Boltzmann Transport Equation For Electrons Research Articles

Discrete unified gas kinetic scheme for the solution of electron Boltzmann transport equationwith Callaway approximation.

Electrons are the carriers of heat and electricity in materials and exhibit abundant transport phenomena such as ballistic, diffusive, and hydrodynamic behaviors in systems with different sizes. The electron Boltzmann transport equation(eBTE) is a reliable model for describing electron transport, but it is a challenging problem to efficiently obtain the numerical solutions of the eBTE within one unified scheme involving ballistic, hydrodynamics, and/or diffusive regimes. In this work, a discrete unified gas kinetic scheme (DUGKS) in the finite-volume framework is developed based on the eBTE with the Callaway relaxation model for electron transport. By reconstructing the distribution function at the cell interface, the processes of electron drift and scattering are coupled together within a single time step. Numerical tests demonstrate that the DUGKS can be adaptively applied to multiscale electron transport, across different regimes.

A Model for Characteristic X-Ray Emission in Electron Probe Microanalysis Based on the (Filtered) Spherical Harmonic () Method for Electron Transport

Classical$k$-ratio models, for example, ZAF and$\phi ( \rho z)$, used in electron probe microanalysis (EPMA) assume a homogeneous or multilayered material structure, which essentially limits the spatial resolution of EPMA to the size of the interaction volume where characteristic X-rays are produced. We present a new model for characteristic X-ray emission that avoids assumptions on the material structure to not restrict the resolution of EPMAa priori. Our model bases on the spherical harmonic ($P_{\rm N}$) approximation of the Boltzmann equation for electron transport in continuous slowing down approximation.$P_{\rm N}$models have a simple structure, are hierarchical in accuracy and well-suited for efficient adjoint-based gradient computation, which makes our model a promising alternative to classical models in terms of improving the resolution of EPMA in the future. We present results of various test cases including a comparison of the$P_{\rm N}$model to a minimum entropy moment model as well as Monte-Carlo (MC) trajectory sampling, a comparison of$P_{\rm N}$-based$k$-ratios to$k$-ratios obtained with MC, a comparison with experimental data of electron backscattering yields as well as a comparison of$P_{\rm N}$and MC based on characteristic X-ray generation in a three-dimensional material probe with fine structures.

Nonnegative magnetoresistance in hydrodynamic regime of electron fluid transport in two-dimensional materials

Electron transport in ultraclean two-dimensional materials has received much attention. However, the sign of the magnetoresistance effect in various electron flow regimes remains controversial. In this work, the complete electron Boltzmann transport equation is numerically solved with the discrete ordinate method in the real space to clarify the condition of the negative magnetoresistance effect under a weak magnetic field. It turns out from the numerical results that this effect occurs only within the ballistic regime under a low electric field rather than the hydrodynamic regime. It is noteworthy that the existence of momentum-conserving scattering dramatically reverses the sign of magnetoresistance in the ballistic regime. When the electric field becomes strong enough compared to the magnetic field, its effect on the deflection of the electrons is not negligible and will lead to positive magnetoresistance in the whole parameter domain. The possible influence of boundary conditions and internal electric field models on the sign of magnetoresistance is also discussed. Our work provides insight into electron fluid transport under electromagnetic fields.

Coupling Nonequilibrium Electron Transport with Maxwell Equations for Modelling Nonlinear Optical Properties

Numerical approach to modelling nonlinear optical properties is presented. The model is based on a self-consistent solution of the Boltzmann transport equation for electrons and the Maxwell equations.

