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The purpose of the ACUTE tool-based curriculum is to introduce interested scientists from academia and industry to the advanced methods of simulation needed for the proper modeling of state-of-the-art nanoscale devices. The multiple scale transport in doped semiconductors is summarized in the figure below, in terms of the transport regimes, relative importance of the scattering mechanisms, and possible applications.

ACUTE begins with a discussion of the energy band structure that enters as an input to any device simulator. The next section offers a discussion of simulators that involve the drift-diffusion model, and then simulations that involve hydrodynamic and energy-balance transport, and conclude the semi-classical transport modeling with application of particle-based device simulation methods.

After the study and utilization of the semiclassical simulation tools and their applications, the next step includes quantum corrections into the classical simulators. The final set of tools is dedicated to the far-from equilibrium transport, where the concept of pure and mixed states and the distribution function is introduced. Several tools that utilize different methods will be used for that purpose, such as tools that use the recursive Green’s-function method and its variant, the Usuki method, as well as the Contact Block Reduction tool, as the most efficient and complete way of solving the quantum-transport problem because this method allows users to simultaneously calculate source-drain current and gate leakage (which is not the case, for example, with the Usuki and the recursive Green’s function techniques that are in fact quasi one-dimensional in nature for transport through a device). A table that shows the advantages and the limitation of various semi-classical and quantum-transport simulation tools is presented below.

More details on the actual tool design and information on commercial tool usage can be found on the web pages:

Computational Electronics HW Set

## Energy Bands and Effective Masses

### Piecewise Constant Potential Barrier Tool in ACUTE– Open Systems

The Piecewise Constant Potential Barrier Tool in ACUTE allows users to calculate the transmission and the reflection coefficient of arbitrary five, seven, nine, eleven and 2n-segment piecewise constant potential energy profile. For the case of a multi-well structure, it also calculates the quasi-bound states. Thus the Piecewise Constant Potential Tool can be used as a simple demonstration tool for the formation of energy bands.

Other uses include: 1) in the case of stationary perturbation theory, as an exercise to test the validity of the first-order and the second-order correction to the ground state energy of the system due to small perturbations of the confining potential, and 2) as a test of the validity of the Wentzel–Kramers–Brillouin (WKB) approximation for triangular potential barriers.

Exercises:

### Periodic Potential Lab in ACUTE

The Periodic Potential Lab in ACUTE solves the time-independent Schrödinger Equation in a one-dimensional spatial potential variation. Rectangular, triangular, parabolic (harmonic), and Coulomb potential confinements can be considered. The user can determine energetic and spatial details of the potential profiles, compute the allowed and forbidden bands, plot the bands in a compact as well as an expanded zone, and compare the results against a simple effective-mass parabolic band. Transmission is also calculated. This lab also allows students to become familiar with the reduced-zone and expanded-zone representation of the dispersion relation (i. e. the E-k relation for carriers).

Exercises:

### Band Structure Lab in ACUTE

In solid-state physics, the electronic band structure (or simply band structure) of a solid describes ranges of energy that an electron is “forbidden” or “allowed” to have. It is due to the diffraction of the quantum mechanical electron waves in the periodic crystal lattice. The band structure of a material determines several characteristics, in particular its electronic and optical properties. The Band Structure Lab in ACUTE enables the study of bulk dispersion relationships of silicon (Si), gallium arsenide (GaAs), and indium arsenide (InAs). Plotting the full dispersion relation of different materials, students first get familiar with a band structure of direct bandgap (gallium arsenide and indium arsenide) and indirect band-gap semiconductors (silicon). In the case of multiple conduction band valleys, the user has first to determine the Miller indices of one of the equivalent valleys and then from that information it immediately follows, e. g., how many equivalent conduction bands one has in silicon and germanium (Ge).

In advanced applications, the users can apply tensile and compressive strain and observe the variation in the band structure, bandgaps, and effective masses. Advanced users can also study band-structure effects in ultra-scaled (thin body) quantum wells, and nanowires of different cross sections. Band Structure Lab uses the *sp3s*d5* tight binding method to compute dispersion (E-k) for bulk, planar, and nanowire semiconductors.

Exercises:

## Drift-Diffusion and Energy Balance Simulations

### PADRE Tool in ACUTE—Modeling of silicon-based devices

PADRE Tool in ACUTE is a two-dimensional/three-dimensional simulator for electronic devices, such as MOSFET transistors.

PADRE Tool in ACUTE is a 2D/3D simulator for electronic devices, such as MOSFETs. With PADRE, users can simulate physical structures of arbitrary geometry—including heterostructures—with arbitrary doping profiles, which can be obtained using analytical functions or directly from such multidimentional process simulators as Prophet For each electrical bias, PADRE Tool in ACUTE solves coupled sets of partial differential equations (PDEs). The variety of PDE systems supported in PADRE form a hierarchy of accuracy: 1) electrostatic (Poisson equations), 2) drift-diffusion, including carrier continuity equations, 3) energy balance, including carrier temperature, and 4) electrothermal, including lattice heating.

Listed below are tools, exercises, and sets of problems that utilize the PADRE Tool in ACUTE:

Supplemental documentation:

A set of course notes on computational electronics with detailed explanations on band structure, pseudopotentials, numerical issues, and drift diffusion is also available.

### SILVACO Simulator—Modeling of Silicon-Based and III-V Devices

In preparation.

## Particle-Based Simulators

### Bulk Monte Carlo Lab in ACUTE

The Bulk Monte Carlo Lab in ACUTE calculates the bulk values of the electron drift velocity, electron average energy, and electron mobility for electric fields applied in arbitrary crystallographic direction in both column 4 (silicon and germanium) and III-V (gallium arsenide, silicon carbide and gallium nitride) materials. All relevant scattering mechanisms for the materials being considered have been included in the model.

