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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)

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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

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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:

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

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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

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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

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

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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

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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 )

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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

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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

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

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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

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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)

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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)

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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:

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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)

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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

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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

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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

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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

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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)

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

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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 ]

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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

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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 !!!

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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:

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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)

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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

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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

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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

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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

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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)

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