Tuning thermoelectric efficiency of monolayer indium nitride by mechanical strain

Tuning the thermoelectric efficiency of a material is a complicated task as it requires the control of interrelated parameters. In this respect, various methods have been suggested to enhance the figure of merit (ZT), including the utilization of low-dimensional systems. Motivated by the effect of strain on intrinsic properties of two-dimensional materials, we examine the thermoelectric response of monolayer indium nitride (h-InN) under low biaxial strain (±1%) by using ab initio methods together with solving Boltzmann transport equations for electrons and phonons. Our results indicate that among the critical parameters, while the Seebeck coefficient is not affected prominently, electrical conductivity can increase up to three times, and lattice thermal conductivity can decrease to half at −1% strain where valence band convergence is achieved. This results in significant enhancement of ZT, especially for p-type h-InN, and it reaches 0.50 with achievable carrier concentrations (∼1013 cm−2) at room temperature. Thermoelectric efficiency further increases with elevated temperatures and rises up to 1.32 at 700 K, where the system remains to be dynamically stable, suggesting h-InN as a promising material for high-temperature thermoelectric applications.

Strongly Out-of-Equilibrium Simulations for Electron Boltzmann Transport Equation Using Modal Discontinuous Galerkin Approach

In this study, we present simulations of strongly out-of-equilibrium conditions for the one-dimensional electron Boltzmann transport equation (BTE) in semiconductor devices. An explicit modal discontinuous Galerkin method is employed to solve the electron BTE along with the simplest collisional model–relaxation time approximation. The electron BTE system is discretized in momentum and time, and applied to the description of the dynamics of non-linear electron transport under a strong static electric field. A third-order explicit SSP-RK scheme is adopted for the temporal discretization of the resulting semi-discrete ODE. For the steady-state electron BTE, the analytical solutions are derived at low and high electric fields which are used to validate the numerical approach. The computed solution show a good agreement with derived analytic solution in wide range of tested parameters and regimes. An extensive range of numerical simulations has also been performed to investigate the impact of exploited flow parameters on the electron BTE solutions. Ultimately, these results predict that the modal DG approach is particularly effective in treating the strongly out-of-equilibrium regimes encountered like in ultrafast transport dynamics.

Combining theory and experiment to determine thermoelectrics of tellurium doped CoSb3

First, we solve a semi-classical electronic Boltzmann transport equation and acquire the thermoelectric coefficients for a pure unit cell using the rigid-band approximation. Then, combined with the existing experimental results, we not only reproduce the Seebeck coefficients and electrical conductivity but also yield the carrier concentrations and relaxation time, which cannot be obtained by either experiments or theories alone. Our results are in good agreement with the experiment for all doping samples. Furthermore, we show that Wiedemann-Franz law fails to connect the electronic thermal conductivity with the electrical conductivity. We also show that the Hall effect fails to measure carrier concentration in this study. We attribute these failures to the breakdown of the parabolic band. We think that our approach can be extended to other alloy or compound thermoelectric materials that utilize electronic transport property because they all have problems identifying relaxation time and carrier concentrations.

Tweaking the Physics of Interfaces between Monolayers of Buckled Cadmium Sulfide for a Superhigh Piezoelectricity, Excitonic Solar Cell Efficiency, and Thermoelectricity.

