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

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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)


Support of the physically based transport modelling to the generalized scaling scenario
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


Multisubband monte carlo msmc approach for mos transistors

y generalized scaling scenario

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


Multisubband monte carlo msmc for n mos transistors electron inversion layers
Multisubband Monte Carlo generalized scaling scenario (MSMC) for n-MOS transistors(electron inversion layers)

D.Esseni, University of Udine


Msmc for n mos transistors 1 effective mass approximation

y generalized scaling scenario

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


y generalized scaling scenario

x

z

MSMC for n-MOS transistors (2) (Effective Mass Approximation)

VG2

VD

VG1

Energy dispersion:

Driving force:

Velocity:

D.Esseni, University of Udine


T ransport in the msmc approach 2d carrier gas
T generalized scaling scenarioransport in the MSMC approach(2D carrier gas)

Force:

Band structure

Kinematics:

Rates of scattering

D.Esseni, University of Udine


Bandstructure for a hole inversion layer
Bandstructure for a hole inversion layer: generalized scaling scenario

  • 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


y generalized scaling scenario

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


Msmc for p mosfets
MSMC for generalized scaling scenariopMOSFETs

  • 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


Semi analytical model for 2d holes
Semi-analytical model for 2D holes generalized scaling scenario

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


Semi analytical model for 2d holes1
Semi-analytical model for 2D holes generalized scaling scenario

1) Bottom of the 2D subbands (the relatively easy part)

D.Esseni, University of Udine


Semi-analytical model for 2D holes generalized scaling scenario(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


Semi analytical model for 2d holes2

2) Energy dependence on k (the generalized scaling scenarioby no means easy part)

Semi-analytical model for 2D holes

D.Esseni, University of Udine


Semi-analytical model for 2D holes generalized scaling scenario(energy dispersion is anisotropic)

k·presults

Si(100)

  • Strongly anisotropic

  • Periodic of p/2

D.Esseni, University of Udine


k·p generalized scaling scenarioresults

Semi-analytical model for 2D holes(energy dispersion is non-parabolic)

Analytical dispersion in the symmetry directions:

D.Esseni, University of Udine


Semi-analytical model for 2D holes generalized scaling scenario(angular dependence)

Fourier series expansion:

A, B, C calculated with no additional fitting parameters:

D.Esseni, University of Udine


Semi analytical model for 2d holes3

Semi-analytical model for 2D holes

D.Esseni, University of Udine


Msmc for p mosfets1
MSMC for generalized scaling scenariopMOSFETs

  • Semi-analytical model for 2D holes

    • Calibration and validation

  • Implementation in a Monte Carlo tool

  • p-MOSFETs: Simulation results

D.Esseni, University of Udine


Calibration of the semi-analytical model generalized scaling scenario(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


Calibration of the semi-analytical model generalized scaling scenario(non parabolicity along symmetry directions)

Si(100), Fc=0.3MV/cm

  • Good results with the proposed non parabolic expression:

D.Esseni, University of Udine


Validation of the semi-analytical model generalized scaling scenario(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


Si(001) generalized scaling scenario

Validation of the semi-analytical model(2D Density Of States - DOS)

Acoustic Phonon scattering:

D.Esseni, University of Udine


Validation of the semi-analytical model generalized scaling scenario(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


Msmc for p mosfets2
MSMC for generalized scaling scenariopMOSFETs

  • 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


Msmc implementation integration of motion during free flights 1

F generalized scaling scenariox1

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


Msmc implementation integration of motion during free flights 2

F generalized scaling scenariox1

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


MSMC Implementation generalized scaling scenario(integration of motion: validation)

  • Trajectories in the phase space validate the approach to the motion equation

2)

1)

D.Esseni, University of Udine


Msmc for p mosfets3
MSMC for generalized scaling scenariopMOSFETs

  • Semi-analytical model for 2D holes

  • Implementation in a Monte Carlo tool

  • p-MOSFETs: Simulation results

D.Esseni, University of Udine


p- generalized scaling scenarioMOSFETs: MSMC Simulation results (Mobility calibration and validation)

  • Phonon and roughness parameters calibrated at 300k good agreement at different temperatures

D.Esseni, University of Udine


p- generalized scaling scenarioMOSFETs: MSMC Simulation results (IDS-VGS and ballisticity ratio)

  • Ballisticity ratios comparable to n-MOSFETs

D.Esseni, University of Udine


Conclusions
Conclusions: generalized scaling scenario

  • 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


MSMC generalized scaling scenario 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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