Research

An Analytical Model for Square Gate-All-Around (GAA) MOSFETs Including Quantum Effects

The main contribution of this work is the development of an analytical model for square gate-all-around (GAA) MOSFETs, which includes quantum effects. This model allows for the precise description of the inversion charge distribution in these devices, which is crucial for compact modeling. Additionally, the inversion charge distribution functions are used to calculate important parameters such as the centroid of the inversion charge and the gate-to-channel capacitance, making this model fundamental for optimizing these devices in future sub-22 nm integrated circuit technologies.

The following equation is presented:

\[ \rho(y, z) = q N_{inv} \left| A_0 \sin\left(\frac{\pi y}{t_{Si}}\right)^{\frac{1}{2}} \sin\left(\frac{\pi z}{t_{Si}}\right)^{\frac{1}{2}} \left(e^{-\frac{b(t_{Si} - y)}{t_{Si}}} + e^{-\frac{by}{t_{Si}}}\right) \left(e^{-\frac{b(t_{Si} - z)}{t_{Si}}} + e^{-\frac{bz}{t_{Si}}}\right) \right|^2 \]

This equation is the result of a creative process in which an analytical representation of the charge density \(\rho(y, z)\) is constructed based on the gate voltage and the geometry of the device. This equation describes how the charge carriers (electrons or holes) are distributed within the structure of the multi-gate MOSFET device, depending on the geometric and electrical parameters. The way the carriers are distributed is deeply influenced by the electric potential generated, which is a solution to Poisson's equation, governing the relationship between the charge density and the electrostatic potential. At nanometric scales, the situation is complicated by quantum effects. Here, carriers cannot occupy any position within the device, but must obey the restrictions imposed by the Schrödinger equation, which governs the probability of finding a carrier at a particular location within the confined geometry of the device.

This process leads to the quantization of the available states for the carriers, meaning that carriers can only occupy certain energy levels and specific positions within the device. The presented equation reflects this quantization, as it describes how carriers are distributed based on probabilities that depend on both the silicon geometry (\(t_{Si}\)) and the applied potential, through the sinusoidal and exponential functions in \(y\) and \(z\).

The main contribution of this equation is its ability to accurately integrate quantum effects into the modeling of nanoscale devices, representing how charge carriers respond to the interaction between the classical Poisson solution and the quantum dictates of Schrödinger. It is a significant advance in understanding multi-gate MOSFET devices in sub-nanometric technologies.

Click here to download the paper.

An Analytical Model for Square Gate-All-Around (GAA) MOSFETs Solving the 2D Poisson Equation

This work is, to date, the best contribution I have made in the field of research. This is mainly because we managed to solve an estimated non-linear partial differential equation (PDE), which represents a significant advance in modeling Gate-All-Around (GAA) MOSFET devices.

The key differential equation we solved, the 2D Poisson equation, includes the inversion charge density, which allowed us to accurately model the electric potential inside these advanced devices. Below is the main equation presented in the article:

\[ \frac{\partial^2 \psi(x, y)}{\partial x^2} + \frac{\partial^2 \psi(x, y)}{\partial y^2} = \frac{q}{\epsilon_{Si}} n_i e^{\frac{q \psi(x, y)}{kT}} \]

Where \( q \) is the electron charge, \( \epsilon_{Si} \) is the permittivity of silicon, \( k \) is the Boltzmann constant, \( T \) the temperature, and \( n_i \) the intrinsic electron density. This equation is fundamental for calculating the electric potential and inversion charge density in square geometry GAA MOSFETs, which had not been modeled with such precision before.

This advance allowed for the development of a compact model that facilitates the design and optimization of MOSFET devices at nanometric scales, particularly in low-power and high-frequency applications. The integration of quantum effects and the precise solution of this equation are a significant contribution to the scientific community, especially in the modeling of sub-22 nm devices.

Click here to download the paper.

