Self-consistent Solver Research Articles

BORIS – Micromagnetic, Spin Transport and Multiscale Atomistic Software for Modelling Magnetic Information Storage

A brief review of BORIS is given here, together with a review of recent works using this software, including applications to modelling magnetic hard-disk-drive read heads, ultrafast magnetization processes, computation of thermodynamic equilibrium states using Monte Carlo algorithms, and modelling skyrmions as information carriers. BORIS is a state-of-the-art multi-physics and multi-scale research software designed to solve three-dimensional magnetization dynamics problems, coupled with a self-consistent charge and spin transport solver, heat flow solver with temperature-dependent material parameters, and elastodynamics solver including thermoelastic and magnetoelastic/magnetostriction effects, in arbitrary multi-layered structures and shapes. Both micromagnetic and atomistic models are implemented, also allowing multi-scale modelling where computational spaces may be configured with multiple simultaneous micromagnetic and atomistic discretization regions. The software allows multi-GPU computations on any number of GPUs in parallel, in order to accelerate simulations and allow for larger problem sizes compared to single-GPU computations – this is the first magnetization dynamics software to allow multi-GPU computations, enabling large problems encompassing billions of cells to be simulated with unprecedented performance.

A Fully Coupled Scheme for a Self-Consistent Poisson-NEGF Solver

This work presents a fully coupled scheme for a self-consistent Poisson-nonequilibrium Green function (NEGF) solver. In this solver, responses of the Green functions to the potential perturbation are rigorously calculated. As a result, the detailed dependence of the electron density on the electrostatic potential can be accurately considered. When applied to the steady-state (DC) simulation, the convergence behavior is much improved compared with the conventional Gummel-type approach. As a numerical example, real-space NEGF results are shown for a highly scaled nanosheet MOSFET under the ballistic transport approximation.

Kinetic energy-free Hartree–Fock equations: an integral formulation

We have implemented a self-consistent field solver for Hartree–Fock calculations, by making use of Multiwavelets and Multiresolution Analysis. We show how such a solver is inherently a preconditioned steepest descent method and therefore a good starting point for rapid convergence. A distinctive feature of our implementation is the absence of any reference to the kinetic energy operator. This is desirable when Multiwavelets are employed, because differential operators such as the Laplacian in the kinetic energy are challenging to represent correctly. The theoretical framework is described in detail and the implemented algorithm is both presented in the paper and made available as a Python notebook. Two simple examples are presented, highlighting the main features of our implementation: arbitrary predefined precision, rapid and robust convergence, absence of the kinetic energy operator.

Modelling and simulation of carrier transport in quantum dot memory device for longer data retention and minimized power consumption

The performance of a group III–V material quantum dot (QD) nanostructure memory is investigated using a self-consistent Schrodinger solver, eight-band k·p model, and carrier dynamics modelling. This model is used to explore the information loss due to the carrier emission rate in the QDs as a function of temperature, size and confinement potential. The results reveal the dominant emission mechanisms that should occur at different operating temperatures. To minimize the loss and improve the performance at room temperature, our findings reveal an increase in the carrier storage time and a reduction in the power dissipation with increasing dot size. It is further illustrated that electrons are advantageous as information carriers over holes and that the inclusion of high-bandgap barrier layers favours longer-duration data retention. The model is extended to include trap states in realistic QDs, whose effect is found to become more prominent with performance optimization. The computed results are in close agreement with other experimental data for different QDs along with barrier layer. This validates the efficacy of the model, which can be utilized as a design tool for fabricating nanoscale memories with better data retention capability.

Spin-State Ordering in Metal-Based Compounds Using the Localized Active Space Self-Consistent Field Method.

Quantitatively accurate calculations for spin-state ordering in transition-metal complexes typically demand a robust multiconfigurational treatment. The poor scaling of such methods with increasing size makes them impractical for large, strongly correlated systems. Density matrix embedding theory (DMET) is a fragmentation approach that can be used to specifically address this challenge. The single-determinantal bath framework of DMET is applicable in many situations, but it has been shown to perform poorly for molecules characterized by strong correlation when a multiconfigurational self-consistent field solver is used. To ameliorate this problem, the localized active space self-consistent field (LASSCF) method was recently described. In this work, LASSCF is applied to predict spin-state energetics in mono- and di-iron systems, and we show that the model offers an accuracy equivalent to that of CASSCF but at a substantially lower computational cost. Performance as a function of basis set and active space is also examined.

