Eric Zhu1, Leo Liu1, Hong Guo1,2 2017/4/21 Device Modeling from Atomistic First Principles: theory of the nonequilibrium vertex correction Eric Zhu1, Leo Liu1, Hong Guo1,2 1 Nanoacademic Technologies Inc. Brossard, QC J4Z 1A7, Canada 2 Dept. of Physics, McGill Univ., Montreal, Quebec, H3A 2T8 Canada Introduction: NEGF-DFT; 4 critical issues: disorder averaging, band gap, large sizes, verification; Two examples: localized doping in Si nanoFET; disorder scattering in MRAM; Summary. Continuum model Atomic model 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

Goal: simulate a transistor from atomic first principles 2017/4/21 Goal: simulate a transistor from atomic first principles Picture from Taur and Ning, Fundamentals of Modern VLSI Devices ~100nm ~10nm Doping & disorder, Band gaps, Large sizes, Accuracy. Other physics: phonons, magnons, photons, correlations… Current: L=22nm Next: L=16nm (10 nm)3 chunk of Si has ~64,000 atoms. DFT: ~1,000 atoms 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

… and many other systems with different materials 2017/4/21 … and many other systems with different materials ? This talk: In any real device made of any real material, there is a degree of disorder. Such disorder impacts device operation in serious ways. How do we compute these effects from first principles? Can we calculate ? 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

Ex 1: Dopant fluctuation gives rise to device-to-device variability 2017/4/21 Ex 1: Dopant fluctuation gives rise to device-to-device variability Huge device to device variability. If every transistor behaves differently, difficult to design a circuit. F.L. Yang et al., in VLSI Technol. Tech. Symp. Dig., pp. 208, June 2007. 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

NEGF5, Jyvaskyla, Finland 2017/4/21 Ex 2: roughness scattering increases resistance of Cu interconnects With Daniel Gall of RPI. $: SRC 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

NEGF5, Jyvaskyla, Finland Ex. 3: disorder effect in topological insulator Bi2Se3 Experiment (Hasan etal) Ab initio (Zhao etal) Conductance: Wang, Hu, H.G. PRB 85, 241402 (2012) Calculated spin direction Top surface Bottom surface Zhao, H.G. etal Nano Lett. 11, 2088 (2011). 2012-08-29 NEGF5, Jyvaskyla, Finland

Can we calculate realistic device parameters? 2017/4/21 Can we calculate realistic device parameters? Semi-empirical device modeling, 10,000 to 100,000 atoms device parameters TCAD atomic simulations materials, chemistry, physics quantum mechanics Physics device modeling < 5nm (1000 atoms) New science engineering Quantitative prediction of quantum transport from atomic first principles without any parameter 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

Two ingredients in theory: Quantum statistics and Hamiltonian 2017/4/21 Two ingredients in theory: Quantum statistics and Hamiltonian H = Hleads + Hdevice + Hcoupling Calculating electric current flow driven by a finite bias voltage is a non-equilibrium problem, it is important to correctly determine non-equilibrium statistics of the device scattering region that is embedded in an open environment. A second consideration is the determination of device Hamiltonian H. H provides the energy levels of the device. How to fill these levels is given by the non-equilibrium statistics. Keldysh non-equilibrium Green’s function (NEGF) is a natural approach to determine the non-equilibrium statistics. One may also evolve a non-equilibrium density matrix from some equilibrium initial condition. What kind of H to use is an issue of taste and approximation: effective mass, kp, EH,TB, DFTB, HF, DFT, GW, higher-functionals, QMC, CI… In the end, one has to compare to experimental data without adjusting free parameters. 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

Method - NEGF-DFT: non-equilibrium density matrix 2017/4/21 Method - NEGF-DFT: non-equilibrium density matrix NEGF-DFT ‘DFT’: density functional theory NEGF: Keldysh nonequilibrium Green’s function DFT NEGF-DFT `DFT’ in NEGF-DFT is not the usual ground state DFT: density matrix of NEGF-DFT is constructed at non-equilibrium. No variational solution. Jeremy Taylor, Hong Guo and Jian Wang, Phys. Rev. B 63, 245407 (2001). M. Brandbyge, J.-L. Mozos, P. Ordejon, J. Taylor, and K. Stokbro, PRB 65, 165401 (2002). 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