Difficulties in Solving Non-linear Partial Differential Equations

Non-linear partial differential equations (PDEs) are considerably more difficult to solve than linear equations for several fundamental reasons:

• Superposition not applicable: In linear PDEs, the principle of superposition can be applied, allowing complex solutions to be constructed by adding simpler solutions. This is not possible with non-linear PDEs, where non-linear terms introduce complex interactions between variables.
• Unpredictable behavior: Non-linear equations often exhibit chaotic behavior or are highly sensitive to initial and boundary conditions. Small differences in input data can generate completely different results, making it very difficult to find exact or approximate solutions.
• No general methods: While many linear PDEs have well-established analytical methods (such as separation of variables, Fourier or Laplace transforms), there is no general method for solving non-linear PDEs. Each type of equation may require specific techniques that are usually more complex, and in many cases, only approximate or numerical solutions are possible.
• Complex phenomena: Non-linear PDEs often represent complex phenomena in nature, such as turbulence in fluids, the propagation of non-linear waves, or the dynamics of biological systems. These phenomena do not have simple or intuitive solutions, as interactions between variables tend to be non-trivial.

The Work of Mathematician I. K. Sabitov

I. K. Sabitov, a prominent Russian mathematician, is known for his contributions to the solution of non-linear partial differential equations, as demonstrated in his influential work "Solutions of \[\Delta u = f(x, y)e^{cu}\] in some special cases" "https://istina.msu.ru/publications/article/6764725/". Sabitov explored advanced methods for solving complex partial differential equations, applying techniques that have significantly advanced the understanding of non-linear systems.

His work on these equations, in particular, has provided the scientific community with valuable tools to address non-trivial problems in various fields. Sabitov's ability to apply special and holomorphic functions in the complex plane was essential for better understanding how to solve non-linear partial differential equations under specific conditions.

Thanks to I. K. Sabitov's work on holomorphic functions in the complex plane, it was possible to apply his methods to my research, which allowed me to successfully solve the non-linear partial differential equation presented in this work.

On the other hand, the work of the Russian mathematician S. Yu. Savitóv should not be confused with I. K. Sabitov.

S. Yu. Savitóv was a Russian mathematician and physicist known for his contributions to the theory of non-linear waves and solitons, phenomena that are strongly associated with non-linear partial differential equations.

One of the most important problems he addressed was the study of solitary waves, which are stable solutions of certain non-linear PDEs, such as the Korteweg-de Vries (KdV) equation. These waves travel without dispersing, despite the presence of non-linearities, and have found applications in fields such as optics, fluid theory, and plasma physics.

Savitóv's work particularly focused on the development of mathematical methods to analyze the stability and formation of solitons in non-linear systems. Among Savitóv's key advances is his use of analytical and numerical methods to demonstrate how soliton-like solutions emerge in systems where non-linear equations describe wave interaction. His research not only provided a better understanding of the properties of these systems but was also crucial for applying solitons in modern technology, such as data transmission over fiber optics.

Time-domain Numerical Modeling of THz Photoconductive Antennas

This article presents an innovative computational procedure to simulate the time-domain behavior of photoconductive antennas (PCAs) made of semiconductor and metallic materials. The study addresses one of the key challenges in THz technology: accurately modeling the interaction between charge carriers and electromagnetic fields in the terahertz (THz) regime. The importance of this model lies in its ability to precisely represent the electromagnetic radiation from these devices, a crucial aspect for various applications, such as spectroscopy in the terahertz range.

One of the most notable contributions is the development of a detailed set of explicit numerical equations, derived using finite-difference time-domain (FDTD) techniques, which couple Poisson's and Maxwell's equations with the charge carrier drift-diffusion model. Through this approach, the distribution of carriers in both transient and steady-state conditions is modeled, enabling the evaluation of the electromagnetic fields generated by the acceleration of these carriers within the PCAs. This procedure shows excellent correlation with previously reported experimental data, underscoring the model's accuracy.

Main Contributions

• Development of a robust numerical simulator that couples Maxwell's equations and the drift-diffusion model, allowing for the simulation of both generation and radiation processes in THz photoconductive antennas. This approach is particularly useful for time-domain spectroscopy applications in the terahertz range.
• Introduction of a detailed procedure to resolve the steady-state condition of charge carriers in PCAs, providing a precise foundation for calculating radiated fields. This steady-state condition is crucial as it directly affects radiation efficiency and antenna behavior.
• Validation of the numerical model through comparison with experimental measurements, demonstrating the simulator's ability to accurately predict the real behavior of PCAs under various operating conditions.
• The study also delves into the importance of considering non-homogeneous, field-dependent mobilities in semiconductors, which significantly improves the accuracy of the models compared to more simplified approaches that do not include these spatial variations.