Can Density Matrix Embedding Theory with the Complete Activate Space Self-Consistent Field Solver Describe Single and Double Bond Breaking in Molecular Systems?

Density matrix embedding theory (DMET) [ Phys. Rev. Lett. 2012, 109, 186404] has been demonstrated as an efficient wave-function-based embedding method to treat extended systems. Despite its success in many quantum lattice models, the extension of DMET to real chemical systems has been tested only on selected cases. Herein, we introduce the use of the complete active space self-consistent field (CASSCF) method as a correlated impurity solver for DMET, leading to a method called CAS-DMET. We test its performance in describing the dissociation of H-H single bonds in a H10 ring model system and an N═N double bond in azomethane (CH3-N═N-CH3) and pentyldiazene (CH3(CH2)4-N═NH). We find that the performance of CAS-DMET is comparable to CASSCF with different active space choices when single-embedding DMET corresponding to only one embedding problem for the system is used. When multiple embedding problems are used for the system, the CAS-DMET is in good agreement with CASSCF for the geometries around the equilibrium, but not in equal agreement at bond dissociation.

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.

Enhancement of graphene’s third-harmonic generation with localized surface plasmon resonance under optical/electro-optic Kerr effects

Although graphene’s particularly strong third-order susceptibility has drawn intensive attention in theoretical and experimental studies, its low bulk nonlinear response heavily emphasizes the nanostructure’s design for a sufficient magnitude of third-harmonic generation (THG). Meanwhile, currently few tools are available for accurate theoretical analyses of graphene’s nonlinear performance within a relatively complex structure, which renders the design of graphene-based nonlinear optoelectronic devices even more challenging. In this work, a high-accuracy self-consistent numerical solver based on the boundary-integral spectral element method is first proposed for the THG problem. Starting from the coupled vector wave equations, the proposed solver solves for the fundamental frequency field and third-harmonic field together iteratively, and it covers the optical/electro-optic Kerr effects ignored by most previous THG studies. After validating the proposed method with the comparison between numerical results and experimental data, we extend our study to the THG enhancement strategy with ultrastrong localized surface plasmon resonances (LSPRs) and Kerr effects. For both optical and electro-optic Kerr effects, the systematic simulation is performed for graphene’s THG within the incident spectra of 400–1000 nm. Compared with the THG of floating single-atom-layer graphene, numerical results show that under specific LSPR engineering, graphene’s THG backward emission is enhanced by 4.4×105 times. Simultaneously applying the electro-optic Kerr process can further boost the THG emission. However, its contribution is only secondary compared with LSPR. This study is also extended to bilayer and trilayer graphene models under strong LSPR.

Physical model for high indium content InGaN/GaN self-assembled quantum dot ridge-waveguide lasers emitting at red wavelengths (λ ~ 630 nm).

We present a physical model for recently demonstrated high indium content self-assembled In0.4Ga0.6N/GaN quantum dot (QD)-based ridge-waveguide lasers emitting at red wavelengths. The strain distribution in the QD is calculated using linear elastic theory with the application of shrink-fit boundary condition at the InGaN/GaN material interface, and the electronic states are evaluated using a single-band effective mass Hamiltonian. A Schrödinger-Poisson self-consistent solver is used to describe the effect of charge screening under current injection. Our theoretical result shows a good match to the measured Hakki-Paoli gain spectrum. Combining the calculated gain spectrum and cavity properties, we have developed a device-level simulation to successfully explain the electrical and optical characteristics of this specific laser. Possible solutions to improving the device performance have been explored.

