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 luca.selmi@uniud.it

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

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

Decoupling lateral transport and transverse quantization ky 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

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

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

Multi subband Monte Carlo x 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

Schroedinger equation x 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

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

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

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

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

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

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

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

Transport with scattering ky 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

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

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

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

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

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

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

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

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