Semi-Classical Transport Theory

Outline:What is Computational Electronics?

Semi-Classical Transport TheoryDrift-Diffusion SimulationsHydrodynamic SimulationsParticle-Based Device Simulations

Inclusion of Tunneling and Size-Quantization Effects in Semi-Classical SimulatorsTunneling Effect: WKB Approximation and Transfer Matrix ApproachQuantum-Mechanical Size Quantization EffectDrift-Diffusion and Hydrodynamics: Quantum Correction and Quantum Moment MethodsParticle-Based Device Simulations: Effective Potential Approach

Quantum TransportDirect Solution of the Schrodinger Equation (Usuki Method) and Theoretical Basis of the Greens Functions Approach (NEGF)NEGF: Recursive Greens Function Technique and CBR ApproachAtomistic Simulations The Future

Prologue

Direct Solution of the Boltzmann Transport Equation Particle-Based Approaches Spherical Harmonics Numerical Solution of the Boltzmann-Poisson Problem

In here we will focus on Particle-Based (Monte Carlo) approaches to solving the Boltzmann Transport Equation

Ways of Solving the BTE Using MCTSingle particle Monte Carlo TechniqueFollow single particle for long enough time to collect sufficient statisticsPractical for characterization of bulk materials or inversion layersEnsemble Monte Carlo TechniqueMUST BE USED when modeling SEMICONDUCTOR DEVICES to have the complete self-consistency built inCarlo Jacoboni and Lino Reggiani, The Monte Carlo method for the solution of charge transport in semiconductors with applications to covalent materials, Rev. Mod. Phys. 55, 645 - 705 (1983).

Path-Integral Solution to the BTEThe path integral solution of the Boltzmann Transport Equation (BTE), where L=Nt and tn=nt, is of the form:

K. K. Thornber and Richard P. Feynman, Phys. Rev. B 1, 4099 (1970).

The two-step procedure is then found by using N=1, which means that t=t, i.e.:Intermediate function that describesthe occupancy of the state (p+eEt)at time t=0, which can be changeddue to scattering events (SCATTER) Integration over a trajectory, i.e.probability that no scattering occurred within time integral t (FREE FLIGHT)+

Monte Carlo Approach to Solving the Boltzmann Transport EquationUsing path integral formulation to the BTE one can show that one can decompose the solution procedure into two components:Carrier free-flights that are interrupted by scattering events Memory-less scattering events that change the momentum and the energy of the particle instantaneously

Particle Trajectories in Phase Space

Carrier Free-FlightsThe probability of an electron scattering in a small time interval dt is (k)dt, where (k) is the total transition rate per unit time. Time dependence originates from the change in k(t) during acceleration by external forces

where v is the velocity of the particle.The probability that an electron has not scattered after scattering at t = 0 is:

It is this (unnormalized) probability that we utilize as a non-uniform distribution of free flight times over a semi-infinite interval 0 to . We want to sample random flight times from this non-uniform distribution using uniformly distributed random numbers over the interval 0 to 1.

Generation of Random Flight TimesHence, we choose a random number Ith particlefirst random numberWe have a problem with this integral!

We solve this by introducing a new, fictitious scattering process which does not change energy or momentum:

Generation of Random Flight Times

Self-Scattering

Self-Scattering

Free-Flight Scatter Sequence for Ensemble Monte Carlo = collisionsHowever, we need a second time scale, which provides the times at which the ensemble is stopped and averages are computed.Particle time scale

Free-FlightScatterSequence

R. W. Hockney and J. W. Eastwood, Computer Simulation Using Particles, 1983.

Choice of Scattering Event Terminating Free Flight

At the end of the free flight ti, the type of scattering which ends the flight (either real or self-scattering) must be chosen according to the relative probabilities for each mechanism.

Assume that the total scattering rate for each scattering mechanism is a function only of the energy, E, of the particle at the end of the free flight

where the rates due to the real scattering mechanisms are typically stored in a lookup table.

A histogram is formed of the scattering rates, and a random number r is used as a pointer to select the right mechanism. This is schematically shown on the next slide.

