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 Acknowledg: NoE. SINANO (EU), PullNano (EU)

SINANO Workshop, 2007 D.Esseni, University of Udine Support of the physically based transport modelling to the generalized scaling scenario Band-structure calculation and optimization: –Carrier velocity and maximum attainable current I BL –Scattering rates, hence real current I ON and BR=(I ON /I BL ) Link the properties and advantages of: Mobility in Long MOSFETs (Uniform transport) I ON in nano-MOSFETs (far from equilibr. transport) Provide sound interpretation to characterization

SINANO Workshop, 2007 D.Esseni, University of Udine VSVS 1D SchrödingerSolve 1D Schrödinger equation in the Z direction   i (x) along the channel Multisubband Monte Carlo (MSMC) approach for MOS transistors VDVD V G1 V G2 z x z y Driving Force in each subband:

SINANO Workshop, 2007 D.Esseni, University of Udine Multisubband Monte Carlo (MSMC) for n-MOS transistors (electron inversion layers)

SINANO Workshop, 2007 D.Esseni, University of Udine MSMC for n-MOS transistors (1) ( Effective Mass Approximation) Schr Ö dinger-like equation : Energy dispersion versus k: VDVD V G1 V G2 m x, m y, m z expressed in terms of m t and m l of the bulk crystal x z y Subband “i” Subband “j” X

SINANO Workshop, 2007 D.Esseni, University of Udine Energy dispersion: Driving force: Velocity: MSMC for n-MOS transistors (2) ( Effective Mass Approximation) VDVD V G1 V G2 x z y

SINANO Workshop, 2007 D.Esseni, University of Udine Transport in the MSMC approach (2D carrier gas) Band structure Kinematics: Rates of scattering Force:

SINANO Workshop, 2007 D.Esseni, University of Udine Bandstructure for a hole inversion layer: Single-band effective mass approx. is not viable: –Three almost degenerate bands at the   point –Spin-orbit interaction k·p method for hole inversion layers

SINANO Workshop, 2007 D.Esseni, University of Udine VDVD V G1 V G2 Finite differences method: √ section and √ in-plane k: eigenvalue problem 6N z x6N z Entirely numerical description of the energy dispersion   Computationally very heavy for simulations of pMOSFETs x z y k·p method for inverted layers: VSVS   Simplified models for energy dispersion of 2D holes Differently from EMA: one eigenvalue problem for each in-plane (k x,k y )

SINANO Workshop, 2007 D.Esseni, University of Udine 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

SINANO Workshop, 2007 D.Esseni, University of Udine Three groups of subbands: Semi-analytical model for 2D holes 1.Calculation of the eigenvalues  v,i 2.New analytical expression for in-plane energy E p (k) k·p results

SINANO Workshop, 2007 D.Esseni, University of Udine Semi-analytical model for 2D holes 1) Bottom of the 2D subbands (the relatively easy part)

SINANO Workshop, 2007 D.Esseni, University of Udine m,z fitted using triangular wells Schrödinger equation as in EMA (m z ): Semi-analytical model for 2D holes (bottom of the 2D subbands) Good agreement also in square well

SINANO Workshop, 2007 D.Esseni, University of Udine Semi-analytical model for 2D holes 2) Energy dependence on k (the by no means easy part)

SINANO Workshop, 2007 D.Esseni, University of Udine Semi-analytical model for 2D holes (energy dispersion is anisotropic) k·p results 1.Strongly anisotropic 2.Periodic of   Si(100)

SINANO Workshop, 2007 D.Esseni, University of Udine Semi-analytical model for 2D holes (energy dispersion is non-parabolic) k·p results Analytical dispersion in the symmetry directions:

SINANO Workshop, 2007 D.Esseni, University of Udine A, B, C calculated with no additional fitting parameters: Fourier series expansion: Semi-analytical model for 2D holes (angular dependence)

SINANO Workshop, 2007 D.Esseni, University of Udine Semi-analytical model for 2D holes 1) Bottom of the 2D subbands 2) Energy dependence on k

SINANO Workshop, 2007 D.Esseni, University of Udine MSMC for pMOSFETs Semi-analytical model for 2D holes –Calibration and validation Implementation in a Monte Carlo tool p-MOSFETs: Simulation results

SINANO Workshop, 2007 D.Esseni, University of Udine m,z fitted using triangular wells Schrödinger equation in the EMA (m z ): Calibration of the semi-analytical model (bottom of the 2D subbands) Good agreement also in square well

SINANO Workshop, 2007 D.Esseni, University of Udine Good results with the proposed non parabolic expression: Calibration of the semi-analytical model (non parabolicity along symmetry directions) Si(100), F c =0.3MV/cm

SINANO Workshop, 2007 D.Esseni, University of Udine Calculation conditions: Triangular well: F C =0.3 MV/cm E-  0 =75 meV Si(001) The model seems to grasp fairly well the complex, anisotropic energy dispersion Validation of the semi-analytical model (overall energy dependence on k)

SINANO Workshop, 2007 D.Esseni, University of Udine Acoustic Phonon scattering: Validation of the semi-analytical model (2D Density Of States - DOS) Si(001)

SINANO Workshop, 2007 D.Esseni, University of Udine Analytical Model: k·p results (numerical determination): Validation of the semi-analytical model (average hole velocity: v x, v y )  Analytical expression for: Average: [0,   P inv =5.6x10 12 [cm -2 ]

SINANO Workshop, 2007 D.Esseni, University of Udine MSMC for pMOSFETs Semi-analytical model for 2D holes Implementation in a Monte Carlo tool –Integration of the motion equation p-MOSFETs: Simulation results

SINANO Workshop, 2007 D.Esseni, University of Udine MSMC Implementation (integration of motion during free flights) (1) Constant electric field F x in each section: F x1 F x2 No simple expressions for:  No analytical integration of the motion !!!

SINANO Workshop, 2007 D.Esseni, University of Udine MSMC Implementation (integration of motion during free flights) (2) Constant electric field F x in each section: F x1 F x2 No analytical integration of:

SINANO Workshop, 2007 D.Esseni, University of Udine MSMC Implementation (integration of motion: validation) Trajectories in the phase space validate the approach to the motion equation 1) 2)

SINANO Workshop, 2007 D.Esseni, University of Udine MSMC for pMOSFETs Semi-analytical model for 2D holes Implementation in a Monte Carlo tool p-MOSFETs: Simulation results

SINANO Workshop, 2007 D.Esseni, University of Udine p-MOSFETs: MSMC Simulation results (Mobility calibration and validation) Phonon and roughness parameters calibrated at 300k  good agreement at different temperatures

SINANO Workshop, 2007 D.Esseni, University of Udine p-MOSFETs: MSMC Simulation results ( I DS -V GS and ballisticity ratio ) Ballisticity ratios comparable to n-MOSFETs

SINANO Workshop, 2007 D.Esseni, University of Udine 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

SINANO Workshop, 2007 D.Esseni, University of Udine Ball Scatt Virtual Source S D 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)