Towards Atomistic Simulations of the Electro-Thermal Properties of Nanowire Transistors Mathieu Luisier and Reto Rhyner Integrated Systems Laboratory ETH Zurich, Switzerland
Outline Motivation Electron Transport Simulation Approach Mobility Calculation in Si NW Phonon Transport Simulation Approach Anharmonic Ph-Ph Scattering Thermal Conductivity in Si NW Outlook and Conclusion
Motivation
Motivation: Transistor Evolution Source: Intel Source: Intel Source: IBM Planar to 3-D CS Scaling Ultimate FET? d 5nm L g 10nm
Motivation: Ballistic Transport? Si GAA NWFET d=3nm L g =5nm 5nm 5nm source Ballistic spectral current drain Spectral current with e-ph scatt. e-ph scattering still matters in L g <10nm Si FETs
Electron Transport
Electron/Hole Bandstructure Model Empirical Nearest-Neighbor Tight-Binding Method GOOD: Si Electron Bandstructure bulk CB and VB fitted extension to nanostructures atomistic description BAD: high computational effort empirical parametrization U Γ X Γ
Electron Transport Solve 1D/2D/3D Schrödinger equations Modified form to account for open boundary conditions Solution ingredients Atomic Orbitals s/s* Numerical Methods Non-equilibrium Green s Function (NEGF) or Wave Function (WF) Massive Parallelization p x d z 2 -r 2
GAA NW FET Simulation Objective: Electron-Phonon in Si NW FETs, extract e-ph limited mobility µ ph Approach: Tight-binding (sp 3 d 5 s*) description of the electron/hole properties Equilibrium phonon population Quantum transport with NEGF Model returns electron mobility of 1630 cm 2 /Vs for bulk Si @T=300K (no fitting parameter) Results and Impacts: Drain current reduction (larger in transistor ON-state) Change in the shape of the electrostatic potential x=<100> d=3nm L g =15nm
Mobility Extraction Technique R(L)=ΔV/I d (L) µ eff (top) N inv (top) Calculation Method R(L)=ΔV/I d (L) R(L)=R 0 +R ph (L) ρ 1D =dr(l)/dl µ ph =1/(q*ρ 1D *N inv ) 1. Perform self-consistent simulation of nanowire FET including electron-phonon scattering at a small drain voltage ΔV 2. Calculate the channel resistance R(L) for different gate lengths 3. Extract the 1-D inversion charge N inv at the top-of-the-barrier
Phonon-limited Channel Resistance <100> <110> R(L)=R 0 +R ph (L) ρ 1D =dr(l)/dl µ ph =1/(q*ρ 1D *P inv ) µ ph =1/(q*ρ 1D *N inv ) <111> Phonon-limited Channel resistance mobility of of d=3nm Si NW Si NW FETs FETs Mobility Comparison extraction of p-type based and on n-type dr/dl devices method <110>: Crystal best orientations electron <100>, and hole <110>, compromise <111>
Phonon Transport
Phonon Bandstructure Model Valence Force Field (VFF) Method with Empirical Potential Features: modified Keating Model 4 bond interactions extension to nanostructures 1 2 3 Δr ΔΘ Δr Δr Si Phonon Bandstructure Sim. Exp. ΔΘ ΔΘ 4 1. bond stretching 2. bond bending 3. cross bond stretching 4. coplanar bond bending
Phonon Transport Model Solve 1D/2D/3D lattice dynamics equations Modified form to account for open boundary conditions Solution ingredients Bond Interactions Δr ΔΘ Numerical Methods Non-equilibrium Green s Function (NEGF) or Wave Function (WF) Massive Parallelization ΔΘ ΔΘ
Anharmonic Phonon Decay In the case of ballistic transport, each phonon enters and leaves a simulation domain with the same energy: Energy x In reality, high energy phonon can decay into two particles with lower energy (ph-ph scattering): Energy E+E E E x
Anharmonic Model Verification Requirement: phonon transport model should be able to reproduce available experimental data Test: lattice thermal conductivity and mean free path for scattering of bulk Si
Application: Si NW Structures d 3 nm L The thermal current flowing through Si nanowires with a diameter d=3 nm, varying lengths L, a n d d i f f e r e n t c r o s s sections is simulated. <100> <110> <111>
Ballistic vs. Dissipative Thermal Current Thermal current through L=50 nm Si nanowires at different temperatures and for different transport directions. 1.7x 4x
Nanowire Thermal Conductivity Since phonon transport in the presence of anharmonic phonon decay is diffusive, a thermal resistivity ρ th and ph-ph limited conductivity κ th can be extracted.
Outlook and Conclusion
Coupled Electron-Phonon Transport Current status: electron/phonon transport solver Separate, but implemented in the same tool Electron Transport OMEN Phonon Transport Coupling through scattering self-energies Numerical implementation very complicated
Conclusion Important of e-ph scattering Even in ultra-short Si devices Electron transport Good reproduction of bulk µ e Modification of NW electrostatics 30% reduction of NW ON-current Phonon transport Good reproduction of bulk κ th Change in thermal current shape Reduction of thermal conductivity