1. Within NEGF-DFT: solving the disorder averaging problem 2017/4/21 1. Within NEGF-DFT: solving the disorder averaging problem Doping and disorder scattering from atomic principles Acknowledgements: Dr. Youqi Ke, Dr. Ke Xia, Dr. Ferdows Zahid, Dr. Eric Zhu, Dr. Lei Liu, Dr. Yibin Hu NEGF-DFT/CPA-NVC Drs. Eric Zhu, Leo Liu, and Yibin Hu: development of the NEGF-DFT/CPA-NVC first principles package Nanodsim (nano-device-simulator) – Nanoacademic Technologis Inc. (www.nanoacademic.com). Youqi Ke, Ke Xia and Hong Guo PRL 100, 166805 (2008); Youqi Ke, Ke Xia and Hong Guo, PRL 105, 236801 (2010); Ferdows Zahid, Youqi Ke, Daniel Gall and Hong Guo, PRB 81, 045406 (2010); Eric Zhu, Lei Liu and Hong Guo, preprint (2012). 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

A tough problem of atomic calculations: disorder scattering 2017/4/21 A tough problem of atomic calculations: disorder scattering Generating many configurations, compute each, and average result very time consuming (Small x, large N) T. Dejesus, Ph.D thesis, McGill University, 2002. For any theoretical calculation, disorder averaging must be done. How to do it in atomistic calculations at non-equilibrium? 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

To build intuition, let’s solve a toy problem exactly 2017/4/21 To build intuition, let’s solve a toy problem exactly 1 2 1D tight binding chain nearest neighbor coupling on-site energy 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

Self-energies for the leads: 2017/4/21 Self-energies for the leads: 1 2 SL SR How to handle half-infinite chain ? Self energy: The problem is reduced to 3 sites plus self-energies 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

Physical quantities and NEGF 2017/4/21 Physical quantities and NEGF Express physical quantities in terms of NEGF: average over disorder configurations L R The problem is reduced to calculate disorder averaged NEGF 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

NEGF5, Jyvaskyla, Finland 2017/4/21 1 2 SL SR Disorder average can be done exactly for the 3-site model 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

Exact solution of the 3-site toy model: 2017/4/21 Exact solution of the 3-site toy model: In general, the number of configuration is 2N (N is the number of disorder sites). It is impossible to enumerate and compute all configurations for large N. We need a better “statistical approach” Coherent potential Approx. 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

CPA - well established formalism 2017/4/21 CPA - well established formalism When there are impurities, translational symmetry is broken. Coherent Potential Approximation (CPA) is an effective medium theory that averages over the disorder and restores the translational symmetry. So, an atomic site has x% chance to be occupied by A, and (1-x)% chance by B. P. Soven, Phys. Rev. 156, 809 (1967). B. Velicky, Phys. Rev. 183 (1969). Rev. Mod. Phys. 46, 466 (1974) 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

NEGF5, Jyvaskyla, Finland 2017/4/21 CPA: CPA picture: effective media and are solved from CPA equation Implementation: needs a method that does one atom at a time: LMTO, KKR, etc.. 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

Non-equilibrium density matrix: nonequilibrium vertex 2017/4/21 Non-equilibrium density matrix: nonequilibrium vertex NEGF-DFT Average over random disorder: X specular part diffusive part Take Home message #1: multiple disorder scattering at non-equilibrium is solved by the non-equilibrium vertex correction theory (NVC) and implemented in NEGF-DFT software Nanodsim. Youqi Ke, Ke Xia and Hong Guo PRL 100, 166805 (2008) 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

Essence of Nonequilibrium Vertex Correction (NVC) 2017/4/21 Essence of Nonequilibrium Vertex Correction (NVC) X Conventional vertex correction, i.e. that appears in computing Kubo formula in disordered metal, is done at equilibrium. NVC is done at non-equilibrium: it is related not only to multiple impurity scattering, but also to the non-equilibrium statistics of the device scattering region. Implementation: LMTO with atomic sphere approximation, plus CPA and NVC, within NEGF-DFT. 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

NVC Equation: some complicated technical details 2017/4/21 NVC Equation: some complicated technical details Youqi Ke, Ke Xia and Hong Guo PRL 100, 166805 (2008). 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

Consistency check: CPA-NVC identity 2017/4/21 Consistency check: CPA-NVC identity NVC CPA CPA The CPA-NVC identity can also be proved analytically at non-equilibrium: CPA and NVC are consistent approximations (Eric Zhu and H.G., 2012). The identity is tested numerically: strong check of the code. 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

NVC solution for the 3-site toy model 2017/4/21 NVC solution for the 3-site toy model and are solved from NVC equation 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

Comparison for the 3-site toy model: 2017/4/21 Comparison for the 3-site toy model: specular part diffusive part Excellent ! 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

NEGF5, Jyvaskyla, Finland 2017/4/21 Non-toy system: At equilibrium, fluctuation-dissipation theorem holds. Left hand side has NVC; right hand side does not. This gives a very strict check to the NVC formalism as well as to the numerical implementation. no NVC NVC exact 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