Key Insights

• The model describes how charge carriers, induced by a laser pulse, interact with the electromagnetic field generated in the photoconductive antenna, allowing the analysis of PCA behavior in the THz regime. The generation and recombination of electron-hole pairs play a crucial role in the antenna dynamics and the emission of electromagnetic waves.
• The study highlights how the choice of different geometric configurations of the electrodes (such as face-to-face dipoles and transmission line dipoles) affects PCA performance, providing valuable insights for improving antenna design.
• The article also introduces a novel methodology for analyzing the transient and steady-state phases of electric and magnetic fields, using a double-update procedure for carrier distributions. This approach ensures numerical stability and model accuracy throughout the time-domain simulations.
• One of the most interesting features of the model is its ability to reproduce the hysteresis effect due to preconditioning in the semiconductor, which allows for more accurate simulation of transient and steady-state phenomena in PCAs. This consideration had not been previously included in other models.

Click here to download the full paper.

On the Numerical Modeling of Terahertz Photoconductive Antennas

This paper explores the significant impact of mobility models on the description of carrier dynamics for the analysis of radiative semiconductor photoconductive devices in the terahertz (THz) regime. The authors developed a simulator that self-consistently solves both the semiconductor device physics and Maxwell's equations to study the radiated electromagnetic fields. A key focus of this work is on the importance of accurately modeling the steady-state regime of the semiconductor device, which is crucial for the precise calculation of radiated fields, particularly in the broadside direction.

One of the primary contributions of this paper is the demonstration of how an accurate steady-state description of the electric potential, field distributions, and local mobility is essential for achieving realistic results in terahertz photoconductive antenna (PCA) simulations. The study shows that previous models, which did not consider detailed steady-state regimes, failed to capture the full complexity of the carrier interactions and their effects on the radiated electromagnetic fields.

Main Contributions

• The paper introduces a comprehensive numerical model that integrates both Poisson's and Maxwell's equations with a drift-diffusion model, providing a highly accurate description of semiconductor carrier dynamics in the THz regime.
• It emphasizes the importance of accurate mobility models in steady-state conditions, showing that even slight inaccuracies in the steady-state regime can lead to significant errors in the predicted radiated fields.
• The study demonstrates, through numerical simulations, the need for a detailed consideration of steady-state mobility distributions, electric fields, and the influence of doping concentrations on device behavior.
• It is the first time that a detailed steady-state consideration has been shown to significantly influence the outcomes of PCA simulations, marking a shift in how such devices are modeled in the terahertz range.

Key Insights

• The simulation results are validated against experimental data, confirming the superior accuracy of the numerical models presented in the study compared to simpler, previous models.
• The authors highlight the importance of accurately modeling the carrier mobility in the semiconductor. In particular, the field-dependent mobility model used in this work allows for a much more precise description of the device behavior under different operational conditions.
• This paper also presents a novel approach to considering the preconditions (steady-state regime) before the laser pulse excitation, introducing the concept of hysteresis in THz devices for the first time.
• This hysteresis effect is shown to play a critical role in the transient behavior of the devices, influencing the final radiated electromagnetic fields and leading to a better understanding of the device performance.

Click here to download the full paper.

Performance Comparison of THz Photoconductive Antennas

This document is one of the most important in my thesis, as it provides, from a relatively simple model, a description of the non-linear behaviors necessary to adequately model photoconductive antennas in the terahertz (THz) regime. At the time, methods such as Monte Carlo or finite elements were suggested as essential for this type of modeling due to the intrinsic complexity of the phenomena involved. However, this study demonstrates that using a simplification based on the dependence of the carrier mobility on the electric field can yield precise and highly efficient experimental results.

The article explores the influence of bias electrode geometry on the performance of photoconductive antennas. It presents a methodology to numerically calculate the operating bandwidth and radiation efficiency of PCAs (photoconductive antennas). The numerical results are validated through comparisons with experimental measurements, lending credibility to the simulations presented.

Main Contributions

• A numerical model based on the finite-difference time-domain (FDTD) method is introduced, which self-consistently solves the drift-diffusion and Maxwell equations.
• The study highlights how the choice of electrode geometry can double the efficiency of the antennas, with a bandwidth penalty generally less than 10%.
• The influence of factors such as bias voltage, semiconductor doping, and incident optical power on the overall performance of the antennas is presented.
• For the first time, it is demonstrated how the use of optimal electrode configurations can significantly increase radiation efficiency in PCAs.