Interfaces of heterostructures are routinely studied for different applications. Interestingly, monolayers of the same material when interfaced in an unconventional manner can bring about novel properties. For instance, CdS monolayers, stacked in a particular order, are found to show unprecedented potential in the conversion of nanomechanical energy, solar energy, and waste heat into electricity, which has been systematically investigated in this work, using DFT-based approaches. Moreover, stable ultrathin structures showing strong capabilities for all kinds of energy conversion are scarce. The emergence of a very high out-of-plane piezoelectricity, |d33| ≈ 56 pm/V, induced by the inversion symmetry broken in the buckled structure helps to supersede the previously reported bulk wurzite GaN, AlN, and Janus multilayer structures of Mo- and W-based dichalcogenides. The piezoelectric coefficients have been found to be largely dependent on the relative stacking between the two layers. CdS bilayer is a direct band gap semiconductor, with its band edges straddling the water redox potential, thereby making it thermodynamically favorable for photocatalytic applications. Strain engineering facilitates its transition from type I to type II semiconductor in CdS bilayer stacked over monolayer boron phosphide, and the theoretically calculated power conversion efficiency (PCE) in the 2D excitonic solar cell exceeds 27% for a fill factor of 0.8, which is much higher than that in ZnO/CdS/CuInGaSe solar cell (20% efficiency). Thermoelectric properties have been investigated using semi classical Boltzmann transport equations for electrons and phonons within the constant relaxation time approximation coupled to deformation potential theory, which reveal ultralow thermal conductivity (κl ≈ 0.78 W m-1 K-1) at room temperature because of the presence of heavy element Cd, strong anharmonicity (high mode Gruneisen parameter at long wavelength, phonon lifetime <5 ps), low phonon group velocity (4 km/s), and low Debye temperature (260 K). Such a low thermal conductivity is lower than that of dumbbell silicene (2.86 W m-1 K-1), SnS2 (6.41 W m-1 K-1) and SnSe2 (3.82 W m-1 K-1), and SnP3 (4.97 W m-1 K-1). CdS bilayer shows a thermoelectric figure of merit (ZT) ≈ 0.8 for p-type and ∼0.7 for n-type doping at room temperature. Its ultrahigh carrier mobility (μe ≈ 2270 cm2 V-1 s-1) is higher than that of single-layer MoS2 and comparable to that in InSe. The versatile properties of CdS bilayer together with its all-round stability supported by ab initio molecular dynamics simulation, phonon dispersion, and satisfaction of Born-Huang stability criteria highlight its outstanding potential for applications in device fabrication and applications in next-generation nanoelectronics and energy harvesting.

Screening Promising Thermoelectric Materials in Binary Chalcogenides through High-Throughput Computations.

The high-throughput (HT) computational method is a useful tool to screen high-performance functional materials. In this work, using the deformation potential method under the single band model, we evaluate the carrier relaxation time and establish an electrical descriptor (χ) characterized by the carrier effective masses based on the simple rigid band approximation. The descriptor (χ) can be used to reasonably represent the maximum power factor without solving the electron Boltzmann transport equation. Additionally, the Grüneisen parameter (γ), a descriptor of the lattice anharmonicity and lattice thermal conductivity, is efficiently evaluated using the elastic properties, omitting the costly phonon calculations. Applying two descriptors (χ and γ) to binary chalcogenides, we HT compute 243 semiconductors and screen 50 promising thermoelectric materials. For these theoretically determined compounds, we successfully predict some previously experimentally and theoretically investigated promising thermoelectric materials. Additionally, 9 p-type and 14 n-type previously unreported binary chalcogenides are also predicted as promising thermoelectric materials. Our work provides not only new thermoelectric candidates with perfect crystalline structure for the future investigations but also reliable descriptors to HT screen high-performance thermoelectric materials.

Electron Monte Carlo simulations of nanoporous Si thin films—The influence of pore-edge charges

Electron transport within nanostructures can be important for various engineering applications, such as thermoelectrics and nanoelectronics. In theoretical studies, electron Monte Carlo simulations are widely used as an alternative approach to solving the electron Boltzmann transport equation, where the energy-dependent electron scattering, exact structure shape, and detailed electric field distribution can be fully incorporated. In this work, such electron Monte Carlo simulations are employed to predict the electrical conductivity of periodic nanoporous Si films that have been widely studied for thermoelectric applications. The focus is on the influence of pore-edge charges on the electron transport. The results are further compared to our previous analytical modeling [Hao et al., J. Appl. Phys. 121, 094308 (2017)], where the pore-edge electric field has its own scattering rate to be added to the scattering rates of other mechanisms.