Choice of Scattering Event Terminating Free-FlightWe can make a table of the scattering processes at the energy of the particle at the scattering time:

Look-up table of scattering rates:Store the total scattering rates in a table for a grid in energy

Sheet1

Choice of the Final State After ScatteringUsing a random number and probability distribution function

Using analytical expressions (slides that follow)

Representative Simulation Results From Bulk Simulations - GaAsSimulation Results Obtained byD. Vasileskas Monte Carlo Code.

Transient Data

Steady-State ResultsGunn Effect

Particle-Based Device SimulationsIn a particle-based device simulation approach the Poisson equation is decoupled from the BTE over a short time period dt smaller than the dielectric relaxation timePoisson and BTE are solved in a time-marching mannerDuring each time step dt the electric field is assumed to be constant (kept frozen)

Particle-Mesh CouplingThe particle-mesh coupling scheme consists of the following steps:

- Assign charge to the Poisson solver mesh- Solve Poissons equation for V(r) Calculate the force and interpolate it to the particle locations - Solve the equations of motion:Laux, S.E., On particle-mesh coupling in Monte Carlo semiconductor device simulation, Computer-Aided Design of Integrated Circuits and Systems, IEEE Transactions on, Volume 15, Issue 10, Oct 1996 Page(s):1266 - 1277

Assign Charge to the Poisson Mesh1. Nearest grid point scheme2. Nearest element cell scheme3. Cloud in cell scheme

Force interpolationThe SAME METHOD that is used for the charge assignment has to be used for the FORCE INTERPOLATION:xp-1xpxp+1

Treatment of the ContactsFrom the aspect of device physics, one can distinguish between the following types of contacts:

(1) Contacts, which allow a current flow in and out of the device - Ohmic contacts: purely voltage or purely current controlled - Schottky contacts

(2) Contacts where only voltages can be applied

Calculation of the CurrentThe current in steady-state conditions is calculated in two ways: By counting the total number of particles that enter/exit particular contact By using the Ramo-Shockley theorem according to which, in the channel, the current is calculated using

Current Calculated by Counting the Net Number of Particles Exiting/Entering a Contact

Device Simulation Results for MOSFETs: Current ConservationVG=1.4 V, VD=1 VCumulative net number of particlesEntering/exiting a contact for a 50 nmChannel length device

Drain contactSource contactCurrent calculated using Ramo-Schockleyformula

X. He, MS, ASU, 2000.

Simulation Results for MOSFETs: Velocity and Enery Along the ChannelVelocity overshoot effect observed throughout the whole channel length of the device non-stationary transport.For the bias conditions used average electron energy is smaller that 0.6 eV which justifies the use of non-parabolic band model.

Simulation Results for MOSFETs: IV CharacteristicsThe differences between the Monte Carlo and the Silvaco simulations are due to the following reasons: Different transport models used (non-stationary transport is taking place in this device structure). Surface-roughness and Coulomb scattering are not included in the theoretical model used in the 2D-MCPS.X. He, MS, ASU, 2000.

Simulation Results For SOI MESFET Devices Where are the Carriers?SOIMOSFETSOIMESFETApplications:Low-power RF electronics.T.J. Thornton, IEEE Electron Dev. Lett., 8171 (1985).Micropowercircuits basedon weakly inverted Implantablecochlea and retinaDigital WatchPacemakerMOSFETs

Proper Modeling of SOI MESFET DeviceGate current calculation:WKB Approximation Transfer Matrix Approach for piece-wise linear potentialsInterface-Roughness:K-space treatmentReal-space treatmentGoodnick et al., Phys. Rev. B 32, 8171 (1985)

Output Characteristics and Cut-off Frequency of a Si MESFET DeviceTarik Khan, PhD, ASU, 2004.

Output Characteristics and Cut-off Frequency of a Si MESFET DeviceTarik Khan, PhD, ASU, 2004.

Modeling of SOI DevicesWhen modeling SOI devices lattice heating effects has to be accounted forIn what follows we discuss the following: Comparison of the Monte Carlo, Hydrodynamic and Drift-Diffusion results of Fully-Depleted SOI Device Structures* Impact of self-heating effects on the operation of the same generations of Fully-Depleted SOI Devices*D. Vasileska. K. Raleva and S. M. Goodnick, IEEE Trans. Electron Dev., in press.

