# Nonlinear k.p Schrödinger-Poisson Equation with the Finite-Difference Method

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## TCAD in an industrial environment

Technology Computer-Assisted Design (TCAD) tools are widely used in industrial environments for modeling electrical characteristics of microelectronics components. This is mostly due to the fact that TCAD technology development can reduce cost and development time, but also secure technology choice [Nier 15]. Additionally, a reliable TCAD tool is required for developing predictive device and process simulation, for supporting the physics coherence and performance of highly scaled devices, for detecting operative limits and also for investigating new device concepts [Sasso 10].
The §1.2 presents the main components of a TCAD device modeling tool. §1.3 exposes numerous comparisons by means of TCAD simulations and explores several topics in its models §1.4 discusses the issues found in TCAD models and proposes a strategy to circumvent such concerns.

TCAD device modeling often describe the carrier distribution and transport inside a given device by solving the Poisson equation coupled to a Density-Gradient (DG) model and also to the Continuity equation and associated transport models. In this section we briefly discuss these approaches.

Poisson equation

The electrostatic solution of the system is determined by means of the Poisson equation, which is essentially based on the Gauss’s law [Maxwell 73]:
and the electric charge density for a given material ( . The details about Poisson Equation are given in chapter 2.Basically,thePoissonequation establishes a relationship between the electrostatic potential

Continuity equation

The carrier transport models can be written in the form of continuity equation, which describes the charge conservation:
The carrier transport models differ in the expressions used to compute the carrier current density .

Transport models

Among the simplest transport models, Drift-Diffusion (DD) has been widely used in industry essentially due to the simplicity of its use and will supposedly continue to be applied in the future, provided that the parameters needed in the simulation are accurately calibrated [Lundstrom 15]. Traditional TCAD tools makes available transport models such as Drift-Diffusion, which treats carrier transport as diffusion and drift processes. For the drift component, the carrier movement is due to an applied electric field, while for the diffusion component, the carrier displacement is due to spatial charge variation. For the DD transport model, the current density is typically computed as follows: where the is the carrier mobility, is the carrier charge density, potential, is the Boltzmann constant, ! is the lattice temperature,  » is the electron charge.
In the DD approach, the electron gas is assumed to be in thermal equilibrium with the lattice temperature. Nevertheless, nanoscale devices are usually under a strong electric field in which carriers gain energy from the field and therefore their temperature is raised and rather non-uniform along the device [Grasser 03].
Moreover, as the device size approaches the nanometer range, carrier transport becomes quasi-ballistic, and non-local effects such as velocity overshoot occur [Sasso 10]. In an attempt to capture these phenomena, more advanced transport models have been proposed, such as hydrodynamic (HD), which can also be found in TCAD solvers.
The following HD model can be found in Sdevice [Synopsys 14], based on Maxwell- Boltzmann statistics: where !) is the carrier temperature and must not be confused with the lattice temperature !, the fitting parameter *+, is the thermal diffusion constant and /011 is the effective mass.
Compared with DD, HD model has extra driving forces accounting for energy distribution on transport of carriers. As shown in 2 , besides the contribution due to spatial variation of electrostatic potential and contribution due to the gradient of carrier concentration, the current density has also a contribution due to the carrier temperature gradient and the spatial variation of the effective mass in hetero-structure devices. The HD model is capable of handling non-uniform temperatures and changes on effective masses along the device (Please note that in Chapter 3 we show that the quantum DD model proposed in this work also presents this capability).
Both the DD and HD models can be viewed as approximations of the Boltzmann transport equation (BTE), which represents a rigorous approach to model carrier transport in semiconductors. Different models taking into account higher moments of the BTE such as Spherical Harmonics Expansion (SHE) can be found in the literature [Hong 09] [Jungemann 06] [Grasser 04], but they are not in the scope of this work.
Further details about pertinent transport models are given in Chapter 3. In addition, extra models accounting for mobility calculation in TCAD can be found in Chapter 4 and models accounting for Generation-Recombination can be found in the Appendix.