Nonlocal transport hydrodynamic model for laser heated plasmas

The interaction of lasers with plasmas, whether pre-formed or due to ablation processes, very often takes place under nonlocal transport conditions. The nonlocality affects the transport of particles, mostly electrons, as much as it does radiation. In this study, the nonlocal transport is investigated for the plasma corona generated due to the deposition of laser energy. The nonlocal theory of the energy transport in radiative plasmas of the arbitrary ratio of the characteristic spatial scale length to the photon and electron mean free paths is applied to define closure relations of the hydrodynamic system. The corresponding transport phenomena cannot be described accurately with the usual fluid approach dealing only with local values and derivatives. Thus, the usual diffusive energy flux is instead calculated directly by solving a simplified transport equation allowing one to take into account the effect of long-range particle transport. The key feature of the proposed hydrodynamic closure is a direct solution of the simplified Bhatnagar-Gross-Krook form of the Boltzmann transport equation for electrons and the proper form of the radiation transport equation.

A deterministic model of electron transport for electron probe microanalysis

Within the last decades significant improvements in the spatial resolution of electron probe microanalysis (EPMA) were obtained by instrumental enhancements. In contrast, the quantification procedures essentially remained unchanged. As the classical procedures assume either homogeneity or a multi-layered structure of the material, they limit the spatial resolution of EPMA. The possibilities of improving the spatial resolution through more sophisticated quantification procedures are therefore almost untouched. We investigate a new analytical model (M1-model) for the quantification procedure based on fast and accurate modelling of electron-X-ray-matter interactions in complex materials using a deterministic approach to solve the electron transport equations. We outline the derivation of the model from the Boltzmann equation for electron transport using the method of moments with a minimum entropy closure and present first numerical results for three different test cases (homogeneous, thin film and interface). Taking Monte Carlo as a reference, the results for the three test cases show that the M1-model is able to reproduce the electron dynamics in EPMA applications very well. Compared to classical analytical models like XPP and PAP, the M1-model is more accurate and far more flexible, which indicates the potential of deterministic models of electron transport to further increase the spatial resolution of EPMA.

An Energy Transport Model Describing Electro-Thermal Transport in Silicon Carbide Semiconductors

ABSTRACTBy taking the moments of the coupled Boltzmann transport equations for electrons and phonons, an extended hydrodynamic model describing the electron transport in silicon carbide semiconductors has been obtained. This model is coupled with the heating of the crystal lattice. All the transport coefficients and the constitutive equations are determined by using the maximum entropy principle of extended thermodynamics. By using this model with a suitable limit, thermoelectric effects in silicon carbide devices are investigated.

Microscopic Hot-Carrier Degradation Modeling of SiGe HBTs Under Stress Conditions Close to the SOA Limit

We present and validate a physics-based model to describe the underlying mechanisms of hot-carrier degradation in bipolar transistors. Our analysis is based on a deterministic solution of the coupled system of Boltzmann transport equations for electrons and holes. The full-band transport model provides the energy distribution functions of the charge carriers interacting with the passivated Si–H bonds along the oxide interface. The simulation results assert the dominant role of hot holes along the emitter–base spacer oxide interface in the long-term degradation of an n-p-n SiGe heterojunction bipolar transistor under low and high-current conditions at the border of the safe-operating area. The interface trap density is calculated by incorporating an energy driven paradigm for the microscopicmechanisms of defect creation into a reaction-limitedmodelwith dispersive reaction rates. These interface traps increase the forward-mode base current via Shockley–Read–Hall recombination and degrade the overall device performance. The Gummel characteristics of a degraded device and time evolution of the excess base current for different stress conditions are verified versus the experimental data obtained for a state-of-the-art toward-terahertz SiGe HBT.

A Deterministic and Self-Consistent Solver for the Coupled Carrier-Phonon System in SiGe HBTs

A stationary deterministic solver based on a spherical harmonics expansion of the Boltzmann transport equations for electrons, holes, and phonons is presented to study self-heating in ultrascaled bipolar transistors. With the electrothermal device simulator, a state-of-the-art toward-terahertz SiGe heterojunction bipolar transistor is analyzed and the simulation results are verified against experimental data. To investigate nonequilibrium effects for the carrier-phonon system, the impact of hot longitudinal optical phonons on steady-state carrier transport is discussed. Furthermore, the self-consistent and deterministic solution of the coupled set of equations allows to extract the junction temperature by making use of a method based on the simulated DC characteristics. The resultant junction temperature is compared with the value obtained from the temperature profile within the nanoscale device. Good agreement is obtained for the average temperature in the base–emitter junction verifying the analytical approach used to extract the thermal resistance of the device by experiments.

