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Multisubband Monte Carlo simulations for p- MOSFETs

Multisubband Monte Carlo simulations for p- MOSFETs. David Esseni DIEGM, University of Udine (Italy) Many thanks to: M.De Michielis, P.Palestri, L.Lucci, L.Selmi. Acknowledg : NoE. SINANO (EU), PullNano (EU).

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Multisubband Monte Carlo simulations for p- MOSFETs

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  1. Multisubband Monte Carlo simulations for p-MOSFETs David Esseni DIEGM, University of Udine (Italy) Many thanks to: M.De Michielis, P.Palestri, L.Lucci, L.Selmi Acknowledg: NoE.SINANO (EU), PullNano (EU)

  2. Support of the physically based transport modelling to the generalized scaling scenario • Band-structure calculation and optimization: • Carrier velocity and maximum attainablecurrent IBL • Scattering rates, hence realcurrent ION and BR=(ION/IBL) • Link the properties and advantages of: Mobility in Long MOSFETs (Uniform transport) ION in nano-MOSFETs (far from equilibr. transport) • Provide sound interpretation to characterization D.Esseni, University of Udine

  3. y x z Multisubband Monte Carlo (MSMC) approach for MOS transistors VG2 • Solve 1D Schrödinger equation in the Z direction ei(x) along the channel VS VD VG1 • Driving Force in each subband: z D.Esseni, University of Udine

  4. Multisubband Monte Carlo (MSMC) for n-MOS transistors(electron inversion layers) D.Esseni, University of Udine

  5. y x X z MSMC for n-MOS transistors (1) (Effective Mass Approximation) VG2 Subband “j” VD Subband “i” VG1 • SchrÖdinger-like equation: • Energy dispersion versus k: • mx, my, mz expressed in terms of mt and mlof the bulk crystal D.Esseni, University of Udine

  6. y x z MSMC for n-MOS transistors (2) (Effective Mass Approximation) VG2 VD VG1 Energy dispersion: Driving force: Velocity: D.Esseni, University of Udine

  7. Transport in the MSMC approach(2D carrier gas) Force: Band structure Kinematics: Rates of scattering D.Esseni, University of Udine

  8. Bandstructure for a hole inversion layer: • Single-band effective mass approx. is not viable: • Three almost degenerate bands at the Gpoint • Spin-orbit interaction k·pmethod for hole inversion layers D.Esseni, University of Udine

  9. y x z k·pmethod for inverted layers: VG2 Differently from EMA: one eigenvalue problem for each in-plane (kx,ky) VS VD • Finite differencesmethod: • √ section and√in-planek: • eigenvalue problem 6Nzx6Nz VG1 • Entirely numerical description of the energy dispersion Computationally very heavy for simulations of pMOSFETs Simplified models for energy dispersion of 2D holes D.Esseni, University of Udine

  10. MSMC for pMOSFETs • Semi-analytical model for 2D holes • Basic idea and full development of the model • Implementation in a Monte Carlo tool • Simulation results D.Esseni, University of Udine

  11. Semi-analytical model for 2D holes Three groups of subbands: • Calculation of the eigenvalues ev,i • New analytical expression for in-plane energy Ep(k) k·presults D.Esseni, University of Udine

  12. Semi-analytical model for 2D holes 1) Bottom of the 2D subbands (the relatively easy part) D.Esseni, University of Udine

  13. Semi-analytical model for 2D holes(bottom of the 2D subbands) Schrödinger equation as in EMA (mz): Good agreement also in square well mn,z fitted using triangular wells D.Esseni, University of Udine

  14. 2) Energy dependence on k (the by no means easy part) Semi-analytical model for 2D holes D.Esseni, University of Udine

  15. Semi-analytical model for 2D holes(energy dispersion is anisotropic) k·presults Si(100) • Strongly anisotropic • Periodic of p/2 D.Esseni, University of Udine

  16. k·presults Semi-analytical model for 2D holes(energy dispersion is non-parabolic) Analytical dispersion in the symmetry directions: D.Esseni, University of Udine

  17. Semi-analytical model for 2D holes(angular dependence) Fourier series expansion: A, B, C calculated with no additional fitting parameters: D.Esseni, University of Udine

  18. Bottom of the 2D subbands • Energy dependence on k Semi-analytical model for 2D holes D.Esseni, University of Udine

  19. MSMC for pMOSFETs • Semi-analytical model for 2D holes • Calibration and validation • Implementation in a Monte Carlo tool • p-MOSFETs: Simulation results D.Esseni, University of Udine

  20. Calibration of the semi-analytical model(bottom of the 2D subbands) Schrödinger equation in the EMA (mz): Good agreement also in square well mn,z fitted using triangular wells D.Esseni, University of Udine

  21. Calibration of the semi-analytical model(non parabolicity along symmetry directions) Si(100), Fc=0.3MV/cm • Good results with the proposed non parabolic expression: D.Esseni, University of Udine

  22. Validation of the semi-analytical model(overall energy dependence on k) Si(001) • Calculation conditions: • Triangular well: FC=0.3 MV/cm • E-e0=75 meV • The model seems to grasp fairly well the complex, anisotropic energy dispersion D.Esseni, University of Udine

  23. Si(001) Validation of the semi-analytical model(2D Density Of States - DOS) Acoustic Phonon scattering: D.Esseni, University of Udine

  24. Validation of the semi-analytical model(average hole velocity: vx, vy) • Analytical Model:  Analytical expression for: Pinv=5.6x1012[cm-2] Average: [0,p/4] • k·presults(numerical • determination): D.Esseni, University of Udine

  25. MSMC for pMOSFETs • Semi-analytical model for 2D holes • Implementation in a Monte Carlo tool • Integration of the motion equation • p-MOSFETs: Simulation results D.Esseni, University of Udine

  26. Fx1 Fx2 MSMC Implementation (integration of motion during free flights) (1) Constant electric field Fx in each section: No simple expressions for:  No analytical integration of the motion !!! D.Esseni, University of Udine

  27. Fx1 Fx2 MSMC Implementation (integration of motion during free flights) (2) No analytical integration of: Constant electric field Fx in each section: D.Esseni, University of Udine

  28. MSMC Implementation (integration of motion: validation) • Trajectories in the phase space validate the approach to the motion equation 2) 1) D.Esseni, University of Udine

  29. MSMC for pMOSFETs • Semi-analytical model for 2D holes • Implementation in a Monte Carlo tool • p-MOSFETs: Simulation results D.Esseni, University of Udine

  30. p-MOSFETs: MSMC Simulation results (Mobility calibration and validation) • Phonon and roughness parameters calibrated at 300k good agreement at different temperatures D.Esseni, University of Udine

  31. p-MOSFETs: MSMC Simulation results (IDS-VGS and ballisticity ratio) • Ballisticity ratios comparable to n-MOSFETs D.Esseni, University of Udine

  32. Conclusions: • 2D hole bandstructure is main the issue in the development of a MSMC for p-MOSFETs • New semi-analytical, non-parabolic, anisotropic bandstructure model and implementation in a self-consistent MSMC for p-MOSFETs • Results for mobility, on-currents, ballisticity ratios Future work: • Extension of the approach to different crystal orientations and strain D.Esseni, University of Udine

  33. MSMC for n-MOS transistors (3) (Effective Mass Approximation) • Development of a complete • MSMS simulator for n-MOSFETs • (L.Lucci et al., IEDM 2005, TED’07) Ball S VirtualSource D Scatt D.Esseni, University of Udine

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