Full-band Monte Carlo Simulations Research Articles

Quantum mechanical model of crossing and anti-crossing points in 3D full-band Monte Carlo simulations

This work presents a 3D quantum mechanics based model to address the physics at band structure crossing/anti-crossing points in full band Monte Carlo (FBMC) simulations. The model solves the Krieger and Iafrate (KI) equations in real time using pre-computed coefficients at k-points spatially sampled within the first Brillouin zone. Solving the KI equations in real time makes this model applicable for all electric fields, which enables its use in FBMC device simulations. In this work, a two-level refinement scheme is used to aggressively sample regions in proximity to band crossings for accurate solutions to the KI equations and coarsely sample everywhere else to limit the number of k-points used. The presented sampling method is demonstrated on the band structure of silicon but is effective for the band structure of any semiconductor material. Next, the adaptation of the fully quantum KI model into the semi-classical FBMC method is discussed. Finally, FBMC simulations of hole transport in 4H silicon carbide with and without the KI model are performed. Results along different crystallographic directions for a wide range of electric fields are compared to previously published simulation and experimental values.

(Invited) A Closer Look at Impact Ionization Coefficients in Wide Band Gap Semiconductors: Theoretical Models and Measured Data

The direct measurement of ionization coefficients in wide band gap semiconductor materials is challenging due to the need to operate at high field strengths and the requirement of specific test structures that enable single carrier injection. More often than not, ionization coefficients are inferred from current multiplication data measured in p-n junctions or transistors structures. Unfortunately, sub-par material quality, processing issues and inappropriate measuring techniques have led to ionization coefficients values that are contradictory and with large variation among different datasets.As a result, theoretical models that attempt to predict the values of the carriers’ ionization coefficients are important since they provide an estimate of the expected values and a first order evaluation of the material performance. Developing theoretical models presents its own set of challenges. The established approach is based on a full electronics structure description of the semiconductor material and a suitable model to quantify the interaction of carriers with phonons, impurities and material imperfections. While this methodology has been successful with conventional elemental and compound semiconductors, and it has been applied to some wide band gap materials such as GaN, 4H-SiC [1,2] and diamond, a number of open issues still exists. The main difficulty is the determination of the carrier-phonon scattering strength at high electric field strengths that cannot be inferred from low-field mobility measurements.Among all wide band gap semiconductors, 4H-SiC [3] and GaN [4] are the material for which ionization coefficients have been measured by several groups. In the case of GaN, different measured ionization coefficients datasets are available and seems to provide a consistent indication of the expected values. Corresponding theoretical models based on full-band Monte Carlo simulations have been able to predict the correct trends, namely the fact that holes dominate the ionization process, but a quantitative agreement between predicted and measured values has only been achieved recently [5].This talk will provide an overview of the currently available experimentally measured ionization coefficients for wide band gap semiconductors and compare them to the ones obtained with theoretical models of different complexity. Specifically the development of the theoretical models for high-field transport for 4H-SiC, GaN will be discussed. Three different approaches, one based on empirical carrier-phonon scattering rates, one using the rigid pseudo ion model, and the direct evaluation of the interaction parameters based on a ab-initio DFT approach will be compared, and the outcome for each one of them benchmarked against the measured values. Additionally, the models will be applied to determine the ionization coefficients in AlGaN alloys, cubic BN and diamond.[1] E. Bellotti et al., J. Appl. Phys., 87, (8), p.3864, 15 April 2000.[2] F. Bertazzi et al., J. Appl. Phys., Vol.106, N6, p.063719, 15 September 2009.[3] A.O. Kostantinov, Appl. Phys. Lett. 71, 90, 1997, T. Hatakeyama, Appl. Phys. Lett. 85, 8, 2004[4] Cao et al., APL, N.112, 2018, Maeda et al., JAP, N.129, 2021, McClintock et al., APL, N.90, 2007, Ji et al., APL, N115, 2019[5] E. Bellotti and M. Matsubara, unpublished.