(Invited) Mobility Enhancement of Uniaxially Strained Germanium Nanowire MOSFETs

Introduction High mobility channel materials are extensively explored to replace strained-Si in MOSFET devices to reduce power consumption by lowering supply voltages, V dd, without degrading circuit performance. Among the various high mobility channel materials, compressively strained-Ge[1-4] is recognized as the most promising option for p-channel FETs due to its significantly high hole-mobility and compatibility with Si-CMOS process. In this paper, Ge nanowire MOSFETs with a uniaxial compressive strain as high as 3.9% were demonstrated by 2-step Ge-condensation technique. Record-high hole mobility (μ eff = 1922 cm2/Vs) and record-low off-current (2.7x10-9A/μm at V d = -0.5V) were achieved among scaled (sub-100nm L g) Ge MOSFET for the device with the L gof 45nm. Device fabrication Uniaxially strained-Ge nanowire channel was formed by the two-step Ge condensation technique, which induced uniaxial stress along the channel direction[3]. As a gate insulator, 5 nm thick Al2O3 was grown by using ALD. Partly, in order to improve the interface characteristics and hole mobility, MMT-plasma oxidized GeOx / 3.2 nm-Al2O3 stacked gate insulator were adopted[4,5]. Ni(Si)Ge-metal S/D structures were formed by SALICIDE-like process. Detailed device fabrication process was shown in [4]. The Relationship between the wire width, W wire and strain applied to the channel was measured by an immersion high-NA Raman spectroscopy[6] and a NBD measurement. The dependence of strain along the channel direction, ε xx on wire width is shown in Fig.1. Extremely high (-3.8%) strain was evaluated for W wire= 20 nm device, which was almost identical with the value obtained from the NBD measurement. Moreover, extremely high (over -3.9%) strain, which is almost identical with the misfit strain between Ge and Si, was evaluated by strained-Ge nanowire channel formed by improved two-step Ge condensation technique[4]. Mobility enhancement by uniaxial compressive strain In order to estimate the mobility enhancement factors in uniaxially strained-Ge and Si1-x Ge x inversion channels, Poisson-Schrodinger self-consistent solver[7] was used to calculate the effective mass (m eff) values. Here, 6x6 k*p method was used. Figure 2 shows the calculated Ge concentration dependence of inversion layer effective hole mass along [110] channel direction. For Ge (x=1), m eff was calculated to be 0.055m 0 at fully strained condition, i.e., the lattice constant is identical with that of Si in [110] directions. Considering the ratio of m eff, 12 times higher mobility that for unstrained Si can be expected on uniaxially strained-Ge. At these highly strained condition, orientation dependence among (001), (112) and (110) is diminished due to the band deformation. Device characterization Hole mobility was extracted by a split CV method for long-channel and multi-wire devices. Figure 3(a) shows hole-mobility of the fabricated multi-wire MOSFETs. The W wire = 40nm device, which has 95% Ge channel and GeOx / Al2O3 stacked gate insulator, showed a peak hole mobility of 1922 cm2/Vs at N s = 1.7x1012cm-2, which exceeds that of -3.8% strained-Ge nanowire MOSFET with Al2O3 gate stack. Moreover, the obtained mobility at N s = 5x1012cm-2 is corresponding to 5.8 times of the counterpart of a state-of-the-art strained-Si pMOSFETs[1]. Figure 3(b) shows that mobility values for fixed N s increased with Ge concentration, x. The tendency is consistent with the m eff calculation in Fig.2. Figure 4 shows I d-V g characteristics of fabricated strained-Ge nanowire pMOSFET with GeOx / Al2O3 stacked gate insulator. Ion/Ioff ~ 105 has also been attained. And a minimum drain leakage current (I d,off), as low as current 2.7x10-9A/μm at V d = -0.5V, were achieved with L g = 45 nm. This I d,off value is the lowest minimum drain leakage among the previously reported sub-100nm L g strained-Ge devices even though that has a shortest L g. Conclusion s Ge nanowire MOSFETs with a uniaxial compressive strain as high as 3.9% were demonstrated by 2-step Ge-condensation technique. Record high hole mobility (μ eff = 1922 cm2/Vs) and record-low off-current (2.7x10-9A/μm at V d = -0.5V) were achieved among scaled (sub-100nm L g) Ge MOSFET for the device with the L gof 45nm. These results indicate that strained-Ge channels have a potential to serve as pFET channel of scaled future CMOS.

GaN-Based Dual-Color LEDs With $p$-Type Insertion Layer for Controlling the Ratio of Two-Color Intensities

In this paper, a novel GaN-based dual-color LED for phosphor-free white-light generation has been demonstrated. By inserting p-type layers with different p-type doping density and thickness into active regions of dual-color GaN LEDs, we can control the ratio of output light intensities from quantum-wells near n- and p-sides. With an optimum sheet charge density of such insertion layer, the intensities of these two colors can be balanced under a much lower driving-current density (45 versus 450 A/cm2) compared with that of reference device without such insertion layer. A 2-D finite-element Poisson and drift-diffusion self-consistent solver including the indium fluctuation is used to design and simulate the device performances. The experimental and simulation results match very well.

