An efficient, mixed semiclassicalquantum mechanical model to simulate planar and wire nanotransistor

1
An efficient, mixed semiclassical/quantum
mechanical model to simulate planar and wire
nano-transistors

• L.Selmi, P.Palestri, D.Esseni,
• L.Lucci, M.De Michielis
• DIEGM-IUNET, University of Udineluca.selmi_at_uniud.
  it

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FET switches the workhorse of electronics
3
FET Technology Boostersin the ITRS roadmap
public.itrs.net
High-K
STRAIN
high µ
BULK
Materials Architec.
Alternative Materials
Alternative Architectures
4
Decoupling lateral transport and transverse
quantization
ky
kx
S
D
E
VS
x
Strong size and bias induced quantization in the
vertical direction (z)
Little or no quantization in the transport plane
(x-y) but ..
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Carrier motion in the channel
Quasi ballistic transport few scatterings
determine the current
Ballistic transport
Source
ITRS 2005 Edition
Modeling and simulation needs to be enhanced to
deal with the key innovations requested by the
PIDS section, including enhanced mobility, high-k
dielectrics, metal gate electrodes, non
classical CMOS
Real device
Ideal device
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nano-FET modeling approaches

• Drift Diffusion or Hydrodynamic models
• commercial tools
• inadequate for nano-FETs
• Monte Carlo solver of the 3D BTE
• far from equilibrium transport
• no vertical or lateral quantization effects
• N.E.G.F.
• 2D quantization in real space
• computationally heavy
• difficult to include all relevant scattering
  mech.
• Multi-Subband Monte Carlo (MSMC)
• accurate treatment of vertical quantization
• efficient semiclassical treatment of far from
  equilibrium transport
• computationally affordable

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Multi subband Monte Carlo
x
VG2
z

• Boltzman Transport Equation in transport
  directionSchrödinger Equationin quantization
  direction
• Solve 1D Schrödinger equation in each section of
  the device
• Solve the BTE in each subband
• The solution of the BTEs are coupled by
  scatterings

VS
VD
VG1
z
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Schroedinger equation
VG2
Subband j
VD
Subband i
VG1

• SchrÖdinger-like equation

• Energy dispersion versus k

• my, mx, mz expressed in terms of mt and ml of
  bulk crystal

• Force

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Band Structure (electrons)
Effective mass approximation

• Non-parabolic elliptical bands
• Any number of ?, L, ? valleys
• Strain additional valley splitting
• Arbitrary crystal orientation
• Subbands with different quantization and
  transport masses
• Various semiconductor materials implementedSi,
  Ge

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Extraction of band parameters

• For a given device
• parametric representation of the bands at a
  given bias
• extraction of eff. masses

UTB silicon (Tsi5nm), (001) Full Band LCBB
calculation
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BTE in quantized systems

• A BTE for each sub-band

? sub-band index
Dim(K) lt3

• Sub-bands are coupled by inter-subband
  scattering
• Degeneration implemented by rejecting the
  scattering according to the occupation of the
  final state

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Scattering Theory of the 2D gas

• Phonons (Price, 1980)
• Ionized impurities (Ando, 1983)
• Surface roughness (Esseni, 2003)
• S.O.phonons in high-k materials

• Matrix elements and scattering rates computed
  from eigenvalues and wave-functions
• Fermi Golden Rule
• Anisotropic scattering (SR, II) is screened with
  the dielectric function of the 2D electron gas

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Model flowchart
Poisson Equation (2D)
electron density n(x,z)
Potential V(x,z)
MonteCarlo (BTE)
Schrödinger equation (1D)
Eigenstates Yn,n,i(z) En,n,i
Scattering Rates
Scattering Theory 2D elecron gas
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Degeneration in thin film SOI

• degeneration plays a major role UTB MOSFETs

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Ballistic transport
ky
DG SOI, NS/D5 1020, EOT 0.7nm, Lg14nm, Tsi4nm
kx
S
D
Phonon scattering in source and drain, no
scattering in the channel
transport plane (x-y)
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Transport with scattering
ky
DG SOI, NS/D5 1020, EOT 0.7nm, Lg14nm, Tsi4nm
kx
S
D
Phonon scattering in source and drain, Phonon,
Surface roughness and Tsi Fluctuations in the
channel
transport plane (x-y)
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Mobility effect of surface orientation
Lucci, IEEE T-ED, p.1156, 2007

• Same model parameters of (001) and (111)
  orientations
• Adjustment of SR spectrum for (110)

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Transport in biax. strained-Si devices
QUANTIZATION DIRECTION
TRANSPORT DIRECTION
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Mobility in biax. strained-Si devices
?CB0.67x eV Rashed, IEDM 1995
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Extension to nanowire FETs
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What are we missing ?

• Surface roughness / interface effects
• Tunneling through the Source barrier
• Scattering mechanisms
• Atomistic effects

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Conclusions

• A new Monte Carlo code based on the theory of the
  two dimensional carrier gas has been developed
  for n- and p-type MOSFETs
• Quasi ballistic transport in ultra thin body DG
  SOI devices has been investigated
• Importance of a correct modeling of scattering in
  the channel and of carrier degeneration has been
  highlighted
• The modularity of the code and the parametric
  description of the band structure make the
  simulator suitable for extensions to devices with
  different channel material and crystal orientation

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Acknowledgements

• EU Nestor (5FP), SiNano (6FP), PullNano (6FP)
  projects
• Italian FIRB 2001 and PRIN 2004 projects
• MS and PhD students Nicola Barin, Marco
  Braccioli, Simone Eminente, Andrea Ghetti, Davide
  Ponton, Ivan Riolino, Massimiliano Zilli and all
  the IU.NET ARCES partners

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Device modeling approaches
Fundamental Theory of transport
Ballistictransport
Velocity overshoot
Verticalquantization
Lateralquantization
Degeneration
Scattering
Full Band
Sub-threshold
Availability
(Densitygradientcorrection)
Near Equilibrium
Drift Diffusion
Possible
NO
NO
m, vs
NO
YES
Possible
Comm
(Densitygradientcorrection)
DisplacedMaxwellian
m, vsT
Possible
Possible
Hydrodynamic
NO
YES
NO
YES
Comm
Classical (3D) Monte Carlo
(S/D tunnelingcorrection)
(Effectivepotentialcorrection)
Univ / Comm
Boltzmann Transport eq.
NO
YES
YES
YES
YES
Possible
(S/D tunnelingcorrection)
Multi Sub Band Monte Carlo
BTE 2D SE 1D
YES
YES
YES
YES
NO
Possible
Univ
Included
Greens Function
YES
YES
YES
YES
YES
Phon
Univ
Schrodinger eg.
Included
Included