Key Insights

• Numerical results are validated through comparisons with experimental measurements, confirming the accuracy of the simulations and their ability to predict the real performance of PCAs under different operating conditions.
• The study also shows how increasing optical power can cause a saturation effect in the antennas due to the accumulation of photogenerated carriers near the electrodes.
• Additionally, it analyzes how the mobility of the carriers is affected by the applied electric field, which directly impacts radiation efficiency.

Click here to download the full paper.

Time-domain Numerical Modeling of Terahertz Receivers Based on Photoconductive Antennas

This paper marked a turning point in my research, published after my PhD defense, and played a significant role in advancing the understanding of photoconductive antennas (PCAs) in the terahertz (THz) domain. It laid the foundation for the process of both emission and reception in near-field PCAs, which would later be presented at a prominent conference in Tucson, Arizona https://doi.org/10.1109/IRMMW-THz.2014.6956333.

The paper focuses on developing a simulator that couples semiconductor charge transport equations with Maxwell's equations to study the performance of terahertz receivers based on PCAs. This model allowed for an accurate characterization of PCAs, confirming experimental results through simulations. The key breakthrough was the detailed analysis of how a photoconductive receiver antenna detects THz radiation by convolving the photoconductivity of the receiver with the electric field generated by an emitter PCA. This simulation tool was critical for understanding and optimizing THz time-domain spectroscopy (THz-TDS) systems.

Main Contributions

• The development of a robust simulator to model photoconductive antennas, essential for THz-TDS applications.
• The introduction of a simple yet highly accurate numerical approach to model the reception processes in PCAs, balancing the trade-off between simplicity and accuracy.
• Numerical results were validated against experimental data, showcasing the simulator's precision in reproducing real-world measurements.
• The simulation tool allowed for the study of key factors such as carrier lifetime, mobility, and noise levels, which are crucial for the signal-to-noise ratio (SNR) in THz detectors.

Key Insights

• The model accurately simulated the current generated in a PCA when exposed to an incident electric field, highlighting how factors such as photoconductivity and incident field strength influence the detected signal.
• The study demonstrated how omitting certain components, like the PCA lens and semi-insulating substrate, could still yield reliable results in specific scenarios, reducing computational complexity.
• This paper established the drift-diffusion model as a viable approach for simulating PCA behavior, offering faster computation times compared to Monte Carlo methods while maintaining accuracy.
• The paper also introduced a new method to model the PCA reception process coherently, providing both amplitude and phase information without requiring additional complex techniques like Kramers-Kronig.

Click here to download the full paper.

A precise analytical model for drift velocity and mobility in InGaAs

This article introduces an analytical model to accurately describe the drift velocity and mobility of electrons and holes in In0.53Ga0.47As under different electric field conditions and dopant concentrations. Using data simulated by Monte Carlo methods, the model combines mathematical simplicity with precision, making it ideal for implementation in compact and efficient simulations.

The following equation (Equation 1) models the drift velocity of electrons as a function of the electric field:

\[ v_e(|\vec{E}|) = \left\{ \begin{array}{ll} \frac{A_e\left(\sin\left( \frac{\pi|\vec{E}|}{10}\right)\right)^{b_e}}{e^{c_e|\vec{E}|^2}} & \text{if } |\vec{E}| < E_{c,e} \\ \frac{D_e|\vec{E}|}{\left(1+\frac{|\vec{E}|-E_{c,e}}{3}\right)^f_e} & \text{if } |\vec{E}| \geq E_{c,e} \end{array} \right. \]

This equation describes how the electron velocity increases with the electric field until it reaches a saturation point, effectively capturing the transitions between different transport regimes.

Similarly, Equation 3 is used to describe the drift velocity of holes as a function of the electric field:

\[ v_h(|\vec{E}|) = \left\{ \begin{array}{ll} \frac{A_h\ln(|\vec{E}|+1)}{(|\vec{E}|+1)^{b_h}} & \text{if } |\vec{E}| < E_{c,h} \\ \frac{C_h\tanh(|\vec{E}|)}{\ln(|\vec{E}|+d_n)^{f_h}} & \text{if } |\vec{E}| \geq E_{c,h} \end{array} \right. \]

Both equations accurately capture the carrier dynamics in the InGaAs material, facilitating fast and precise simulations of electronic devices at the macroscopic level.