Intrinsic limits of subthreshold slope in biased bilayer graphene transistor

In this work, we investigate the intrinsic limits of subthreshold slope in a dual gated bilayer graphene transistor using a coupled self-consistent Poisson-bandstructure solver. We benchmark the solver by matching the bias dependent band gap results obtained from the solver against published experimental data. We show that the intrinsic bias dependence of the electronic structure and the self-consistent electrostatics limit the subthreshold slope obtained in such a transistor well above the Boltzmann limit of 60 mV/decade at room temperature, but much below the results experimentally shown till date, indicating room for technological improvement of bilayer graphene.

AQUASOL: An efficient solver for the dipolar Poisson-Boltzmann-Langevin equation.

The Poisson-Boltzmann (PB) formalism is among the most popular approaches to modeling the solvation of molecules. It assumes a continuum model for water, leading to a dielectric permittivity that only depends on position in space. In contrast, the dipolar Poisson-Boltzmann-Langevin (DPBL) formalism represents the solvent as a collection of orientable dipoles with nonuniform concentration; this leads to a nonlinear permittivity function that depends both on the position and on the local electric field at that position. The differences in the assumptions underlying these two models lead to significant differences in the equations they generate. The PB equation is a second order, elliptic, nonlinear partial differential equation (PDE). Its response coefficients correspond to the dielectric permittivity and are therefore constant within each subdomain of the system considered (i.e., inside and outside of the molecules considered). While the DPBL equation is also a second order, elliptic, nonlinear PDE, its response coefficients are nonlinear functions of the electrostatic potential. Many solvers have been developed for the PB equation; to our knowledge, none of these can be directly applied to the DPBL equation. The methods they use may adapt to the difference; their implementations however are PBE specific. We adapted the PBE solver originally developed by Holst and Saied [J. Comput. Chem. 16, 337 (1995)] to the problem of solving the DPBL equation. This solver uses a truncated Newton method with a multigrid preconditioner. Numerical evidences suggest that it converges for the DPBL equation and that the convergence is superlinear. It is found however to be slow and greedy in memory requirement for problems commonly encountered in computational biology and computational chemistry. To circumvent these problems, we propose two variants, a quasi-Newton solver based on a simplified, inexact Jacobian and an iterative self-consistent solver that is based directly on the PBE solver. While both methods are not guaranteed to converge, numerical evidences suggest that they do and that their convergence is also superlinear. Both variants are significantly faster than the solver based on the exact Jacobian, with a much smaller memory footprint. All three methods have been implemented in a new code named AQUASOL, which is freely available.

Simulation of Hole Mobility in DGSOI Transistors

In this paper we study carrier mobility in double gate SOI p-channel devices. A self-consistent solver for the six-band k•p and Poisson equations is employed to compute the wavefunctions and dispersion relations of hole subbands. Hole mobility is evaluated employing the Kubo-Greenwood formalism. Both phonon and surface roughness scattering mechanisms are taken into account. In the case of surface roughness scattering, an extension of a model previously developed for n-channel double gate devices has been developed. To do so, new assumptions were necessary to maintain reasonable simulation times as k•p approach is considered. The results show that volume inversion has a lower impact in DGSOI p-channel devices than the one shown for DGSOI n-channel devices.

Effects of Spatial and Spectral Characteristics of Electrons on Quantum Capacitance Oscillations in Silicon NanoWire-Based MOS Structure

Capacitance-gate voltage [C(V)] characteristics of inversion layer of the cylindrical silicon nanowire (SiNW) MOS are simulated using a two-dimensional Schrödinger-Poisson (SP) self-consistent solver with explicit cylindrical coordinates and quantum-mechanical treatment within the effective mass approximation. Our simulation results demonstrate strong effects of confinement of electron wavefunctions on spatial and spectral characteristics of excess (inversion) electrons in SiNW-based MOS device, which eventually yield in significant oscillations in C(V) dependencies, simulated at temperatures below 40 K and SiNW radii below 8 nm.

Simulation of hole mobility in two-dimensional systems

We develop a fully self-consistent solver for the six-band k ⋅ p Schrödinger and Poisson equations to compute the valence-band structure of Si and Ge devices with arbitrary substrate orientation and uniaxial or biaxial strain. This allows us to compute the potential, charge distribution and subband energy dispersion relation for hole inversion layers in different devices and, using a simplex Monte Carlo simulator, to evaluate the low-field mobility. New procedures have been developed to calculate the scattering rates. The results obtained in the case of a (0 0 1) Si MOSFET device are compared with experimental mobility curves and a very good agreement is found. Then, hole mobility curves for different structures and crystallographic orientations both with strained and unstrained materials are evaluated.