2. Within NEGF-DFT: solving the band gap problem 2017/4/21 2. Within NEGF-DFT: solving the band gap problem The band gap problem … Acknowledgements: Dr. Youqi Ke, Dr. Wei Ji, Mathieu Cesar, Dr. Eric Zhu, Dr. Lei Liu, Dr. Zetian Mi, Dr. Ferdows Zahid NEGF-DFT-CPA-NVC 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

The band gap problem of local functionals in DFT 2017/4/21 The band gap problem of local functionals in DFT DFT calculation of band gaps: MBJ computation time is ~LDA 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

Some relevant band gaps for transistors materials: 2017/4/21 Some relevant band gaps for transistors materials: 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

Some relevant effective masses 2017/4/21 Some relevant effective masses Take home message 2: the band gap problem is practically resolved by MBJ semi-local exchange within LMTO implementation of NEGF-DFT. 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

NEGF5, Jyvaskyla, Finland 2017/4/21 Experimental data Calculated data MBJ potential + CPA: works. Good agreement with experimental data. 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

3. Within NEGF-DFT: solving the large size problem 2017/4/21 3. Within NEGF-DFT: solving the large size problem Solving large problems from self-consistent first principles. Acknowledgements: Dr. Eric Zhu, Dr. Lei Liu (Nanoacademic Technologies Inc.) Dr. Yibin Hu, Mohammed Harb, Vincent Michaud-Roux (McGill) J. Maassan, E. Zhu, V. Michaud-Vioux, M. Harb and H.G., to appear in IEEE Proceedings (2012). 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

Locality: the principle underlying all O(N) methods 2017/4/21 Locality: the principle underlying all O(N) methods Locality: the properties of a certain observation region comprising one or a few atoms are only weakly influenced by factors that are spatially far away from this observation region. S. Geodecker Rev. Mod. Phys. (1999) Equilibrium density matrix exhibits decaying property: insulator metal LMTO (nanodsim) LCAO (nanodcal) Example: Si bulk 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

Density matrix computation: 2017/4/21 Density matrix computation: DFT – computes potential and energy levels of the device; NEGF – non-equilibrium statistics that fills levels; The self-consistent loop – NEGF-DFT algorithm we use. Practically, density matrix is divided into two parts: equilibrium and non-equilibrium parts: no locality locality 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

NEGF5, Jyvaskyla, Finland 2017/4/21 Roadmap for locality-less computation of density matrix of large systems no preconditioner with preconditioner qmr SOR ILU H-Matrix Jacobi gmres bicgstab algorithms iterative method direct method MCS Large: ~20,000 atoms Do not depend on locality 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

NEGF5, Jyvaskyla, Finland 2017/4/21 Roadmap (cont.) single wall double wall thin & long thick & short algorithms iterative method direct method nested dissection principal layer pardiso 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

Performance: nano-device simulator (nanodsim) 2017/4/21 Performance: nano-device simulator (nanodsim) Nanodsim has been fully parallelized and optimized for both speed and memory costs. The speed is made nearly O(N) along the transport direction. speed performance memory performance Lx = periodic Ly = 10 nm Lz = 5, 10 ,15, 20 ,25, 30nm 160 cores Benchmark: 160 cores, 3GB / core, 12,800 atomic sites, <30 min/step 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

Solving ~20,000 atomic spheres for open devices at nonequilibrium 2017/4/21 Solving ~20,000 atomic spheres for open devices at nonequilibrium Open device structure of Si: parallel NEGF-DFT run on 480 cores. Structure Size # of atoms NEGF-DFT run Convergence Leads : 1 X 20 X 1 320 atomic spheres (2880 Orbitals) Two probe 1 1 X 20 X 20 6400 atomic spheres (57600 Orbitals) 107 NEGF-DFT steps, 5.5 min/step, total 9.8 hours Potential 1.0 x 10-5 Charge 1.23 x 10-5 per atom Two probe 2 1 X 20 X 40 12800 atomic spheres (115200 Orbitals) 195 NEGF-DFT steps, 14 min/step, total 46 hours Charge 6.2 x 10-6 per atom Two probe 3 1 x 20 x 60 19200 atomic spheres (172800 Orbitals) 267 NEGF-DFT steps, 30 min/step, total 134 hours Same as above Summary: NEGF-DFT modeling has reached realistic device sizes! 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

If using tight binding model: huge systems can be done 2017/4/21 If using tight binding model: huge systems can be done Run on a single computing node with 12 cores and 36 GB memory 1,024,000 Si atoms 10.9 nm ×10.9 nm × 173.8 nm time = 2 days Computation time scales linearly with the channel length Computation time increases 6~7 times if the cross section doubles For Lz = 173.8 nm, 1/3 of computation time is spent on surface Green’s function, and 2/3 spent on transmission calculation 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