Theories of carrier transport can be divided according to whether they are microscopic or macroscopic in character [Ancona 11]. Microscopic theories deal with the individual carriers, such as wave functions, density matrices, etc. The macroscopic theories are devoted to the carrier populations. For the latter, semiconductor devices that are small enough are directly impacted by quantum mechanics effects, such as the phenomena of quantum confinement. Quantum confinement is basically due to the impact on atomic structure as a result of direct effect of nanoscale lengths on the energy band structure [Zhao 04]. For nanostructures, quantum effects become relevant due to the ratio between the size of the device (thin film) or potential well in the case of electrical confinement and the mean free path of the carrier. The Figure 1.1 illustrates the classical distribution and the quantum confinement in a nanoscale device:
SiO2 Si SiO2
DENSITY CLASSICAL
CHARGE
QUANTUM CONFINEMENT
POSITION
Contrary to classical charge distribution, the quantum confinement is described by envelop functions associated to a well-defined number of discrete energy levels and these envelop functions will have their maximum density pushed-away from the Si-SiO2 interface. The low-density zone close to this interface is the so-called “dark space”. For a quantum confinement in the channel thickness direction, electrons are no longer represented as a 3D electron gas (3DEG), but as a 2D electron gas (2DEG) in which the transport is modeled in the two dimensions of the channel plan (further details will be presented in section 2.4).
A simplified TCAD model accounting for quantization effects is given, for instance, by the modified local-density approximation (MLDA) which is a model that calculates the confined carrier distributions occurring near semiconductor-insulator interfaces [Synopsys 14] [Paasch 82]. It can be applied to both inversion and accumulation regimes, and simultaneously to electrons and holes. The confined electron density at a distance 3 from a Si-SiO2 interface is given for the Maxwell-Boltzmann distribution by: 4 3 56 789 :;789 3<=; > ? where 56 is the electron density of states, :; @A @6 < ! in which @A is the quasi-Fermi level and @6 is the conduction band. =; BC>< /D ! is the electron thermal wavelength which depends on the quantization mass /D.
Another model for accounting for quantum confinement effects used in TCAD tools in ultra-scaled devices is the Density-Gradient (DG) model. This model is described in E [Ancona 87] [Ancona 89] and implemented in Sdevice as: 4 56 FG<> H @A J@6!) I;K where the FG<> is the Fermi-Dirac statistics and I; is the quantum potential given by: LC> >O 4 P I; E/,MN O 4 where L is a fitting parameter “weighting factor” for quantum potential (this L must not be confused with the “quantum statistics correction factor” which will be presented in chapter 3). The expressions E and P are for electrons and similar equations are applied to holes. Furthermore, rather more elaborated models must be applied in order to account for extra features such as stress, non-parabolic bands and geometric quantization in thin-layer structures [Penzin 11], and for all these models appropriated calibrations are needed.
The calculation of the electron concentration in the presence of confinement effect requires to solve, for instance, the Poisson and Schrodinger equation self-consistently [Colinge 06]. However, when DG is required, one can also make use of the self-consistent Poisson-Schrödinger solution for a proper calibration of the model [Pons 13].
In this chapter, it will be examined the consistency of traditional TCAD tools for modeling ultimate devices in which all simulations are performed with DG along its analysis of validity and limitations.

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#### TCAD for 14FDSOI and 10FDSOI UTBB

TCAD for industrial purposes is based on an accurate modeling of the process steps. However, since this work is focused on the device itself, simplified 14FDSOI and 10FDSOI devices with ultra-thin body and boxes (UTBB) have been created. These template devices have been carefully set up in order to match the characteristics of process-simulated devices as it will be shown later in this work. The process variations such as Lightly Doped Drain (LDD) extension change and morphological variations are mimicked using analytical functions and parameterized structures.
In this Chapter, we will presented comparisons by means of simulations performed with Sdevice [Synopsys 14] using Density Gradient model with either DD or HD transport model. For both models, the same channel-thickness dependent mobility, the remote Coulomb scattering and the nonlinear piezoelectric strain-dependent models are used. The details on these models and the calibration procedures on advanced solvers can be found in [Nier 13].