Self-consistent full band two-dimensional Monte Carlo two-dimensional Poisson device solver for modeling SiGe p-channel devices

Even now there are no device simulators that can model hole carrier transport in p-channel Si or SiGe metal-oxide-semiconductor field-effect transistors (MOSFETs), by properly addressing the issues of hole band-structure and quantum confinement effects. The purpose of this work is to remedy this situation by presenting an approach to modeling p-channel devices using a two-dimensional (2D) Monte Carlo transport kernel that is coupled self-consistently to a 2D Poisson equation solver and to a six band k∙p band-structure module. The need for full band solver for hole transport is especially true in the case of surface channel strained Si and buried channel strained SiGe p-MOSFETs investigated here. We have paid special attention on properly implementing and investigating the role of interface roughness on the operation of these device structures. We find that SiGe p-channel MOSFETs show performance improvement only for low gate and drain biases; i.e., when the device is operated in subthreshold and weak inversion regime and carrier spillover into the surface channel and surface roughness do not play significant role on the device operation.

Calculation of the direct tunneling current in a metal-oxide-semiconductor structure with one-side open boundary

The leakage current through the oxide of an n-channel metal-oxide-semiconductor (MOS) structure with an open boundary on one side is numerically computed by applying a one-dimensional Schrödinger-Poisson self-consistent solver. By embedding the n-channel MOS structure in a well, which prevents the penetration of particles into the metallic gate, the potential profile, the bounded energy levels, and the spatial distribution of electrons in the quantized levels are calculated in the inversion regime. Penetration of electrons into the metallic gate with an open boundary results in a broadening of the discrete bound states at the interface of the substrate with the oxide, transforming the bounded energy levels to the quasibound states. Starting from the continuity equation, a qualitative formula for the current in terms of the electron lifetime in the quasibound states is derived. Based on the determination of the energy level width corresponding to the wave functions, we suggest a method to compute the lifetime, and subsequently, the tunneling current across the potential barrier. The tunneling current is computed for a MOS structure with SiO2 and Si3N4 gate dielectrics. The computational results are compared with those obtained experimentally for similar structures, yielding an excellent agreement.

Variational density-functional perturbation theory for dielectrics and lattice dynamics

The application of variational density functional perturbation theory (DFPT) to lattice dynamics and dielectric properties is discussed within the plane-wave pseudopotential formalism. We derive a method to calculate the linear response of the exchange-correlation potential in the GGA at arbitrary wavevector. We introduce an efficient self-consistent solver based on all-bands conjugate-gradient minimization of the second order energy, and compare the performance of preconditioning schemes. Lattice-dynamical and electronic structure consequences of space-group symmetry are described, particularly their use in reducing the computational effort required. We discuss the implementation in the CASTEP DFT modeling code, and how DFPT calculations may be efficiently performed on parallel computers. We present results on the lattice dynamics and dielectric properties of $\ensuremath{\alpha}$-quartz, the hydrogen bonded crystal $\mathrm{Na}\mathrm{H}{\mathrm{F}}_{2}$ and the liquid-crystal-forming molecule 5CB. Excellent agreement is found between theory and experiment within the GGA.

Threshold current reduction due to piezoelectric effects in InGaAs/AlGaAs laser diodes

We show that despite expectations the piezoelectric field present, for instance, in strained InGaAs quantum wells (QWs) grown along the (111)B orientation, can lead to a significant threshold current reduction in laser diodes (LDs) using piezoelectric QWs as active medium. We have fabricated a series of LD structures in variable cavity lengths, with a single 10nm-thick InGaAs/AlGaAs active QW, grown either on (100) or (111)B GaAs substrates. We have observed for all temperatures in the range 20–300 K, and for all cavity lengths between 0.5–2 mm, systematically lower threshold currents in the (111)B LDs compared to their (100) counterparts. To interpret this unexpected result, we have employed modeling of the devices based on a self-consistent Schroedinger-Poisson solver, incorporating strain and screening effects, as well as the piezoelectric field dependence of gain and spontaneous emission rate.