Sentaurus TCAD simulator supports five different transport models where each of them can be selected depending on the type of device under study and level of modeling precision required. The five different transport models are Drift-Diffusion,
Thermodynamic, Hydrodynamic, Quantum Transport and Monte Carlo as shown in Figure 2.17.
Figure 2.17 Carrier transport models (“Sentaurus Device User Guide,” 2013)
All carrier transport models can be written in the form of continuity equation, which describe the charge conversation:
∇. 𝐽𝑛 = 𝑞𝑅𝑛𝑒𝑡+ 𝑞𝛿𝑛𝛿𝑡 (𝑎) −∇. 𝐽𝑝 = 𝑞𝑅𝑛𝑒𝑡+ 𝑞𝛿𝑝𝛿𝑡 (𝑏) (2.9)
Where:
𝑅𝑛𝑒𝑡 is the net recombination rate
𝐽𝑛 is the electron current density
𝐽𝑝 is the hole current density
The carrier transport models vary in the expression used to compute Jn and Jp. Brief
explanation on each carrier transport model and selected model justified in this work are highlighted in this section.
(a) Drift-Diffusion Transport
Drift-diffusion (DD) model is the simplest carrier transport model which solves the self-consistently coupled Poisson and carrier (electron or hole or both) continuity equation (Eq. 3.1) in the chosen device region with specified boundary conditions. The current densities for electrons and holes model are given by (2.10):
𝐽𝑛 = −𝑛𝑞𝜇𝑛∇𝜑𝑛 (2.10 (a))
𝐽𝑝 = −𝑝𝑞𝜇𝑝∇𝜑𝑝 (2.10 (b))
Where
𝜇𝑛 and 𝜇𝑝 are the electron and hole mobilities
𝜑𝑛 and 𝜑𝑝 are the electron and hole quasi-Fermi potentials
This model is the isothermal simulation and suitable to be used for low-power density devices with long active regions. Under low-drain bias, it is sufficient to use the DD model. Several studies have been conducted to investigate the suitability of this model for a particular device where most of them require modification to improve the accuracy of the simulation (T. Grasser, Kosina, & Selberherr, 2003).
(b) Thermodynamic Transport
For the thermodynamic model, equation 2.10 (a) and (b) are extended to include the temperature gradient as a driving term:
𝐽𝑛 = −𝑛𝑞𝜇𝑛(∇𝜑𝑛+ 𝑃𝑛∇𝑇) (2.11 (a))
𝐽𝑝 = −𝑝𝑞𝜇𝑝(∇𝜑𝑝+ 𝑃𝑝∇𝑇) (2.11 (b))
Where Pn and Pp are the absolute thermoelectric powers and T is the lattice
temperature.
Thermodynamic model will solve the lattice temperature equation in addition to the Poisson equation and carrier continuity equations. It is suitable for a simulation with high current levels where considerable self-heating can occur, such as for power devices, MOSFET with high gate or drain voltage and open bipolar transistors.
(c) Hydrodynamic Transport
The hydrodynamic transport model solves the carrier temperature and heat flow equations in addition to the Poisson equation and carrier continuity equations. The current densities are defined as:
𝐽𝑛 = 𝑞𝜇𝑛(𝑛∇𝐸𝐶+ 𝑘𝑇𝑛∇𝑛− 𝑛𝑘𝑇𝑛∇𝑙𝑛𝛾𝑛+ 𝜆𝑛𝑓𝑛𝑡𝑑𝑘𝑛∇𝑇 𝑛− 1.5𝑛𝑘𝑇𝑛∇𝑙𝑛𝑚𝑛) (2.12 a) 𝐽𝑝 = 𝑞𝜇𝑝(𝑝∇𝐸𝐶+ 𝑘𝑇𝑝∇𝑝− 𝑝𝑘𝑇𝑝∇𝑙𝑛𝛾𝑝+ 𝜆𝑝𝑓𝑝𝑡𝑑𝑘𝑝∇𝑇 𝑝− 1.5𝑝𝑘𝑇𝑝∇𝑙𝑝𝑚𝑝) (2.12 b)
In the first term of equation 2.12 a and b, the contribution due to the spatial variations of electrostatic potential, electron affinity and the band gap were considered. The remaining terms considered the involvement due to the gradient of concentration, the carrier temperature gradients and the spatial variation of the effective masses mn and mp.
This model is suitable for deep-submicron MOSFETs below 0.18 um generation and as the energy, transport across the heterointerfaces taken into account during simulation, hence it is suitable to be used in heterostructure devices for partially depleted floating-body SOI MOSFETs. The velocity overshoot problem which was not considered in the DD simulation is taken into account in this model. This model also avoids the onset premature breakdown due to the local field assumption in the DD model. Another benefit of this model is where the energy, transport across the heterointerfaces is taken into account during simulation, hence it is suitable to be used for heterostructure devices.
(d) Quantum transport (Quantization model)
When the MOSFET is scaled towards the nanometer regime, some of the features such as oxide thickness and channel width reached the quantum mechanical length scales. With regards to this, the wave nature of electrons or holes must be taken into account in calculating the device characteristics. This model was used in simulating nano scale devices such as fully depleted SOIs, double-gate SOIs and FinFET structures as well as ultra-thin oxide devices. The shrink of device dimension to the order of nanometers allowed the confinement and tunneling to play a significant role. The effect of quantum confinement due to band to band, source to drain and gate tunneling can lead to the shifts of threshold voltage and increased leakage. This impact was significantly observed in the performance of nanowire transistors as a possible device replacement for technology beyond the 22 nm technology node.
There are four quantization models implemented in Sentaurus which differed based on physical sophistication, numeric expenses and robustness. The four quantization models consists of the Van Dort model, 1D Schrodinger model, density gradient (DG)
model and modified local-density approximation (MLDA) model. These quantization models solved the quantum potential equations self-consistently with the Poisson equation and carrier continuity equations.
(e) Monte carlo
Monte Carlo (MC) simulation techniques is a common category of computational algorithm used in recurrent random sampling to model a physical or mathematical system. Following the MC procedure, the microscopic physics of individual carrier motions was tracked resulting in a physical accuracy. The DD method, on the other hand, is based on the average properties of the charge transport system. It is therefore able to yield more accurate result when MC was adopted as compared to DD. However, the MC technique involved larger computational overhead. This required extended simulation times due to the presence of statistical noise which was larger in the sub-threshold regime. Hence it is not efficient as a simulation method as the number of carriers and scattering events are low in the sub-threshold regime.