Detailed derivation of the scattering rates for most of the scattering mechanisms included in the model can be found on Prof. Vasileska personal web-site (look under class EEE534 Semiconductor Transport). Description of the Monte Carlo method used to solve the Boltzmann Transport Equation and implementation details of the tool are given in the

Bulk Monte Carlo Code Described

An A/V presentation is also available:

Ensemble Monte Carlo Method Described

that gives more insight on the implementation details of the Ensemble Monte Carlo technique for the solution of the Boltzmann Transport Equation. Examples of simulations that can be performed with this tool are given below:

Consistent Parameter Set for an Ensemble Monte Carlo Simulation of 4H-SiC

Exercises:

### Quamc2D Lab in ACUTE

QuaMC 2D (pronounced “quam-see”) is a quasi three-dimensional quantum-corrected semi-classical Monte-Carlo transport simulator for conventional and non-conventional MOSFET devices.

A parameter-free quantum field approach has been developed and utilized quite successfully in order to capture the size-quantization effects in nanoscale MOSFETs. The method is based on a perturbation theory around thermodynamic equilibrium and leads to a quantum field-formalism in which the size of an electron depends upon its energy. This simulator uses different self-consistent event-biasing schemes for statistical enhancement in the Monte-Carlo device simulations. Enhancement algorithms are especially useful when the device behavior is governed by rare events in the carrier transport process. A bias technique, particularly useful for small devices, is obtained by injection of hot carriers from the boundaries. Regarding the Monte Carlo transport kernel, the explicit inclusion of the longitudinal and transverse masses in the silicon conduction band is realized in the program using the Herring-Vogt transformation. Intravalley scattering is limited to acoustic phonons. For the intervalley scattering, both g- and f-phonon processes have been included.

Exercises:

### Thermal Particle-Based Device Simulator

In preparation.

Exercises and Other Resources:

## Inclusion of Quantum Corrections in Semiclassical Simulation Tools

### Schred in ACUTE

Schred in ACUTE calculates the envelope wavefunctions and the corresponding bound-state energies in a typical MOS (Metal-Oxide-Semiconductor) or SOS (Semiconductor-Oxide- Semiconductor) structure and a typical SOI structure by solving self-consistently the one-dimensional Poisson and Schrödinger equations.

To better understand the operation of Schred in ACUTE and the physics of MOS capacitors please refer to:

Exercises:

### 1D Heterostructure Tool in ACUTE

The 1D Heterostructure Tool in ACUTE simulates the confined states in one-dimentional heterostructures by self-consistently calculating their charge based on a quantum-mechanical description of the one-dimensional device. Increased interest in high electron mobility transistors (HEMTs) is due to the eventual limitations reached by scaling conventional transistors. The 1D Heterostructure Tool in ACUTE is a very valuable tool for the design of HEMTs because the user can determine such components as the position and the magnitude of the delta-doped layer, the thickness of the barrier, and the spacer layer, for which the user can maximize the amount of free carriers in the channel, which, in turn, leads to a larger drive current.

Exercises:

The most commonly used semiconductor devices for applications in the GHz range now are gallium arsenide based MESFETs, HEMTs and HBTs. Although MESFETs are the cheapest devices because they can be realized with bulk material, i.e. without epitaxially grown layers, HEMTs and HBTs are promising devices for the near future. The advantage of HEMTs and HBTs compared to MESFETs is a higher power density (by a factor of two to three), which leads to a significantly smaller chip size.

HEMTs are field-effect transistors wherein the flow of the current between two ohmic contacts, known as the source and the drain, is controlled by a third contact, the gate. Such gates are usually Schottky contacts. In contrast to ion-implanted MESFETs, HEMTs are based on epitaxial layers with different band gaps.

## Quantum Transport

### Recursive Green’s Function Method for Modeling RTD’s

in preparation.

### nanoMOS in ACUTE

nanoMOS in ACUTE is a two-dimensional simulator for thin body (less than 5 nm), fully depleted, double-gated n-MOSFETs. Five transport models is available (drift-diffusion, classical ballistic, energy transport, quantum ballistic, and quantum diffusive). The transport models treat quantum effects in the confinement direction exactly, and the names indicate the technique used to account for carrier transport along the channel. Each of these transport models is solved self-consistently with Poisson’s equation. Several internal quantities such as subband profiles, subband areal electron densities, potential profiles, and current-voltage (I/V) information can be obtained from the source code.

NanoMOS 3.0 includes an improved treatment of carrier scattering. Some important information about nanoMOS in ACUTE can be found on the following links:

### CBR

in preparation.

## Atomistic Modeling

### NEMO3D in ACUTE

NEMO3D in ACUTE calculates eigenstates in (almost) arbitrarily shaped semiconductor structures in the typical column IV and III-V materials. Atoms are represented by the empirical tight binding model using *s*, *sp3s**, or *sp3d5s** models with or without spin. Strain is computed using the classical valence force field (VFF) with various Keating-like potentials.

Users of NEMO3D in ACUTE can analyze quantum dots, alloyed quantum dots, long-range strain effects on quantum dots, the effects of wetting layers, piezo-electric effects in quantum dots, quantum-dot nuclear-spin interaction, quantum-dot phonon spectra, coupled quantum-dot systems, miscut silicon quantum wells with silicon-germanium alloy buffers, core-shell nanowires, alloyed nanowires, phosphorous impurities in silicon (P:Si qubits), and buck alloys.

Boundary conditions to treat the effects of surface states have been developed. Direct and exchange interactions and interactions with electromagnetic fields can be computed in a post-processing approach based on the NEMO 3D single particle states.

Exercises: