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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)

- 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

x

z

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

D.Esseni, University of Udine

y

x

X

z

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

x

z

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

VG2

VD

VG1

Energy dispersion:

Driving force:

Velocity:

D.Esseni, University of Udine

Force:

Band structure

Kinematics:

Rates of scattering

D.Esseni, University of Udine

- 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

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

- 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

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

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

D.Esseni, University of Udine

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

2) Energy dependence on k (the by no means easy part)

D.Esseni, University of Udine

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

k·presults

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(angular dependence)

Fourier series expansion:

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

D.Esseni, University of Udine

- Bottom of the 2D subbands
- Energy dependence on k

D.Esseni, University of Udine

- 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(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(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(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)

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(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

- 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

Fx1

Fx2

Constant electric field Fx in each section:

No simple expressions for:

No analytical integration of the motion !!!

D.Esseni, University of Udine

Fx1

Fx2

No analytical integration of:

Constant electric field Fx in each section:

D.Esseni, University of Udine

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

- Semi-analytical model for 2D holes
- Implementation in a Monte Carlo tool
- p-MOSFETs: Simulation results

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

D.Esseni, University of Udine

p-MOSFETs: MSMC Simulation results (IDS-VGS and ballisticity ratio)

- Ballisticity ratios comparable to n-MOSFETs

D.Esseni, University of Udine

- 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 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