Книги+1 / 2013 [Chandan_Kumar_Sarkar]_Technology_CAD

FIGURE 5.3

Input and output in ATLAS.

3.Silvaco general structure file: The structure files store the twoand three-dimensional data relating to the values of solution variables such as electric field, electrostatic potentials, etc., within the device for a single bias point.

The log files and the solution structure files are visualized using TONYPLOT. Figure 5.3 summarizes the various input and output techniques available for device simulation using ATLAS.

5.4  Simulation Setup

Once a structure file is defined in ATLAS, the remaining part of the setup (except for result analysis) consists of providing command line instructions that must be used to instruct ATLAS in order to complete a simulation run. The order in which these instructions are given is important and can be divided into five groups of statements outlined in Figure 5.2. Failing to adhere to this order may result in premature termination of a simulation run, or in an error message being produced. The order of statements within each group such as structural definition, model specification, and solution groups is also important. Failing to place these statements in proper order may result in aforementioned complications.

5.5  Brief Review of Electro-Physical

Models Employed in ATLAS

The accuracy of the results obtained by device simulation process using ATLAS depends on the models used in the simulation process. Normally physics-based models are used to account for the complex dependencies of the device properties on dimensions and other process variables. Generally, the model parameters are derived from measurements and characteristics of the devices.

The drift-diffusion (DD) model is the simplest current transport model and is derived from Boltzmann’s transport equation (BTE). It is famous for its simplicity and most efficient approach. Over a long period of time, this model has been at the core of semiconductor device simulation. The robust discretization of the DD equations proposed by Scharfetter and Gummel [9] is still in use today. In isothermal simulation, this simplest DD model includes three partial differential equations (PDEs): Poisson’s equation and current continuity equations for electrons and holes. The total current density at any point (sum of electron and hole current) of the structure is found after solving the free electron and hole concentrations using potential and variable parameters like mobility, electric field and generation-recombination rate, etc.

As technology moves forward, devices scale down to sub-micron regime, without scaling the supply voltages proportionately. This results in a large electric field inside the device. As this large electric field changes quickly over small lengths, it brings about non-local and hot-carrier effects that tend to dominate device performance. Therefore, non-stationary carrier transport phenomenon such as velocity overshoot, which substantially influences the on-current (mainly the drift-current) of the MOSFET should be taken into account in the simulation. This makes the suitability of the DD model highly questionable for the simulation of nanometer MOSFET, as it neglects nonstationary effects.

As a result, transport models have been refined and extended to capture the practical transport phenomena more accurately. To address this issue, extensions of the DD model have been proposed. The extensions add a balance equation for the average carrier energy. They also add a driving term to the current relation, proportional to the gradient of the carrier temperature.

A non-isothermal thermodynamic model is chosen where self-heating effects are involved. Therefore, additional PDEs corresponding to nonisothermal temperature distribution must be coupled [10].

A more complex hydrodynamic (HD) or energy balance model is chosen for deep sub-micron devices, which involves solution of six coupled partial differential equations: three basic semiconductor equations and three

energy balance equations. The individual electron and hole temperatures are calculated from the energy balance equation.

In the HD model the equation set of the DD model is extended by the energy balance equation to allow for non-stationary carrier transport. The disadvantages of the HD model are that it is less stable than DD, it is less efficient in terms of computation time than DD, and it sometimes overestimates the current in MOSFET [11].

The typical hierarchy of the MOSFET simulation approach consists of DD, HD, and Monte Carlo (MC) methods. MC is most reliable for calculation of on-current. Unfortunately, the added accuracy comes at the expense of higher computational complexity as compared to HD and DD. Thus MC is not suitable for investigations where large numbers of different transistors are to be simulated. Even 3D simulations necessary for FinFETs are difficult with MC. Furthermore, because of its statistical nature, the MC method has serious problems in calculating the very low subthreshold currents of MOSFETs accurately.

In a recent study, it was shown that the DD on-current is 10% less than MC-current calculated for a 100-nm MOSFET [12]. For shorter length MOSFET such as 40 nm, the difference of on-current is about 40% [13]. However, the on-current simulated by DD can be improved by adjusting the mobility model used. For the calculation of subthreshold current, the DD model is best suited for both long and short channel MOSFETs [14]. Granzner et al. [15] suggested that modifications of the velocity-field characteristics in the DD simulations are suggested to improve the accuracy of the DD model, and for the simulation of subthreshold current the standard DD model is best suited.

To incorporate quantum-mechanical effects (QMEs) that are significant for highly scaled deca-nanometer devices, the Schrödinger equation should be solved self-consistently in order to obtain the most accurate simulation result.

Therefore, to achieve maximum predictive capability one needs to move toward the quantum transport regime of the hierarchical structure shown in Figure 5.4. The green function’s approach [16] shown in Figure 5.4 is most difficult in terms of complexity of equations and numerical efficiency.

The details of various advanced electro-physical models to characterize the properties and behavior of analyzed semiconductor device structures can be found in the user manual of simulator ATLAS [17].

Physical models are specified in the “MODELS” statement depending on the material used and physical nature of the device. For example, the statement “MODELS CVT SRH FERMIDIRAC” enables the Lombardi mobility model (constant voltage and temperature, CVT), Shockley-Read-Hall recombination with fixed carrier lifetimes (SRH) and Fermi-Dirac statistics (FERMIDIRAC) for common long-channel Si-MOSFET simulation.