Near-junction phonon thermal spreading in GaN HEMTs: A comparative study of simulation techniques by full-band phonon Monte Carlo method

Accurate thermal simulation is essential for the near-junction thermal management and electro-thermal co-design of GaN HEMTs. While various methods have been employed to simulate phonon thermal transport in GaN, a comprehensive evaluation of their performance and reliability has yet to be conducted. In this work, first-principle-based steady-state full-band phonon tracing Monte Carlo (MC) simulations are conducted to study the thermal spreading resistance in GaN HEMTs. The results of full-band MC serve as a standard against which the applicability, accuracy, and computational efficiency of three widely-used approaches to simulate the near-junction phonon transport in GaN are thoroughly examined. The simulation techniques compared in this study include MC simulations with empirical isotropic phonon dispersion (isotropic MC), MC simulations with gray-medium approximation (gray MC), and finite-element methods (FEM) with effective thermal conductivities (FEM with keff). It is found that isotropic MC largely overestimates the thermal resistance due to the empirical model’s overestimation of phonon mean free path (MFP) distributions. By selecting an appropriate average MFP, gray MC can approximate the full-band results well, but due to its inability to reflect the contributions of different phonon modes, discrepancies are inevitable for some geometric parameters. For FEM-based analysis, although the diffusive nature of Fourier’s law precludes the reproduction of channel temperature distributions, the influence of phonon ballistic effects on the junction temperature can be accurately reflected in the well-chosen effective thermal conductivities. The comparison highlights the importance of directly incorporating first-principles-calculated phonon properties into device thermal simulations, and the paper can provide a clearer understanding of near-junction thermal transport in GaN and can be useful for thermal simulations of GaN-based devices.

Phonon transport mechanism of HfO2 ultrathin film with temperature-correction full-band Monte Carlo simulation

Hafnium dioxide (HfO2) has been widely used in microelectronics nowadays and commonly withstands extremely high temperatures, so the investigation of its thermodynamic properties is particularly essential. This paper develops a temperature-correction full-band Monte Carlo (TFMC) method to investigate the HfO2 ultrathin film. The phonon dynamics parameters of HfO2 are calculated based on the first-principles method. TFMC can better describe the phonon cumulative distribution function in different temperatures by modifying the phonon relaxation time and heat capacity. The thermal conductivity of HfO2 ultrathin film is calculated based on the above method and is in good agreement with the literature. It is observed that the optical phonons in HfO2 ultrathin film are prominent in the phonon heat transport, which is quite different from the mechanism in common semiconductor materials. Combined with the full-band diffuse mismatch model, the Si-based HfO2 ultrathin film is studied. It is found that the existence of the interface with substrates improves the thermodynamic properties of the ultrathin film, which provides a reference for the selection of substrate materials.

Full-band Monte Carlo simulation of two-dimensional electron gas in (AlxGa1−x)2O3/Ga2O3 heterostructures

β -Gallium oxide (Ga2O3) is an extensively investigated ultrawide-bandgap semiconductor for potential applications in power electronics and radio frequency switching. The room temperature bulk electron mobility (∼200cm2V−1s−1) is comparatively low and is limited by the 30 phonon modes originating from its 10-atom primitive cell. The theoretically calculated saturation velocity in bulk is 1–2×107cms−1 (comparable to GaN) and is limited by the low field mobility. This work explores the high field electron transport (and hence the velocity saturation) in the 2DEG based on the first principles calculated parameters. A self-consistent calculation on a given heterostructure design gives the confined eigenfunctions and eigenenergies. The intrasubband and the intersubband scattering rates are calculated based on the Fermi’s golden rule considering longitudinal optical (LO) phonon–plasmon screening. The high field characteristics are extracted from the full-band Monte Carlo simulation of heterostructures at 300 K. The overall system is divided into a 2D and a 3D region mimicking the electrons in the 2DEG and the bulk, respectively. The electron transport is treated through an integrated Monte Carlo program which outputs the steady state zone population, transient dynamics, and the velocity–field curves for a few heterostructure designs. The critical field for saturation does not change significantly from bulk values, however, an improved peak velocity is calculated at a higher 2DEG density. The velocity at low 2DEG densities is impacted by the antiscreening of LO phonons which plays an important role in shaping the zone population. A comparison with the experimental measurements is also carried out and possible origins of the discrepancies with experiments is discussed.

