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 Udine

2. FET switches: the workhorse of electronics
Gate
Substrate
Source
Drain
Current
SLONANO 2007
L.Selmi, University of Udine

3. FET Technology Boosters in the ITRS roadmap [public.itrs.net]
High-K
STRAIN
high μ
BULK
Materials & Architec.
Alternative Materials
Alternative Architectures
SLONANO 2007
L.Selmi, University of Udine

4. Decoupling lateral transport and transverse quantizationky L kx S D VD
VG1
VG2 VS E VS x
what is new in these devices ?
Strong size and bias induced quantization in the vertical direction (z)
Little or no quantization in the transport plane (x-y) but …..
SLONANO 2007
L.Selmi, University of Udine

5. 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 […]
Ideal device
Real device
SLONANO 2007
L.Selmi, University of Udine

6. 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
unificare con slide 6 semplificando
SLONANO 2007
L.Selmi, University of Udine

7. Multi subband Monte Carlox
VG2 z
Boltzman Transport Equation in transport direction Schrödinger Equation in 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
The Monte Carlo method solves the semi-classical Boltzmann Transport Equation without a-priori assumptions on the carrier distribution function in k-space.
Ideally suited for the simulation of off-equilibrium effects in nano MOSFETs where the transport is quasi-ballistic.
Advantage: modularity with respect to the implementation of new models for the particle dynamics or for the collision term.
Disadvantage: most quantum-mechanical effects are not intrinsically taken into account (but fully quantum-mechanical are still unpractical)
Compromise: combine a partial treatment of quantization effects with the Monte Carlo solution of the semi-classical BTE
Claim: today the MC method features the best trade-off between accuracy and numerical efficiency for device simulation z
SLONANO 2007
L.Selmi, University of Udine

8. Schroedinger equationx z y
VG2
Subband “j” VD
Subband “i”
VG1 X
SchrÖdinger-like equation:
Energy dispersion versus k:
Three relevant masses;
My is the quantization mass that affect the eigenvalues at each section;
Mx and mz are the in-plane or transport masses that afect the carrier velocity and the 2D density of states, hence the scattering rates.
my, mx, mz expressed in terms of mt and ml of bulk crystal
Force:
SLONANO 2007
L.Selmi, University of Udine

9. 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 implemented Si, Ge …
SLONANO 2007
L.Selmi, University of Udine

10. Extraction of band parameters
For a given device:
parametric representation of the bands at a given bias
extraction of eff. masses
below 5 nm parameters change. Quantization is becoming very relevant, as confirmed by threshold voltage and mobility data.
UTB silicon (Tsi=5nm), (001)
Full Band LCBB calculation
SLONANO 2007
L.Selmi, University of Udine

11. BTE in quantized systems
A BTE for each sub-band:
: sub-band index
Dim(K) <3
Sub-bands are coupled by inter-subband scattering
Degeneration implemented by rejecting the scattering according to the occupation of the final state
ingredienti campo, scattering rates
SLONANO 2007
L.Selmi, University of Udine

12. 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
MATRIX ELEMENT SR
SLONANO 2007
L.Selmi, University of Udine

13. 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
SLONANO 2007
L.Selmi, University of Udine

14. Degeneration in thin film SOIVD
VG1 x z
VG2 VS ky kx k VS
degeneration plays a major role UTB MOSFETs
SLONANO 2007
L.Selmi, University of Udine

15. L.Selmi, University of Udine
Ballistic transport ky
DG SOI, NS/D=5 1020,
EOT 0.7nm, Lg=14nm,
Tsi=4nm kx S D
Phonon scattering in source and drain,
no scattering in the channel
transport plane (x-y)
SLONANO 2007
L.Selmi, University of Udine

16. Transport with scatteringky
DG SOI, NS/D=5 1020,
EOT 0.7nm, Lg=14nm,
Tsi=4nm kx S D
Phonon scattering in source and drain,
Phonon, Surface roughness and Tsi Fluctuations in the channel
transport plane (x-y)
SLONANO 2007
L.Selmi, University of Udine

17. 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)
SLONANO 2007
L.Selmi, University of Udine

18. Transport in biax. strained-Si devices
QUANTIZATION DIRECTION
TRANSPORT DIRECTION
essderc
SLONANO 2007
L.Selmi, University of Udine 18 18

19. Mobility in biax. strained-Si devices
CB=0.67x [eV] [Rashed, IEDM 1995]
SLONANO 2007
L.Selmi, University of Udine 19 19

20. Extension to nanowire FETs
SLONANO 2007
L.Selmi, University of Udine

21. L.Selmi, University of Udine
What are we missing ?
Surface roughness / interface effects
Tunneling through the Source barrier
Scattering mechanisms
Atomistic effects
source
3 nm
drain
gate
oxide
SLONANO 2007
L.Selmi, University of Udine

22. L.Selmi, University of Udine
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
DA RIVEDERE PIU’ AVANTI
SLONANO 2007
L.Selmi, University of Udine

23. L.Selmi, University of Udine
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
SLONANO 2007
L.Selmi, University of Udine

24. Device modeling approaches
Fundamental Theory of transport
Ballistic transport
Velocity overshoot
Availability
Vertical quantization
Lateral quantization
Degeneration
Scattering
Full Band
Sub-threshold
(Density gradient correction)
Drift Diffusion
Near Equilibrium
Comm
Possible
Possible NO
m, vs NO NO
YES
(Density gradient correction)
Hydrodynamic
Displaced Maxwellian
Possible
Possible
Comm NO
m, vs T
YES NO
YES
Classical (3D) Monte Carlo
(Effective potential correction)
(S/D tunneling correction)
Univ /
Comm
Boltzmann
Transport eq. NO
YES
YES
YES
YES
Possible
Multi Sub Band Monte Carlo
(S/D tunneling correction)
BTE 2D + SE 1D
YES
YES
YES
YES NO
Possible
Univ
Included
Green’s Function
Univ
YES
YES
Phon
YES
YES
YES
Schrodinger eg.
Included
Included
SLONANO 2007
L.Selmi, University of Udine