Multisubband Monte Carlo simulations for pMOSFETs

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

• Band-structure calculation and optimization
• Carrier velocity and maximum attainable current
  IBL
• Scattering rates, hence real current 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

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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
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Multisubband Monte Carlo (MSMC) for n-MOS
transistors(electron inversion layers)
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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 ml of the
  bulk crystal

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MSMC for n-MOS transistors (2) (Effective Mass
Approximation)
VG2
VD
VG1
Energy dispersion
Driving force
Velocity
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Transport in the MSMC approach(2D carrier gas)
Force
Band structure
Kinematics
Rates of scattering
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Bandstructure for a hole inversion layer

• Single-band effective mass approx. is not viable
• Three almost degenerate bands at the G point
• Spin-orbit interaction

kp method for hole inversion layers
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kp method for inverted layers
VG2
Differently from EMA one eigenvalue problem for
each in-plane (kx,ky)
VD
VS

• Finite differences method
• v section and v in-plane k
• 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
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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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Semi-analytical model for 2D holes
Three groups of subbands

• Calculation of the eigenvalues ev,i
• New analytical expression for in-plane energy
  Ep(k)

kp results
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Semi-analytical model for 2D holes
1) Bottom of the 2D subbands (the relatively
easy part)
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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
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Semi-analytical model for 2D holes
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Semi-analytical model for 2D holes(energy
dispersion is anisotropic)
kp results
Si(100)

• Strongly anisotropic
• Periodic of p/2

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Semi-analytical model for 2D holes (energy
dispersion is non-parabolic)
Analytical dispersion in the symmetry directions
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Semi-analytical model for 2D holes(angular
dependence)
Fourier series expansion
A, B, C calculated with no additional fitting
parameters
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Semi-analytical model for 2D holes
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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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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
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Calibration of the semi-analytical model (non
parabolicity along symmetry directions)
Si(100), Fc0.3MV/cm

• Good results with the proposed non parabolic
  expression

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Si(001)

• Calculation conditions
• Triangular well FC0.3 MV/cm
• E-e075 meV

• The model seems to grasp fairly well the complex,
  anisotropic energy dispersion

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

• Acoustic Phonon scattering

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Validation of the semi-analytical model (average
hole velocity vx, vy)

• Analytical Model

? Analytical expression for
Pinv5.6x1012cm-2
Average 0,p/4

• kp results (numerical
• determination)

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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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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 !!!
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MSMC Implementation (integration of motion
during free flights) (2)
No analytical integration of
Constant electric field Fx in each section
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MSMC Implementation (integration of motion
validation)

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

2)
1)
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MSMC for pMOSFETs

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

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

• Ballisticity ratios comparable to n-MOSFETs

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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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MSMC for n-MOS transistors (3) (Effective Mass
Approximation)

• Development of a complete
• MSMS simulator for n-MOSFETs
• (L.Lucci et al., IEDM 2005, TED07)

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