Книги+1 / 2013 [Chandan_Kumar_Sarkar]_Technology_CAD

simulation proceeds, output data for each of the electrodes (currents, voltages, and charges) are saved to the current file after each step and therefore the electrical characteristic is obtained. This can be plotted using Inspect. The Solve section is shown below.

Solve {

Poisson

Coupled {Poisson Electron}

Quasistationary (Goal {Name=“gate” Voltage=2}) {Coupled {Poisson Electron}}

}

•Poisson: This specifies that the initial solution is of the nonlinear Poisson equation only. Electrodes have initial electrical bias conditions as defined in the Electrode section.

•Coupled {Poisson Electron}: The second step introduces the continuity equation for electrons, with the initial bias conditions applied. In this case, the electron current continuity equation is solved fully coupled to the Poisson equation, taking the solution from the previous step as the initial guess. The fully coupled or “Newton” method is fast and converges in most cases.

•Quasistationary (Goal {Name = “gate” Voltage = 2})

•{Coupled {Poisson Electron}}

The Quasistationary statement specifies that quasi-static or steady-state “equilibrium” solutions are to be obtained. A set of Goals for one or more electrodes is defined in parentheses. In this case, a sequence of solutions is obtained for increasing gate bias up to and including the goal of 2 V. A fully coupled (Newton) method for the self-consistent solution of the Poisson and electron continuity equations is specified in braces. Each bias step is solved by taking the solution from the previous step as its initial guess. If Extrapolate is specified in the Math section, the initial guess for each bias step is calculated by extrapolation from the previous two solutions.

4.5.2  Physical Models

The Physics section allows a selection of the physical models to be applied in the device simulation [2]. The physical phenomena that actually occur within semiconductor devices are very complicated and are generally described using differential equations (partial and full) of different levels of complexity. The coefficients and the boundary conditions required for solving the equations depend on the structure of the device, the principle of action, and the applied bias. Sentaurus Device allows for arbitrary combinations of transport equations and physical models.

4.5.2.1  Transport Equations

Depending on the device required to be simulated and the level of modeling accuracy required, four different simulation models can be selected:

•Drift-diffusion isothermal simulation: Described by basic semiconductor equations and is suitable for low-power density devices with long active regions.

•Thermodynamic: Accounts for self-heating and is suitable for devices with low thermal exchange, particularly, high-power density devices with long active regions.

•Hydrodynamic: Accounts for energy transport of the carriers. Suitable for devices with small active regions.

•Monte Carlo: Allows for full band Monte Carlo device simulation in the selected window of the device.

4.5.2.2 Poisson Equation and Continuity Equations

The three fundamental equations that dictate the charge transport in semiconductor devices are the Poisson equation and the electron and hole continuity equations. The Poisson equation is given as:

where ε is the electrical permittivity, q is the elementary electronic charge, n and p are the electron and hole densities, ND is the concentration of ionized donors, NA is the concentration of ionized acceptors, and ρtrap is the charge density contributed by traps and fixed charges. The keyword for the Poisson equation is Poisson. The keywords for the electron and hole continuity equations are electron and hole, respectively. They are written as:

→

where Rnet is the net electron–hole recombination rate, Jn is the electron current density, and J→p is the hole current density.

4.5.2.3  Drift-Diffusion Model

The drift-diffusion model is widely used for the simulation of carrier transport in semiconductors and is defined by the Poisson and continuity equations, see (4.1) and (4.2), where current densities for electrons and holes are given by:

where μn and μp are the electron and hole mobilities, and Φn and Φp are the electron and hole quasi-Fermi potentials, respectively.

The thermodynamic or non-isothermal model extends the drift-diffusion model to account for electrothermal effects. It assumes that the charge carriers are in thermal equilibrium with the lattice. In this model the electron and hole temperatures are assumed to be equal to the lattice temperature. The thermodynamic model is described by (4.1), (4.2), and lattice heat flow equations. Because the size of power devices is extremely large compared to that of CMOS devices, the drift-diffusion model including thermodynamic effects is usually sufficient in terms of accuracy. The drift-diffusion transport model, however, fails to describe the internal and external characteristics of deep submicron semiconductor devices. In particular, the drift-diffusion approach cannot reproduce velocity overshoot and often overestimates the impact ionization generation rates. The Monte Carlo method for the solution of the Boltzmann kinetic equation is the most general approach. However, it suffers from high computational requirements. Hence, it cannot be used for the routine simulation of devices in an industrial setting. The hydrodynamic (or energy balance) model is a good compromise. In the hydrodynamic transport model, carrier temperatures are allowed to be different from the lattice temperature.