Main Contributions

• Development of a simple and precise analytical model for mobility in In0.53Ga0.47As, applicable to compact simulations.
• The model accounts for both electron and hole drift velocity, offering a comprehensive solution for charge transport in this semiconductor material.
• Model validation through comparisons with Monte Carlo-based simulations and experimental data.

Key Aspects

• The model is easy to implement in electronic device simulations, providing quick and accurate results on charge carrier behavior.
• This model captures both low and high electric field transport regimes, offering a more realistic description of transport in InGaAs.
• The model's simplicity ensures low computational cost, allowing its use in intensive simulations of circuits and devices.

Click here to download the full article.

Full Explicit Numerical Modeling in Time-Domain for Nonlinear Electromagnetics Simulations in Ultrafast Laser Nanostructuring

This paper introduces a fully explicit finite-difference time-domain (FDTD) method for modeling nonlinear electromagnetics, focusing on its application to ultrafast laser nanostructuring. Here we developed a stable algorithm capable of handling complex nonlinear phenomena such as Kerr and Raman effects, plasma generation, and light interactions at metal-dielectric interfaces. The algorithm’s accuracy and stability were theoretically proven, making it a powerful tool for simulating laser-material interactions at the nanometric scale.

One of the most significant contributions of this work is the detailed study of numerical stability, particularly the identification of stability conditions that ensure convergence. The research highlights how to optimize the framework to maintain convergence when dealing with highly nonlinear effects in nanostructured materials.

The nonlinear effects considered include multiphoton ionization, free-electron plasma generation, and metal dispersion. Additionally, this paper also addresses the stability conditions for the FDTD algorithm. Appendix A of the article provides a detailed derivation of the Von Neumann stability criteria and the Routh-Hurwitz criterion, ensuring the robustness of the numerical scheme across various nonlinear scenarios.

Main Contributions

• Development of a fully explicit FDTD method capable of handling nonlinear electromagnetic effects in ultrafast laser nanostructuring.
• Accurate simulation of complex phenomena such as Kerr effect, Raman effect, and plasma generation at nanometric metal-dielectric interfaces.
• Introduction of a stable algorithm with theoretical proofs of energy conservation and numerical stability.

Key Insights

• The research establishes clear stability frameworks for nonlinear simulations, ensuring the convergence of results under various physical conditions.
• The method efficiently models free-electron plasma generation and Kerr-induced nonlinearities in nanostructured dielectrics.
• This approach significantly enhances the accuracy of simulations involving light-matter interactions, especially at the nanoscale.

Click here to download the full paper.

Comparative study of grids based on the cubic crystal system for the FDTD solution of the wave equation

This article introduces a comparative analysis of different grids based on the cubic crystal system for the explicit solution of the wave equation using the finite difference time domain (FDTD) method. The grids studied include the simple cubic (SC), body-centered cubic (BCC), face-centered cubic (FCC), and compact packing cubic (CPC).

One of the most important contributions of this work is the detailed study of numerical stability conditions and the identification of frameworks where the grids are most efficient and accurate. Special emphasis is placed on how each grid offers advantages in terms of physical dispersion error and relative anisotropy depending on the characteristics of the problem being solved.

The article also addresses the study of computational complexity, evaluating the cost associated with implementing each grid in terms of simulation time and computational load. The results show that the BCC grid, though complex, offers the best trade-off between accuracy and computational costs for specific applications.

Main Contributions

• Development of a comprehensive comparative analysis of cubic grids (SC, BCC, FCC, and CPC) for the solution of the wave equation using FDTD.
• Detailed study of numerical stability and convergence conditions, based on the analysis of the Hessian matrix.
• Analysis of the minimum physical dispersion error and relative anisotropy in the different grids under the same number of points per wavelength.
• Evaluation of the computational cost associated with each grid, demonstrating that the BCC grid offers an optimal balance between accuracy and efficiency.

