Global TCAD Solutions

Predictions based on Physics

Schrödinger-Poisson Simulation of FinFETs

Fig.1: Structure generation based on a TEM cross-section published by chipworks
Fig.2a.: VSP model structure
Fig.2b.: Element-based FVM discretization
Fig.3: 3D FinFET simulation: Wave functions
Fig.4: Hole dispersion relation in the p-type MOS capacitor calculated using a k⋅p band structure at inversion VG=-2V
Fig.5: S/P electron concentration and dispersion relation for axial stress

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]:

Hamiltonian as suggested by J. C. Hensel, H. Hasegawa, and M. Nakayama

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]:

H Hamiltonian as suggested by T. Manku and A. Nathan

H-+ Hamiltonian as suggested by T. Manku and A. Nathan

Hbc Hamiltonian as suggested by T. Manku and A. Nathan

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.


With ever-smaller device features, considering quantum effects is vital for avoiding design risk and understanding device physics. In contrast to empirical descriptions used in most available tools, VSP provides a physical description and insight relying on well-known material properties.

VSP is the first commercially available tool capable of accurate simulation of real-world 2D/3D device geometries on a triangular / tetrahedral mesh using a K⋅P Schrödinger Poisson Hamiltonian, including strain effects.

VSP enables you to explore, understand and specifically address quantum-effects in recent and future devices, significantly improving efficiency in the design process.




  • Explore design space:
    • non-planar geometries
    • crystal orientation
    • strain
  • Meaningful calibration of quantum corrections like density gradient

Download Application Flyer: Non-Planar Geometries

File link icon for GTS-App-Nonplanar-Web_03.pdf


GTS Framework Application Flyer: Non-Planar Geometries (web version)

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Further Information

To take a closer look on VSP and its easy-to-use interface, please see Schrödinger Poisson, which shows how simulation of nanowires can be carried out.

For a feature summary, please refer to the VSP product description.


[Auth12] C. Auth et. al., "A 22nm high performance and low-power CMOS technology featuring fully-depleted tri-gate transistors, self-aligned contacts and high density MIM capacitors," VLSI Technology (VLSIT), 2012 Symposium on , vol., no., pp.131,132, 12-14 June 2012.

[Hensel65] J. C. Hensel, H. Hasegawa, and M. Nakayama, "Cyclotron Resonance in Uniaxially Stressed Silicon. II. Nature of the Covalent Bond", Phys. Rev., vol. 138, no. 1A, A225-A238, Apr. 1965.

[Manku93] T. Manku and A. Nathan, "Valence energy-band structure for strained group-IV semiconductors", J. Appl. Phys., vol. 73, no. 3, pp. 1205-1213, Feb. 1993.

[Bir74] G. Bir and G. Pikus, Symmetry and Strain-induced E ects in Semiconductors. Wiley, New York, 1974.

[Stanojevic11] Z. Stanojevic, M. Karner, K. Schnass, Ch. Kernstock, O. Baumgartner, H. Kosina, "A Versatile Finite Volume Simulator for the Analysis of Electronic Properties of Nanostructures",
Proceedings of the 16th International Conference on Simulation of Semiconductor Processes and Devices", (2011), ISBN: 978-1-61284-418-3; 143 - 146.

[Stanojevic13] Z. Stanojevic, O. Baumgartner, K. Schnass, M. Karner, H. Kosina, "VSP - a Quantum Simulator for Engineering Applications",
Proceedings of the 16th International Workshop on Computational Electronics (IWCE 2013)", (2013), 132 - 133.