Analysis of uniaxial stress impact on drift velocity of 4H-SiC by full-band Monte Carlo simulation

SiC is expected to be the next-generation semiconductor material especially for power devices, and some have been put into practical use. However, its stress response has not been completely elucidated. In this paper, we analyzed the stress response of the drift velocity of 4H-SiC with a combination of first principles calculation and full-band Monte Carlo simulation. The response decreases with an increase in the electric field except for a hump around 1 MV/cm. The decreasing trend is explained by increasing scattering rates which diminish the effect of the change of the effective mass due to stress. The hump comes from the transition of electrons to band minima with a lighter effective mass.

Accurate Nonlocal Impact Ionization Models for Conventional and Staircase Avalanche Photodiodes Derived by Full Band Monte Carlo Transport Simulations

We present a procedure to extract the nonlocal impact ionization coefficients in Avalanche Photodiodes (APDs) operating in the linear regime from Full Band Monte Carlo simulations. The Monte Carlo calculations have been calibrated on existing experimental data for GaAs p-i-n APDs with different thickness of the intrinsic region. Inspection of impact ionization generation rate in p-i-n and staircase GaAs APDs led us to identify the limitations of existing nonlocal-history dependent impact ionization models. The introduction of an energy dependent relaxation length for the computation of the effective fields significantly improves the model accuracy in predicting the gain and noise associated to conduction and valence band steps in staircase APDs without additional computational burden. This improved nonlocal-history dependent model is thus a powerful tool to design and optimize APDs with different architectures.

Modeling Approaches for Gain, Noise and Time Response of Avalanche Photodiodes for X-Rays Detection

We report on a suite of modeling approaches for the optimization of Avalanche Photodiodes for X-rays detection. Gain and excess noise are computed efficiently using a non-local/history dependent model that has been validated against full-band Monte Carlo simulations. The (stochastic) response of the detector to photon pulses is computed using an improved Random-Path-Length algorithm. As case studies, we consider diodes consisting of AlGaAs/GaAs multi-layers with separated absorption and multiplication regions. A superlattice creating a staircase conduction band structure is employed in the multiplication region to keep the multiplication noise low. Gain and excess noise have been measured in devices fabricated with such structure and successfully compared with the developed models.

Massively parallel FDTD full-band Monte Carlo simulations of electromagnetic THz pulses in p-doped silicon at cryogenic temperatures

In semiconductor samples with a low density of free charge carriers the current response to an intense electromagnetic pulse is weak and can be treated as a perturbation in the finite-difference-time-domain calculation of the electromagnetic field. Up to first order the electric field that drives the particles does not depend on their density and the particles can be simulated independently. This opens the door for massively parallel Monte Carlo simulations that can be processed efficiently on a computer cluster and a large number of particles can be simulated resulting in a very low noise level. By this full-band Monte Carlo finite-difference-time-domain approach the generation of higher harmonics in p-doped silicon at cryogenic temperatures is investigated. It is found that the higher harmonics are mostly due to the warped valence bands, which are discretized on adaptive unstructured tetrahedral grids in the k-space to avoid numerical artifacts.

Study of phonon transport across Si/Ge interfaces using Full-Band phonon Monte Carlo simulation

A Full Band Monte Carlo simulator has been developed to consider phonon transmission across interfaces disposed perpendicularly to the heat flux. This solver of the Boltzmann transport equation does not require any assumption on the shape the phonon distribution and can naturally consider all phonon transport regimes from the diffusive to the fully ballistic regime. This simulator is used to study single and double Si/Ge heterostructures from the micrometer scale down to the nanometer scale, i.e. in all phonon transport regime from fully diffusive to ballistic. A methodology to determine the thermal conductivity at thermal interfaces is presented. It is also shown that the different transport regimes are correlated to different spectral contributions of the phonon modes to the heat flux along the devices. This local indicator of the transport regime gives new insights into the out-of-equilibrium phonon transport near the interfaces.