4.5.2.4  Quantization Models

Some features of current MOSFET devices such as oxide thickness, channel width, etc., have reached their quantum-mechanical length scales. Therefore, the wave nature of electrons and holes can no longer be neglected. Quantization effects in a classical device simulation are included by incorporating potential, like quantity Λn in the classical carrier density formula as follows:

An analogous quantity Λp is used for holes. Sentaurus Device implements four quantization models—that is, four different models for Λn and Λp. They differ in accuracy, computational expense, and robustness:

•The van Dort model is a numerically robust, fast, and proven model. However, it is only suited to bulk MOSFET simulations. The important terminal characteristics are well described by this model, but it does not give the correct density distribution in the channel.

•The 1D Schrödinger equations make up the most accurate quantization model. It can be used for MOSFET simulation, and quantum well and ultrathin silicon-on-insulator (SOI) simulation. However, the simulation procedure is slow and often leads to convergence problems that restrict its use to situations with small current flow. It is used mainly for the validation and calibration of other quantization models.

176Technology Computer Aided Design: Simulation for VLSI MOSFET

•The density gradient model is numerically robust but significantly slower than the van Dort model. It can be applied to MOSFETs, quantum wells, and SOI structures. It gives reasonable description of terminal characteristics and charge distribution inside a device. Compared to the other quantization models, it can describe 2D and 3D quantization effects.

•The modified local-density approximation (MLDA) is a numerically robust and fast model. It can be used for bulk MOSFET simulations and thin SOI simulations. Although it sometimes fails to calculate accurate carrier distribution in the saturation regions because of its one-dimensional characteristic, it is suitable for three-dimensional device simulations because of its numeric efficiency.

Due to the wide usage of the density gradient model, we briefly discuss this below.

The density gradient model for Λn in (4.4) is given by a partial differential equation:

where γ is a fitting parameter. The density gradient equation for electrons and holes is activated by the eQuantumPotential and hQuantumPotential switches in the Physics section. These switches can also be used in region wise or material wise in the Physics section. In metal regions, the equations are never solved. Apart from activating the equations in the Physics section, the equations for the quantum corrections must be solved by using eQuantumPotential or hQuantumPotential, or both in the Solve section. For example:

Physics { eQuantumPotential

}

Plot { eQuantumPotential

}

Solve {

Coupled {Poisson eQuantumPotential} Quasistationary (

Do Zero InitialStep=0.01 MaxStep=0.1 MinStep=1e-5 Goal {Name=“gate” Voltage=2}

){

Coupled {Poisson Electron eQuantumPotential}

}

}

4.5.2.5  Mobility Models

Sentaurus Device provides several options for the description of carrier mobilities. The various causes of mobility degradation can be individually modeled. In the simplest case, the mobility is considered to be a function of the lattice temperature. This is referred to as the constant mobility model and accounts only for phonon scattering. This should only be used for undoped materials. In doped semiconductors, scattering of the carriers by charged impurity ions leads to degradation of the carrier mobility. This is modeled in Sentaurus Device [4]. In the channel region of a MOSFET, the high transverse electric field forces carriers to interact strongly with the semiconductor–insulator interface. Carriers are subjected to scattering by acoustic surface phonons and surface roughness. This phenomenon of mobility degradation at interface is also modeled in Sentaurus Device. The carrier-carrier scattering effect can also be modeled. The carrier-carrier contribution to overall mobility degradation is combined with mobility contributions from other degradation models following Matthiessen’s rule. The Philips unified mobility model unifies the description of majority and minority carrier bulk mobilities. In addition to describing the temperature dependence of the mobility, the model takes into account electron–hole scattering, screening of ionized impurities by charge carriers, and clustering of impurities. The mobility degradation due to high electric field can also be modeled in Sentaurus Device.

4.5.2.5.1  Doping-Dependent Mobility Degradation

The models for the mobility degradation due to impurity scattering are activated by specifying the DopingDependence flag to Mobility:

Physics {Mobility (DopingDependence...)...}

Different models are available and are selected by options to DopingDependence: Physics {Mobility (DopingDependence ([Masetti | Arora |UniBo])...)...}

If DopingDependence is specified without options, Sentaurus Device uses a material-dependent default. The default model used by Sentaurus Device to simulate doping-dependent mobility in silicon was proposed by Masetti et al. This is as follows:

The reference mobilities min 1, min 2, and 1; the reference doping concentrations Pc , Cr Cs ; and the exponents α and β are accessible in the parameter set DopingDependence. The corresponding values for silicon are given in Table 4.1.

