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Multisubband Monte Carlo simulations for p- MOSFETsPowerPoint Presentation

Multisubband Monte Carlo simulations for p- MOSFETs

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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)

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

y generalized scaling scenario

x

z

Multisubband Monte Carlo (MSMC) approach for MOS transistorsVG2

- 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 generalized scaling scenario (MSMC) for n-MOS transistors(electron inversion layers)

D.Esseni, University of Udine

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 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: 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 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 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 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

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

Semi-analytical model for 2D holesD.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

- Bottom of the 2D subbands generalized scaling scenario
- Energy dependence on k

D.Esseni, University of Udine

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 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

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

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 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: 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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