FD-SOI Devices:Monte Carlo vs. Hydrodynamic vs. Drift-DiffusionSilvaco ATLAS simulations performed by Prof. Vasileska.90 nm

FD-SOI Devices: Monte Carlo vs. Hydrodynamic vs. Drift-DiffusionSilvaco ATLAS simulations performed by Prof. Vasileska.25 nm

FD-SOI Devices: Monte Carlo vs. Hydrodynamic vs. Drift-DiffusionSilvaco ATLAS simulations performed by Prof. Vasileska.14 nm

FD-SOI Devices:Why Self-Heating Effect is Important?1. Alternative materials (SiGe)2. Alternative device designs (FD SOI, DG,TG, MG, Fin-FET transistors

FD-SOI Devices:Why Self-Heating Effect is Important?dSL ~ 300nmA. Majumdar, Microscale Heat Conduction in Dielectric Thin Films, Journal of Heat Transfer, Vol. 115, pp. 7-16, 1993.

Conductivity of Thin Silicon Films Vasileska Empirical Formula

Particle-Based Device Simulator That Includes Heating

Heating vs. Different Technology Generation

Higher Order Effects Inclusion in Particle-Based SimulatorsDegeneracy Pauli Exclusion Principle

Short-Range Coulomb Interactions

Fast Multipole Method (FMM)V. Rokhlin and L. Greengard, J. Comp. Phys., 73, pp. 325-348 (1987).Corrected Coulomb ApproachW. J. Gross, D. Vasileska and D. K. Ferry, IEEE Electron Device Lett. 20, No. 9, pp. 463-465 (1999).P3M MethodHockney and Eastwood, Computer Simulation Using Particles.

Potential, Courtesy of Dragica Vasileska, 3D-DD Simulation, 1994.MOTIVATION

Length [nm]100110120130140150Width [nm]6080100120140Current Stream Lines, Courtesy of Dragica Vasileska, 3D-DD Simulation, 1994.MOTIVATION

The ASU Particle-Based Device Simulator (1)Corrected Coulomb Approach(2) P3M Algorithm (3)Fast Multipole Method (FMM) Ferrys Effective Potential MethodQuantum Field ApproachStatistical Enhancement: Event Biasing SchemeShort-Range Interactions and Discrete/Unintentional Dopants Quantum Mechanical Size-quantization EffectsBoltzmann Transport Equations(Particle-Based Monte Carlo Transport Kernel)Long-range Interactions(3D Poisson Equation Solver)

Significant Data ObtainedBetween 1998 and 2002

MOSFETs - Standard CharacteristicsThe average energy of the carriers increases when going from the source to the drain end of the channel. Most of the phonon scattering events occur at the first half of the channel.Velocity overshoot occurs near the drain end of the channel. The sharp velocity drop is due to e-e and e-i interactions coming from the drain.W. J. Gross, D. Vasileska and D. K. Ferry, "3D Simulations of Ultra-Small MOSFETs with Real-Space Treatment of the Electron-Electron and Electron-Ion Interactions," VLSI Design, Vol. 10, pp. 437-452 (2000).

MOSFETs - Role of the E-E and E-I mesh forceonlywith e-e and e-iIndividual electron trajectories over time

MOSFETs - Role of the E-E and E-I Mesh force onlyWith e-e and e-iShort-range e-e and e-i interactions push someof the electrons towards higher energiesD. Vasileska, W. J. Gross, and D. K. Ferry, "Monte-Carlo particle-based simulations of deep-submicron n-MOSFETs with real-space treatment of electron-electron and electron-impurity interactions," Superlattices and Microstructures, Vol. 27, No. 2/3, pp. 147-157 (2000).

Degradation of Output Characteristics LG = 35 nm, WG = 35 nm, NA = 5x1018 cm-3, Tox = 2 nm, VG = 11.6 V (0.2 V)The short range e -e and e -i interactions have significant influence on the device output characteristics.There is almost a factor of two decrease in current when these two inte-raction terms are considered.LG = 50 nm, WG = 35 nm, NA = 5x1018 cm-3Tox = 2 nm, VG = 11.6 V (0.2 V)W. J. Gross, D. Vasileska and D. K. Ferry, "Ultra-small MOSFETs: The importance of the full Coulomb interaction on device characteristics," IEEE Trans. Electron Devices, Vol. 47, No. 10, pp. 1831-1837 (2000).