Drag of electrons in graphene by substrate surface polar phonons

It is known that electron scattering by surface polar phonons (SPPs) of the substrate reduces their mobility in supported graphene. However, there is no experimental evidence for contribution of drag of electrons by SPP to thermoelectric phenomena in graphene: graphene thermopower exhibits good agreement with Mott's law, which means that the diffusion contribution to the thermopower is dominant in a wide range of carrier densities and temperatures. Here we develop a complete theory of drag of electrons in graphene by SPP. By solving Boltzmann transport equation for electrons scattered by SPPs we derive SPP drag contribution to the thermopower in graphene. Compared to diffusion thermopower, obtained values appear to be one order of magnitude lower for various substrates. This can be explained by low occupation number of the SPPs and short mean free path of such phonons stemming from their small group velocity. We conclude that experiments on thermopower in graphene can be treated within the framework of Mott's law.

Effect of the reflection coefficients on the conductivity of a thin metal layer in the case of an inhomogeneous time-periodic electric field

In the case of an inhomogeneous time-periodic electric field E, the electrical current distribution is evaluated for a thin metal layer with different reflection coefficients q1 and q2 of its surfaces. The dependence of the conductive function Σ on the frequency of the external inhomogeneous time-periodic electric field and on the coefficient of the inhomogeneous field κ is analyzed. The relaxation time approximation to the kinetic Boltzmann equation for electron transport is used.

Electro-thermal simulation based on coupled Boltzmann transport equations for electrons and phonons

To study the thermal effect in nano-transistors, a simulator solving self-consistently the Boltzmann transport equations for both electrons and phonons has been developed. It has been used to investigate the self-heating effects in a 20 nm-long double-gate MOSFET (Fig. 1). A Monte Carlo solver for electrons is coupled with a direct solver for the steady-state phonon transport. The latter is based on the relaxation time approximation. This method is particularly efficient to provide a deep insight of the out-of-equilibrium thermal dissipation occurring at the nanometer scale when the device length is smaller than the mean free path of both charge and thermal carriers. It allows us to evaluate accurately the phonon emission and absorption spectra in both real and energy spaces.

Group classification of an energy transport model for semiconductors with crystal heating

We consider an energy transport model for semiconductors describing the electro-thermal effects. The unipolar model involves four equations: the continuity and the energy balance equations for the electrons, the thermal diffusion equation for the lattice and the Poisson equation for the electric potential. The model can be derived by moment method from the Boltzmann transport equation for electrons in semiconductors. For this model, we perform a symmetry group classification by the infinitesimal Lie method and exact solutions are found.

Deterministic solvers for the Boltzmann transport equation of 3D and quasi-2D electron and hole systems in SiGe devices

We present a review of recent advances in deterministic solvers for the Boltzmann transport equation for electrons and holes in a 3D and quasi 2D k→-space and demonstrate the capabilities of deterministic solvers by two new examples: a THz SiGe HBT and a quantum well PMOSFET. Compared to the standard approach, the Monte Carlo method, these deterministic solvers have certain advantages. They yield exact stationary solutions and they allow small-signal and noise analysis directly in the frequency range from 0 to THz. Inclusion of magnetic fields, the Pauli principle or rare events causes no problems. Thus, it is now possible to calculate certain key figures of merit for devices based on the Boltzmann transport equation, which was previously very difficult or not possible at all. On the other hand, the deterministic solvers are more memory intensive and more difficult to code than the Monte Carlo method.