Description of the simplified devices

In order to proceed with the modeling investigation, the template device architecture is shown in Figure 1.3 for both nMOS and pMOS:
Figure 1.3: Description of FDSOI (nMOS and pMOS).
The gate stack is composed by High-k material (UVW>, X ) over the Interfacial layer-IL (YZ[5\ X E E). The spacer is composed by a Nitride (]^_`a, X P ?). The Buried Oxide (Box) is composed by Silicon oxide (]^W>, X ( b). The ground plane is composed by Silicon (]^, X P) with Boron doping concentration 8 Gcd/e_ for the nMOS and Phosphorus doping concentration 8 Gcd/e _ for the pMOS. The source and drain regions of nMOS device are composed by Silicon (]^, X P) with Phosphorus doping concentration (f > d/e_. The pMOS device features SiGe epitaxied in-situ doped Source/Drain with Boron doping concentration of (f > d/e_. The SOI <110>-oriented channel has a Boron doping concentration ?8 Ggd/e_ for all cases. The lattice temperature for all simulations in this work is set to 300K.
Assuming that the Ltot=74nm for Lg=24nm, we consider 50nm of S/D region to ensure consistent boundary conditions of the simulations. Lext=4.5nm is given value accounting for source/drain epitaxy and silicidation process. Note that the reference values (in bold) are corresponding to the ones used for the comparison of TCAD results and electrical measurements.

Source and Drain regions

The LDD region has potentially different doping profiles. In order to provide an example of a process variation, we consider a split of the spacer thickness and final anneal temperature split (i.e. the diffusion of the in-situ doped S/D region). Doping profiles obtained from process simulations can be either fitted by a Gaussian or by an analytical expression.

General Introduction
1 TCAD: Application and Physical’s Modeling Validity
1.1 TCAD in an industrial environment
1.2.1 Poisson equation
1.2.2 Continuity equation
1.2.3 Transport models
1.3 TCAD for 14FDSOI and 10FDSOI UTBB
1.3.1 Description of the simplified devices
1.3.2 Device dimensions and physical parameters
1.3.3 Source and Drain regions
1.3.4 Oxide and space charges
1.3.5 Extracted electrical quantities
1.3.6 Saturation regime and model comparison
1.3.7 14nFDSOI results
1.3.1 14pFDSOI results
1.3.3 10nFDSOI predictions
1.3.4 10pFDSOI predictions
1.4 Conclusion
2 Nonlinear k.p Schrödinger-Poisson Equation with the Finite-Difference Method
2.1 Introduction
2.2 The UTOXPP solver
2.3 The Poisson Equation
2.3.1 The Boundary Conditions
2.3.2 Nonlinear Poisson Equation
2.3.3 Kerker Mixing
2.3.4 Predictor-Corrector Scheme
2.3.5 Initial Guess
2.3.6 Matrix System Solving
2.4 The Schrödinger Equation
2.4.1 No Confinement – Gas 3D
2.4.2 Confinement 1D – Gas 2D
2.4.3 Confinement 2D – Gas 1D
2.4.1 1.5D Poisson-Schrödinger
2.4.2 Mode Space
2.5 Arbitrarily Oriented Devices
2.6 Self-Consistent Simulation Flow
2.6.1 Algorithm n°1
2.6.2 Algorithm n°2
2.7 Simulation Results
2.8 Conclusion
3 Transport Models: From Classical to Quantum Drift-Diffusion
3.1 Introduction
3.2 Drift-Diffusion Model
3.2.1 Continuity Boundary Conditions
3.2.2 Self-Consistent Poisson-Continuity
3.3 Quantum Drift-Diffusion Model
3.4 Simulation Results for nFDSOI
3.5 Simulation Results for pFDSOI
3.6 Convergence Study
3.6.1 Comparison FEM vs FDM
3.7 Drift-Diffusion decoupling
3.8 Conclusion
4 Mobility Models
4.1 Introduction
4.2 Empirical Mobility Models
4.2.1 Charge-dependent Models
4.2.2 Field-dependent Models
4.2.3 Access Resistance
4.2.4 Velocity Saturation
4.3 Kubo-Greenwood Mobility Models
4.3.1 Mobility from Linearized Boltzmann Equation
4.3.2 Collision times calculation
4.3.3 Kubo-Greenwood in UTOXPP
4.4 Mobility Models study
4.4.1 Long Channel Mobility
4.4.2 Device Mobility
4.5 NEGF (Mode Space) in planar systems
4.5.1 Transverse -vector integration
4.5.2 Ballistic transport in effective mass: analytic k-points integration
4.5.3 Phonon scattering: numerical k-points integration
4.5.4 Decoupled mode-space approximation
4.6 Mobility extracted from NEGF simulations
4.7 NEGF versus QDD
4.7.1 Ballistic diffusive transport
4.7.2 Saturations effects
4.7.3 Validity of QDD model
4.8 IMEP-Lahc NEGF solver applied to 14FDSOI
4.9 Conclusion
General Conclusion
Bibliography

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