Key Insights

• The work establishes clear frameworks for selecting the appropriate grid based on the characteristics of the problem, highlighting the importance of balancing accuracy and computational cost.
• The BCC grid proved to be the most efficient in terms of physical dispersion error under the same conditions, even surpassing the FCC grid in specific situations.
• The FCC grid remains the best choice in cases where low relative anisotropy must be maintained, while the SC grid is preferred for problems with a dominant single propagation direction.

Click here to download the full article.

Illumination study of a quantum MIM diode for the mid-infrared

This article explores the illumination effects on quantum metal-insulator-metal (MIM) diodes in the mid-infrared range, focusing on improving the harvested tunneling current through optimized illumination techniques. The study investigates the use of a distributed illumination method combined with a Kretschmann and Reather prism-based configuration. This technique extends the tunneling event across the entire diode junction, significantly increasing both the quantum tunneling current and the diode's responsivity.

One of the most notable contributions of this work is the exploration of numerical stability and optimization in diode responsivity. The research highlights how distributed illumination leads to a more uniform electric field across the junction, improving the quantum rectification process compared to traditional methods. The study of nonlinear effects is also emphasized, particularly in the context of how quantum tunneling is enhanced by the applied illumination configuration.

Additionally, this study examines the computational modeling using the ADE-FDTD method to solve Maxwell's equations with quantum tunneling incorporated. This numerical approach helps accurately simulate the interaction between the electromagnetic fields and the MIM diode, providing insights into how different illumination angles and metal thicknesses affect performance.

Main Contributions

• Development of a new distributed illumination technique using a Kretschmann and Reather prism to enhance quantum tunneling across the MIM diode.
• Improvement of the quantum diode's responsivity by increasing the uniformity of the electric field along the junction.
• Use of the ADE-FDTD method to model the interaction of electromagnetic fields with the MIM structure, improving the accuracy of numerical simulations.

Key Insights

• Distributed illumination increases the harvested current significantly, with a responsivity of 0.24 A/W, which is about 12 times higher than previous techniques.
• The Kretschmann prism configuration facilitates stronger electromagnetic field penetration in the MIM junction, leading to more effective quantum rectification.
• Uniform field distribution across the diode junction enhances performance compared to conventional focused illumination techniques.

Click here to download the full article.

BCC-Grid versus SC-Grid in the modeling of a sheet of graphene as a surface boundary condition in the context of ADE-FDTD

This article presents a comparative study between the simple cubic grid (SC-Grid) and body-centered cubic grid (BCC-Grid) for modeling a graphene sheet as a surface boundary condition using the Auxiliary Differential Equation Finite-Difference Time-Domain (ADE-FDTD) method. The study focuses on the intraband and interband contributions of graphene’s conductivity while considering the metal in contact as a dispersive medium.

One of the key contributions of this work is the use of the **BCC-Grid**, which avoids discontinuities in the normal components of the electric and magnetic fields on the graphene surface, proving superior to the traditional SC-Grid for these applications. The BCC-Grid offers computational advantages by reducing complexity while maintaining accuracy, especially in mid-infrared and far-infrared frequencies where volumetric models of graphene become impractical.

Main Contributions

• Development of a surface boundary condition for graphene using ADE-FDTD, incorporating both intraband and interband conductivity contributions.
• Introduction of the BCC-Grid, which ensures continuity of electric and magnetic field components on the graphene surface, improving upon the SC-Grid approach.
• Significant reduction in computational burden with BCC-Grid, achieving the same accuracy as SC-Grid with eight times fewer grid points.
• The ability to represent a three-dimensional object like graphene as a two-dimensional mathematical boundary condition is crucial for computational efficiency and simulation accuracy. This approach reduces processing requirements by avoiding the need to model the entire volume, allowing faster and more manageable simulations.

Key Insights

• The BCC-Grid eliminates the need to handle discontinuities in the electric field at the graphene interface, which are problematic in the SC-Grid.
• The method allows for accurate simulation of metal-graphene interactions, useful for a wide range of electronic and photonic applications.
• The computational efficiency gained by using BCC-Grid makes it ideal for high-performance computing simulations, significantly reducing memory and time requirements.
• Modeling graphene as a mathematic two-dimensional boundary condition is essential for accurately simulating surface phenomena while avoiding the complexity of a full three-dimensional object. This approach enhances precision without sacrificing simulation efficiency.

Click here to download the full article.