Mizuno result:(60% of the fluctuations)Stolk et al. result:(100% of the fluctuations)Fluctuations in thesurface potentialFluctuations in theelectric fieldDepth-Distributionof the charges

MOSFETs - Discrete Impurity Effects [1] T. Mizuno, J. Okamura, and A. Toriumi, IEEE Trans. Electron Dev. 41, 2216 (1994).[2] P. A. Stolk, F. P. Widdershoven, and D. B. Klaassen, IEEE Trans. Electron Dev. 45, 1960 (1998).

Depth Correlation of sVT To understand the role that the position of the impurity atoms plays on the threshold voltage fluctuations, statistical ensembles of 5 devices from the low-end, center and the high-end of the distribution were considered.Significant correlation was observed between the threshold voltage and the number of atoms that fall within the first 15 nm depth of the channel.Number of atoms in the channelNumber of devicesNumber of atoms in the channelDepth [nm]VT correlationThreshold voltage [V]

Fluctuations in High-Field Characteristics Impurity distribution in the channel also affects the carrier mobility and saturation current of the device.Significant correlation was observed between the drift velocity (saturation current) and the number of atoms that fall within the first 10 nm depth of the channel.Number of atoms in the channelDrift velocity [cm/s]Number of atoms in the channelDepth [nm]CorrelationDrain current [mA]VG = 1.5 V, VD = 1 V

Current Issues in NovelDevices Unintentional Dopants

THE EXPERIMENT

Results for SOI DeviceSize Quantization Effect (Effective Potential):S. S. Ahmed and D. Vasileska, Threshold voltage shifts in narrow-width SOI devices due to quantum mechanical size-quantization effect, Physica E, Vol. 19, pp. 48-52 (2003).

Results for SOI DeviceDue to the unintentional dopant both the electrostatics and the transport are affected.

Results for SOI DeviceUnintentional Dopant:D. Vasileska and S. S. Ahmed, Narrow-Width SOI Devices: The Role of Quantum Mechanical Size Quantization Effect and the Unintentional Doping on the Device Operation, IEEE Transactions on Electron Devices, Volume 52, Issue 2, Feb. 2005 Page(s):227 236.

Results for SOI DeviceChannel Width = 10 nmVG = 1.0 VVD = 0.1 V

Results for SOI DeviceChannel Width = 5 nmVG = 1.0 VVD = 0.1 V

Results for SOI DeviceImpurity located at the very source-end, due to the availability of Increasing number of electrons screening the impurity ion, has reduced impact on the overall drain current.

Results for SOI DeviceD. Vasileska and S. S. Ahmed, IEEE Transactions on Electron Devices, Volume 52, Issue 2, Feb. 2005 Page(s):227 236.S. Ahmed, C. Ringhofer and D. Vasileska, Nanotechnology, IEEE Transactions on, Volume 4, Issue 4, July 2005 Page(s):465 471.D. Vasileska, H. R. Khan and S. S. Ahmed, International Journal of Nanoscience, Invited Review Paper, 2005.

Results for SOI DeviceElectron-Electron Interactions: D. Vasileska and S. S. Ahmed, IEEE Transactions on Electron Devices, Volume 52, Issue 2, Feb. 2005 Page(s):227 236.S. Ahmed, C. Ringhofer and D. Vasileska, Nanotechnology, IEEE Transactions on, Volume 4, Issue 4, July 2005 Page(s):465 471.D. Vasileska, H. R. Khan and S. S. Ahmed, International Journal of Nanoscience, Invited Review Paper, 2005.

SummaryParticle-based device simulations are the most desired tool when modeling transport in devices in which velocity overshoot (non-stationary transport) existsParticle-based device simulators are rather suitable for modeling ballistic transport in nano-devicesIt is rather easy to include short-range electron-electron and electron-ion interactions in particle-based device simulators via a real-space molecular dynamics routineQuantum-mechanical effects (size quantization and density of states modifications) can be incorporated in the model quite easily with the assumption of adiabatic approximation and solution of the 1D or 2D Schrodinger equation in slices along the channel section of the device