4.5.2.5.2  Mobility Degradation at Interfaces

In the channel region of a MOSFET, the high transverse electric field forces carriers to interact strongly with the semiconductor–insulator interface. The carriers are thus subjected to scattering by acoustic surface phonons and surface roughness. The models in this section describe mobility degradation caused by these effects. To select the calculation of field perpendicular to the semiconductor–insulator interface, specify the Enormal option to Mobility:

Physics {Mobility (Enormal...)...}

The surface contribution due to acoustic phonon scattering has the form:

And the contribution attributed to surface roughness scattering is given by:

These surface contributions to the mobility are then combined with the bulk mobility according to Mathiessen’s rule:

where Fref = 1 V/cm, Fn = Normal electric field, D = exp (–x⁄ lcrit) (where x is the distance from the interface and lcrit is a fit parameter). In the Lombardi

model, the exponent in (4.8), A* is equal to 2. According to another study, an improved fit to measured data is achieved if A* is given by:

The respective default parameters that are appropriate for silicon are given in Table 4.2.

4.5.2.5.3  High-Field Saturation

In high electric fields, the carrier drift velocity is no longer proportional to the electric field; instead, the velocity saturates to a finite speed vsat. The high-field saturation models include three sub-models: the actual mobility model, the velocity saturation model, and the driving force model. With some restrictions, these models can be freely combined. The actual mobility model is selected by flags eHighFieldSaturation or hHighFieldSaturation. The default is the Canali model whose parameter values for Silicon are given in Table 4.3. The Canali model originates from the Caughey–Thomas formula but has temperature-dependent parameters, which were fitted up to 430 K by Canali et al. [5]:

Enormal

)

Recombination(

SRH(DopingDep)

)

}

Plot{

*— Density and Currents, etc eDensity hDensity

TotalCurrent/Vector eCurrent/Vector hCurrent/Vector eMobility hMobility

eVelocity hVelocity eQuasiFermi hQuasiFermi

*— Temperature

eTemperature Temperature * hTemperature

*— Fields and charges

ElectricField/Vector Potential SpaceCharge

*— Doping Profiles

Doping DonorConcentration AcceptorConcentration

*— Generation/Recombination

SRH Band2Band * Auger AvalancheGeneration eAvalancheGeneration

hAvalancheGeneration

*— Driving forces

eGradQuasiFermi/Vector hGradQuasiFermi/Vector eEparallel hEparallel eENormal hENormal

*— Band structure/Composition BandGap

BandGapNarrowing Affinity

ConductionBand ValenceBand eQuantumPotential

}

Math { Extrapolate Iterations=20 Notdamped=100 RelErrControl

ErRef(Electron)=1.e10

ErRef(Hole)=1.e10

}

182 Technology Computer Aided Design: Simulation for VLSI MOSFET

Solve {

*- Build-up of initial solution:

NewCurrentFile=“init”

Coupled(Iterations=100){Poisson eQuantumPotential}

Coupled{Poisson Electron Hole eQuantumPotential}

*- Bias drain to target bias Quasistationary(

InitialStep=0.01 Increment=1.35 MinStep=1e-5 MaxStep=0.2 Goal{Name=“gate” Voltage=2.0 }

){Coupled{Poisson Electron Hole eQuantumPotential}

}

*- gate voltage sweep NewCurrentFile=“”

Quasistationary(

InitialStep=1e-3 Increment=1.35 MinStep=1e-5 MaxStep=1.1 Goal{Name=“drain” Voltage=2.0}

){Coupled{Poisson Electron Hole eQuantumPotential} CurrentPlot(Time=(Range=(0 1) Intervals=20))

}

* none

}

This will generate output files “crc_des.tdr”, “crc_des.plt”, and “crc_des.log”.

4.6  Tecplot

Tecplot is a plotting software with extensive 2D and 3D capabilities for visualizing data from simulations and experiments. Tecplot can be started at the command prompt without loading any data file:

> tecplot_sv

4.6.1  Input Files

Two types of files can be loaded into Tecplot SV. The first type is the .tdr file. This file is used to describe a device structure, corresponding meshing, and the values of the field quantities existing in the corresponding device. The other type is the .plt file. Datasets included in this file are used by Tecplot SV to generate X-Y plots. Loading can be performed initially when Tecplot SV is started from the command line or interactively after Tecplot SV has started.