared to YAuPb. At 300 K, the calculated values of specific heat capacity per atom at constant volume for LuAuPb and YAuPb compounds are 24.37 Jmol−1 K−1 and 23.95 Jmol−1 K−1 respectively. This is the first quantitative theoretical estimation of these properties.

High throughput combinatorial method for fast and robust prediction of lattice thermal conductivity

The lack of computationally inexpensive and accurate ab-initio based methodologies to predict lattice thermal conductivity, without computing the anharmonic force constants or time-consuming ab-initio molecular dynamics, is one of the obstacles preventing the accelerated discovery of new high or low thermal conductivity materials. The Slack equation is the best alternative to other more expensive methodologies but is highly dependent on two variables: the acoustic Debye temperature, θa, and the Grüneisen parameter, γ. Furthermore, different definitions can be used for these two quantities depending on the model or approximation. In this article, we present a combinatorial approach to elucidate which definitions of both variables produce the best predictions of the lattice thermal conductivity, κl. A set of 42 compounds was used to test the accuracy and robustness of all possible combinations. This approach is ideal for obtaining more accurate values than fast screening models based on the Debye model, while being significantly less expensive than methodologies that solve the Boltzmann transport equation.

Al5BO9: A Wide Band Gap, Damage‐Tolerant, and Thermal Insulating Lightweight Material for High‐Temperature Applications

The electronic structure, chemical bonding, mechanical, and thermal properties of a B‐containing mullite Al5BO9 were investigated using a combination of first‐principles calculations, Debye model, Clack model, and Slack's equation. The results show that Al5BO9 has a wide band gap of 6.4 eV. The bonding in BO3 triangle and AlO4 tetrahedron is covalent‐ionic, whereas that in AlO6 octahedra and AlO5 bipyramids is ionic‐covalent. The maximum (245 GPa) and minimum Young's modulus (161 GPa) are along [100] and [010] direction, respectively. Based on the low shear modulus (78 GPa) and low Pugh's ratio (G/B = 0.497), Al5BO9 is predicted as a damage‐tolerant ceramic. The most likely slip systems are [010](100) and [100](010). The high Debye temperature (956 K) indicates good high‐temperature stability. The minimum thermal conductivity is estimated to be 1.59 W·(m·K)−1, and the thermal conductivity decreases with temperature as 2071.9/T.