The purpose of the ACUTE tool-based curricula is to introduce interested scientists from Academia and Industry in advanced simulation methods needed for 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.

Relationship between various transport regimes and significant length-scales.

We first discuss the energy bandstructure that enters as an input to any device simulator. We then begin with the discussion of simulators that involve the drift-diffusion model, and then move into simulations that involve hydrodynamic and energy balance transport and conclude the semi-classical transport modeling with application of particle-based device simulation methods.

Having discussed and utilized the semiclassical simulation tools and their applications, we then move into inclusion of quantum corrections into classical simulators. The final set of tools is dedicated to the far-from equilibrium transport, where we will utilize the concept of pure and mixed states and the distribution function. Several tools that utilize different methods will be used for that purpose. We will utilize tools that use the recursive Green’s function method and its variant, the Usuki method. Also, we will utilize the Contact Block Reduction tool as the most efficient and most complete way of solving the quantum transport problem since this method allows one 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-1D 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.

## Energy Bands and Effective Masses

### Piece-Wise Constant Potential Barrier Tool – Open Systems

The Piece-Wise Constant Potential Barriers Tool allows calculation of the transmission and the reflection coefficient of arbitrary five, seven, nine, eleven and 2n-segment piece-wise constant potential energy profile. For the case of multi-well structure it also calculates the quasi-bound states so it can be used as a simple demonstration tool for the formation of energy bands.

Also, it can be used in the case of stationary perturbation theory exercises to test the validity of, for example, the first order and the second order correction to the ground state energy of the system due to small perturbations of, for example, the confining potential. The PCPBT tool can also be used to test the validity of the WKB approximation for triangular potential barriers.

Exercises:

- More on the energy bands formation: Cosine bands

### Periodic Potential Lab

The Periodic Potential Lab solves the time independent Schroedinger Equation in a 1-D 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 and an expanded zone,

and compare the results against a simple effective mass parabolic band. Transmission is also calculated. This Lab also allows the students to become familiar with the reduced zone and expanded zone representation of the dispersion relation (E-k relation for carriers).

Exercises:

### Bandstructure Lab

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 tool enables the study of bulk dispersion relationships of Si, GaAs, InAs. Plotting the full dispersion relation of different materials, students first get familiar with a band-structure of direct band-gap (GaAs, InAs) and indirect band-gap semiconductors (Si). For the case of multiple conduction band valleys one has to determine first the Miller indices of one of the equivalent valleys and from that information it immediately follows how many equivalent conduction bands one has in Si and Ge, for example.

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

Exercises:

## Drift-Diffusion and Energy Balance Simulations

### PADRE Simulator – Modeling of Si-based devices

PADRE is a 2D/3D simulator for electronic devices, such as MOSFET transistors.

It can simulate physical structures of arbitrary geometry—including heterostructures—with arbitrary doping profiles, which can be obtained using analytical functions or directly from multidimensional process simulators such as Prophet. For each electrical bias, PADRE solves a coupled set of partial differential equations (PDEs). A variety of PDE systems are supported which form a hierarchy of accuracy: (1) electrostatic (Poisson equation), (2) drift-diffusion (including carrier continuity equations), (3) energy balance (including carrier temperature) and (4) electrothermal (including lattice heating).

Several example problems that utilize Padre are given below:

A variety of supplemental documents are available that deal with the PADRE software and TCAD simulation:

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- Abbreviated First Time User Guide

A set of course notes on Computational Electronics with detailed explanations on bandstructure, pseudopotentials, numerical issues, and drift diffusion is also available.

### SILVACO Simulator – Modeling of Si-based and III-V devices

In preparation.

## Particle-Based Simulators

### Bulk Monte Carlo Code

The Bulk Monte Carlo Tool 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 (Si and Ge) and III-V (GaAs, SiC and GaN) 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 http://www.eas.asu.edu/~vasilesk (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

Available also is a voiced presentation

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:

### QUAMC 2D – Particle-Based Device Simulator

QuaMC (pronunciation: quamsee) 2-D is effectively a quasi three-dimensional quantum-corrected semiclassical Monte Carlo transport simulator for conventional and non-conventional MOSFET devices.

Device structures that can be simulated.

Phenomena that can be explained

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 done 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.

## Inclusion of Quantum Corrections into Semi-Classical Simulation Tools

### Schred

Schred 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 (1D) Poisson equation and the 1D Schrodinger equation.

To better understand the operation of SCHRED tool and the physics of MOS capacitors please refer to:

Exercises:

### 1D Heterostructure Tool

The 1D Heterostructure Tool simulates confined states in 1D heterostructures by calculating charge self-consistently in the confined states, based on a quantum mechanical description of the one dimensional device. The greater interest in HEMT devices is motivated by the limits that will be reached with scaling of conventional transistors. The 1D Heterostructure Tool in that respect is a very valuable tool for the design of HEMT devices as one can determine, for example, the position and the magnitude of the delta-doped layer, the thickness of the barrier and the spacer layer for which one maximizes the amount of free carriers in the channel which, in turn, leads to larger drive current. This is clearly illustrated in the examples below.

Exercises:

The most commonly used semiconductor devices for applications in the GHz range now are GaAs 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 is a factor of 2 to 3 higher power density compared to MESFETs which leads to significantly smaller chip size.

HEMTs are field effect transistors where the current flow between two ohmic contacts, Source and Drain, and it is controlled by a third contact, the Gate. Most often the Gate is a Schottky contact. In contrast to ion implanted MESFETs, HEMTs are based on epitaxially grown layers with different band gaps Eg.

## Quantum Transport

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

in preparation.

### nanoMOS

NanoMOS is a 2-D simulator for thin body (less than 5 nm), fully depleted, double-gated n-MOSFETs. A choice of 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 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 can be found on the following links:

### CBR

in preparation.

## Atomistic Modeling

### NEMO3D

NEMO 3-D 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.

NEMO3D has been used to analyze quantum dots, alloyed quantum dots, long range strain effects on quantum dots, effects of wetting layers, piezo-electric effects in quantum dots, quantum dot nuclear spin interactions, quantum dot phonon spectra, coupled quantum dot systems, miscut Si quantum wells with SiGe alloy buffers, core-shell nanowires, allyed nanowires, phosphorous impurities in Silicon (P:Si qbits), bulk 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 3-D single particle states.

Exercises: