Fully-depleted FinFET devices provide improved transistor electro-statics which result in better performance at lower supply voltages and significantly reduced short-channel effects SCE [Auth12].
Carrier Quantization in Fully Depleted Fins
The fin width must be sufficiently small with low channel doping to be fully depleted; an even narrower fin width is desired for improved SCE. When channel cross-section dimension approaches the free electron wavelength the carriers become geometrically confined. This strongly affects device physics, such as electrostatic properties and carrier transport; for accurate modeling, it is vital to account for this behaviour.
Quantum-Mechanical Models – Ready for Use
GTS Framework's tool set contains physical models for confined carriers, aimed at analysis and optimization of FinFET structures: The Vienna Schrödinger Poisson (VSP) Simulator can accurately capture the physics of such devices in 1D, 2D, and 3D. Devices can be created and edited conveniently in GTS structure (c.f. Fig.1).
Arbitrary Geometries, Triangular or Tetrahedral Mesh
In VSP, classical and quantum-mechanical (QM) solvers are included for self-consistent iteration with the Poisson model (see Fig.2a). The closed boundary QM model allows to calculate a self consistent carrier concentration on FinFETs or nanowire cross-sections including quantum confinement. An element-based finite volume (FVM) discretization method (see Fig.2b) inherently ensures compliance with physical conservation laws. Furthermore, it accurately handles material anisotropy and thus is capable of capturing orientation and strain effects [Stanojevic11].
Numerical Efficiency and Parallel Processing
VSP employs state-of-the-art libraries for solving linear, nonlinear and eigenvalue problems [Stanojevic13]. These are combined with methods that enhance the numerical performance especially for eigenvalue problems, such as the shift-invert technique or subspace deflation. Scalable parallelization is provided, making VSP a highly efficient tool – allowing full 3D Schrödinger Poisson simulation, as shown in Fig.3.
K⋅P Hamiltonian for Non-Parabolic Band Structure
For very narrow channels, the non-parabolic band structure plays an important role. Band structures can be modeled efficiently using K⋅P (K dot P) Hamiltonians. For holes, a six-band K⋅P Hamiltonian including spin orbit coupling is applied, as suggested by [Manku93]:
The Hamiltonian is an expansion around the Γ point of the Brillouin zone. The Hamiltonian includes the perturbation S, and the deformation potential matrix DV for modeling strain effects.
For electrons, VSP uses a two-band K⋅P Hamiltonian according to [Hensel65]:
Here, k=0 corresponds to the X point. The diagonal elements correspond to the effective mass approximation, while the off-diagonal elements add the non-parabolic dispersion.
Fig.4 shows the hole-dispersion relation in the p-type MOS capacitor (1D cut) calculated using a k⋅p band structure at inversion VG = -2.0 V.
Arbitrary Substrate and Channel Orientation with Strain Distribution
Both Hamiltonians (electrons and holes) naturally include strain effects through deformation potentials [Bir74]. The Hamiltonians cover the shift of minima as well as a change in the dispersion relation which results in effective mass variation. This is demonstrated for an axially stressed FinFET cross-section in Fig.5, which shows the self-consistent carrier concentration and the sub-band structure.