On the other hand, de-coupled solutions where a subset of the equation is solved while others are held constant has shown an advantage when the interaction between the equations is small (typically low voltage and current levels). They tolerate a relatively poor initial guess for convergence. They tend to either diverge or take excessive central processing unit (CPU) time once the interaction among the equations becomes stronger. Gummel’s method cannot be used with lumped elements or current boundary conditions. In a typical fully de-coupled procedure, if we choose quasi-Fermi level formulation, at first non-linear Poisson’s equation is solved. The potentials obtained are substituted into continuity equations that are now linear and solved directly. The results in terms of quasi-Fermi levels are substituted back to Poisson’s equation until the convergence is reached, as shown in Figure 5.5.

Ingeneral,Gummel’smethodisusefulwherethesystemofequationsisweakly coupled, but it has only linear convergence. The Newton method is useful when the system of equations is strongly coupled and has quadratic convergence.

Gummel’s method can provide better initial guess to the problems. Therefore, it is possible to start a solution using few Gummel’s iterations to generate a good initial guess and then switch to Newton to complete the simulation [18], as shown in Figure 5.6.

The combined method will solve some equations fully coupled and the remaining others using a de-coupled method. The Block method can provide faster simulation than Newton’s method for solving quantities that are weakly coupled. They involve solving a subgroup of equations in various sequences. In non-isothermal drift-diffusion simulation using Block method,

FIGURE 5.5

Uncoupled numerical solution.

FIGURE 5.6

Steps for numerical solutions.

Newton’s method is solved for a constant lattice temperature to update carrier concentration and potential, after which a heat flow equation with the appropriate continuity equation is solved using a de-coupled method to update carrier temperature and carrier concentration.

5.7  Mobility Models in ATLAS

The choice of carrier mobility models is one of the most important decisions of TCAD device simulation. This is governed by physical parameters, ambience, and operating conditions.

Electrons and holes accelerate by electric fields but lose momentum due to various scattering processes like lattice vibrations (phonons), impurity ions or other carriers, surfaces, and other material imperfections. Various microscopic phenomena affect the macroscopic mobility specified in the transport equation. Hence these mobilities are functions of the local electrical field, doping concentration, lattice temperature, and many other parameters. Based on the operating region, mobility models can generally be classified into four types: (1) low field, (2) high field, (3) bulk semiconductor region, and

(4) inversion region.

From an alternative perspective, mobility models can be categorized into three types: (1) physically based, (2) semi-empirical, and (3) empirical.

Physical-based mobility models are obtained from fundamental calculations using a lot of approximations, and they rarely match the experimental data. In a semi-empirical model, to obtain an agreement between the model and the experimental result, the coefficients of the physical-based models are allowed to vary, keeping the power law dependencies preserved. In contrast, it is possible to vary the power-law dependencies in a purely empirical model, exhibiting the lowest physical content and a narrower range of validity. The most popular mobility models used in ATLAS are listed in Table 5.1.

Low field mobility degrades upon phonon and impurity scattering and is valid for carriers present in the bulk under low electric field. The high

TABLE 5.1

Summary of the Popular Mobility Model Used in ATLAS

electric field dependent parameter decreases with increasing high electric field, making the velocity of the carrier constant known as saturation velocity.

Bulk mobility modeling generally requires:

1.Identification of low field mobility as a function of doping and lattice temperature

2.Identification of saturation velocity as a function of temperature

3.Description of transition between high and low field region

In ATLAS, low field mobility for the bulk region includes:

1.Constant mobility model

2.Caughey and Thomas model (doping and temperature dependent) [19]

3.Arora model (doping, temperature and carrier-carrier scattering dependent) [20]

4.Klaassen unified low-field mobility model (unified minority and majority carrier mobility, includes lattice scattering, carrier-carrier scattering, impurity-clustering effects at high concentrations) [21,22]

It is important to model mobility dependent on the transverse electric field associated with inversion layers in order to obtain accurate results. ATLAS supports five major transverse field-dependent mobility models. They are the CVT (Lombardi) MODEL [23], Watt model [25], Shirahata model [26], Yamaguchi model [27], and Tasch model [28].

In ATLAS, the CVT [23] model is selected by specifying CVT on the model statement. This model overrides any other mobility model specification. In the CVT model, three different components based on transverse field and doping concentrations are combined using Mathiessen’s rule [24]. The com-

ponents are µAC, µsr, and µb. µAC is the surface mobility limited by scattering with acoustic phonons. µsr is a mobility component limited by surface rough-

ness. µb is the mobility component limited by scattering with optical intervalley phonons. According to Mathiessen’s rule the total mobility in a CVT

model is given by T−1 = −AC1 + −sr1 + b−1.

The Watt model (SURFMOB) [25] is a surface mobility model and is activated by specifying the parameter WATT in the MODEL statement. The Watt model takes into consideration the phonon scattering, surface scattering, and charge impurity scattering mechanisms in the inversion layer.

The Shirahata model [26] is enabled by specifying the SH parameter of the model statement. It is a general-purpose MOS mobility model that takes into account the screening effect in inversion layers and perpendicular field dependence for thin gate oxides. When the user enables Shirahata model, ATLAS automatically enables Klaassen’s model [21,22].

The Yamaguchi model [27] and the Tasch model [28] are selected by setting YAMAGUCHI and TASCH, respectively, on the MODEL statement. The model overrides all mobility models other than CVT.

ATLAS invokes filed dependent mobility models if FLDMOB is specified in the MODEL statement. FLDMOB should always be specified unless one of the inversion layer mobility models (exhibits parallel field dependency itself) is specified.

In ATLAS, it is possible to use more than one mobility model. The default mobility model is the constant low field mobility µno and µpo for electron and hole, respectively. Values of µno and µpo may be specified in the MATERIAL statement using the parameters “MUN” and “MUP”. This