Quantitatively predicting modal thermal conductivity of nanocrystalline Si by full-band Monte Carlo simulations

Thermal transport of nanocrystalline Si is of great importance for the application of thermoelectrics. A better understanding of the modal thermal conductivity of nanocrystalline Si will be expected to benefit the efficiency of thermoelectrics. In this work, the variance reduced Monte Carlo simulation with full band of phonon dispersion is applied to study the modal thermal conductivity of nanocrystalline Si. Importantly, the phonon modal transmissions across the grain boundaries which are modeled by the amorphous Si interface are calculated by the mode-resolved atomistic Greens function method. The predicted ratios of thermal conductivity of nanocrystalline Si to that of bulk Si agree well with that of the experimental measurements in a wide range of grain size. The thermal conductivity of nanocrystalline Si is decreased from 54 percent to 3 percent and the contribution of phonons with mean free path larger than the grain size increases from 30 percent to 96 percnet as the grain size decreases from 550 nm to 10 nm. This work demonstrates that the full band Monte Carlo simulation using phonon modal transmission by the mode-resolved atomistic Greens function method can capture the phonon transport picture in complex nanostructures, and therefore can provide guidance for designing high performance Si based thermoelectrics.

3-D Full-Band Monte Carlo Simulation of Hot-Electron Energy Distributions in Gate-All-Around Si Nanowire MOSFETs

The energy distributions of electrons in gate-all-around (GAA) Si MOSFETs are analyzed using full-band 3-D Monte Carlo (MC) simulations. Excellent agreement is obtained with experimental current-voltage characteristics. For these 24-nm gate length devices, the electron distribution features a smeared energy peak with an extended tail. This extension of the tail results primarily from the Coulomb scattering within the channel. A fraction of electrons that enter the drain retains their energy, resulting in an out-of-equilibrium distribution in the drain region. The simulated density and average energy of the hot electrons correlate well with experimentally observed device degradation. We propose that the interaction of high-energy electrons with hydrogen-passivated phosphorus dopant complexes within the drain may provide an additional pathway for interface-trap formation in these devices.

Numerical Study of Thermal Dissipation Processes in Silicon

Abstract Heat dissipation in nanoelectronics has become a major bottleneck to further scaling in next-generation integrated circuits. In order to address this problem and develop more energy-efficient nanoelectronic transistor, sensor, and storage devices, we must understand thermal processes at the atomic scale, which requires numerical simulation of the interaction between electrons and heat, carried by quantized lattice vibrations called phonons. Here we examine in detail the phonon emission and absorption spectra in silicon at several elevated values for the electron temperature. The effect of electric field on the electron distribution and equivalent electron temperature is obtained from full-band Monte Carlo simulation for bulk silicon. The electron distributions are used to numerically compute the phonon emission and absorption spectra and discover trends in their behavior at high electron temperatures. The concept of electron temperature is used to understand the relationship between field and heat emission, and it is found that longitudinal acoustic (LA) phonon emission increases at high electron temperatures. It is also found that emission of slower zone-edge phonons increases for all phonon branches at high electron temperatures. These conclusions at high electric fields can be used to enable heat-conscious design of future silicon devices.

Electron transport analysis of 4H-SiC with full-band Monte Carlo simulation including real-space Coulomb interactions

4H-SiC has been widely used in many applications. All of these benefit from its extremely high critical electric field with good electron mobility. 4H-SiC possesses a critical field ten times higher than that of Si, which allows high-voltage blocking layers composed of 4H-SiC to be approximately a tenth the thickness of a comparable Si device, thus reducing the device on-resistance and power losses, while maintaining the same high blocking capability. Unfortunately, commercial Technology for Computer-Aided Design tools like Sentaurus and Silvaco Atlas are based on effective mass approximation, while most 4H-SiC devices are not operated under a low electric field so the parabolic-like band approximation does not hold anymore. Hence, to get more accurate and reliable simulation results, full-band analysis is needed. The first step in the development of a full-band device simulator is the calculation of the band structure. In this work, the empirical pseudopotential method is adopted. The next task in the sequence is the calculation of the scattering rates. Acoustic, non-polar optical phonon, polar optical phonon, and Coulomb scattering are considered. Coulomb scattering is treated in real space using the particle–particle–particle–mesh approach. The third task is coupling the bulk full-band solver with a 3D Poisson equation solver to generate a full-band device simulator. For proof-of-concept of the methodology adopted here and for simplicity, a 3D resistor is simulated. From the resistor simulations, the low-field electron mobility dependence upon Coulomb scattering in 4H-SiC devices is extracted. The simulated low-field mobility results are in excellent agreement with available experimental data.

Analysis of saturation velocity and energy relaxation time of electrons in Si using full‐band Monte Carlo simulation

AbstractIn this paper, we have investigated the influence of optical phonon energy on electron saturation velocity and energy relaxation time by using a full band Monte Carlo simulator. The energy band structure is obtained using the first principle calculation. The scattering probability is calculated so as to conserve the energy and momentum in the full band structure. It is found that the range of optical phonon energy from 56.2 to 63.9 meV allows experimental saturation velocity and energy relaxation time. Electron saturation velocity is more sensitive than relaxation time to the optical phonon energy.

Full band Monte Carlo simulation of AlInAsSb digital alloys

AbstractAvalanche photodiodes fabricated from AlInAsSb grown as a digital alloy exhibit low excess noise. In this article, we investigate the band structure‐related mechanisms that influence impact ionization. Band‐structures calculated using an empirical tight‐binding method and Monte Carlo simulations reveal that the mini‐gaps in the conduction band do not inhibit electron impact ionization. Good agreement between the full band Monte Carlo simulations and measured noise characteristics is demonstrated.image

Analysis of Saturation Velocity and Energy Relaxation Time of Electrons in Si Using Full-band Monte Carlo Simulation

Analysis of Saturation Velocity and Energy Relaxation Time of Electrons in Si Using Full-band Monte Carlo Simulation

“Hot electrons in Si lose energy mostly to optical phonons”: Truth or myth?

Theoretical studies of heat generation and diffusion in Si devices generally assume that hot electrons in Si lose their energy mainly to optical phonons. Here, we briefly review the history of this assumption, and using full-band Monte Carlo simulations—with electron-phonon scattering rates calculated using the rigid-ion approximation and both empirical pseudopotentials and Harris potentials—we show that, instead, electrons lose as much as 2/3 of their energy to acoustic phonons. The scattering rates that we have calculated have been used to study hot-electron effects, such as impact ionization and injection into SiO2, and are in rough agreement with those obtained using density functional theory. Moreover, direct subpicosecond pump-probe experimental results, some of them dating back to 1994, are consistent with the predictions of our model. We conclude that the study of heat generation and dissipation in nanometer-scale Si devices may require a substantial revision of the assumptions that have been considered “common wisdom” so far.

Computational Study of Geometrical Designs for Source/Drain Contacts to Reduce Parasitic Resistance in Extremely Scaled MOSFETs

Comprehensive simulation results are reported for the source/drain (S/D) geometry effects on the drive current of various n-type (Si, In0.53Ga0.47As, Ge) and p-type (Si, Ge) metal–oxide–semiconductor field-effect transistors (MOSFETs). Full-band Monte Carlo (MC) simulations are run for different S/D structures (“raised” versus “lateral”) at a relevant International Technology Roadmap on Semiconductors node (gate length 14 nm). The “lateral” S/D gives higher drive currents (lower parasitic resistance) than the “raised” S/D because carriers do not need to make large momentum changes through scattering to get injected into the channel or to get collected by the drain, unlike in the “raised” S/D where 90°-turns are required. The S/D geometry effect is significant for novel materials with light effective mass and low scattering rate (e.g., ~20% on-current change in In0.53Ga0.47As) while it is still not negligible (up to ~5%) for conventional Si. The effect is more pronounced with small contact resistivity and low-S/D doping density. The classical model (drift–diffusion) captures only a part of the S/D geometry effect. In MOSFET benchmarking for future technology nodes, it should be critical to consider the S/D structure effect using rigorous models (such as MC) to correctly project the drive current performance, especially for novel materials with high mobility.

Understanding the Average Electron–Hole Pair-Creation Energy in Silicon and Germanium Based on Full-Band Monte Carlo Simulations

The thermalization process of sub-10-eV charge carriers is examined with treating carrier transport with full-band Monte Carlo simulations. The average energy loss (3.69 eV in Si and 2.62 eV in Ge) required to create a thermalized electron–hole pair, obtained from the simulations, is very close to the experimentally measured radiation–ionization energies of Si and Ge irradiated with high-energy particles. These results suggest that only interactions that occur after the radiation-generated charge carriers decay to energies of ~10 eV or less determine the fundamental property of the radiation–ionization energies. In addition to an energy loss equal to the band gap energy via impact ionization, acoustic-phonon emission, which has been omitted in prior work, contributes 30% of the remaining carrier energy loss, while optical-phonon emission contributes the other 70%.