4. How do we know all are well for real devices ? 2017/4/21 4. How do we know all are well for real devices ? Bench-marking the NEGF-DFT atomistic model for device simulations Acknowledgements: Lining Zhang (ECE, HKUST), Dr. Ferdows Zahid (Physics, HKU), Dr. Mansun Chan (ECE, HKUST), Dr. Jian Wang (Physics, HKU), Dr. Jesse Maassen (ECE, Purdue), Dr. Eric Zhu (Nanoacademic). NEGF-DFT/CPA-NVC 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

NEGF5, Jyvaskyla, Finland 2017/4/21 Commercial TCAD tool: 1328 pages of parameter and physics descriptions 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

Hundreds of parameters are needed ! 2017/4/21 Hundreds of parameters are needed ! 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

NEGF-DFT/CPA-NVC versus Sentaurus 2017/4/21 NEGF-DFT/CPA-NVC versus Sentaurus Atomic model: parameter-free Continuum model with external parameters NEGF-DFT/CPA-NVC Sentaurus: Drift-diffusion coupled with Poisson solver in real space grids 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

Potential of Si TFET: p-i-n structure 2017/4/21 Potential of Si TFET: p-i-n structure L. Zhang, F. Zahid, M. Chan, J. Wang, H.G. (2012). p-i-n tunnel FET (TFET) potential: almost perfect agreement Sentaurus (green) versus NEGF-DFT (red) T=300K Intrinsic channel 8nm 12nm 14nm Band gaps; doping; disorder; large sizes; computation; … Doping in the channel does not affect the potential profile due to high doping at S/D 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

Verification for MOSFET channels 2017/4/21 Verification for MOSFET channels Green: Sentaurus. Red: Nanodsim 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

New: non-uniform doping – delta doping 2017/4/21 New: non-uniform doping – delta doping Atomistic treatment of doping (P-doped) within CPA formalism Red – NEGF-DFT Green – Sentaurus with Fermi statistics Black – Sentaurus with Boltzman statistics 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

Full double gate FET simulation: 2017/4/21 Full double gate FET simulation: LG LS LD oxide sS sD gate Tox TSi p n+ 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

Nanodsim (self-consistent atomic) 2017/4/21 I-V characteristics I-V characteristics calculated by atomic model are in good agreement with NanoMos (effective mass model). Atomic model can go much further: surface roughness scattering, inhomogeneous doping, new materials, etc. Nanodsim (self-consistent atomic) NanoMos (Zhibin Ren’s thesis) 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

Example 1: localized doping 2017/4/21 Example 1: localized doping NEGF-DFT-CPA-NVC Localized doping suppresses off-state source-to-drain tunneling and reduces performance variability. Acknowledgement: Dr. Jesse Maassan (ECE, Purdue) Jesse Maassan & H.G. preprint (2012). 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

New idea: suppressing S-to-D off-state tunneling 2017/4/21 New idea: suppressing S-to-D off-state tunneling 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

Example 2: MRAM simulations 2017/4/21 Example 2: MRAM simulations Increasing spin transfer torque (STT) by impurity doping. Acknowledgements: Dr. Youqi Ke, Prof. Ke Xia, Dr. Eric Zhu, Dr. Dongping Liu, Prof. Xiu Feng Han NEGF-DFT-CPA-NVC Youqi Ke, Ke Xia and Hong Guo, PRL 105, 236801 (2010) D.P. Liu, X.F. Han and Hong Guo, PRB 85, 245436 (2012). 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

MTJ - magnetic tunneling junctions 2017/4/21 MTJ - magnetic tunneling junctions Picture from W. Butler, Nature Mat., 3, 845 (2004) Tunnel barrier is a few atomic layers thick. TMR = Spin transfer torque (STT) Problem: for a given bias, STT is too small, or junction resistance too large. 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

Solution: decreasing the junction resistance 2017/4/21 Solution: decreasing the junction resistance Why resistance is large? Because tunnel barrier is an insulator. How to reduce resistance? Dope the insulator with metal atoms. Why it does not work? Because impurity scattering destroys TMR. Youqi Ke, Ke Xia and Hong Guo, PRL 105, 236801 (2010) 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

NEGF5, Jyvaskyla, Finland 2017/4/21 New idea: Can we find a dopant that exponentially decreases resistance, but only linearly decreases TMR? We thus predict that Zn doping into MgO barrier will solve our problem! D.P. Liu, X.F. Han and Hong Guo, PRB 85, 245436 (2012). 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

Newest: CPA to compute variance by evaluating <GGGG> 2017/4/21 Newest: CPA to compute variance by evaluating <GGGG> Huge device to device variability. Eric Zhu & H.G. (2012). F.L. Yang et al., in VLSI Technol. Tech. Symp. Dig., pp. 208, June 2007. 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

NEGF5, Jyvaskyla, Finland 2017/4/21 Summary By solving 4 critical problems: disorder averaging, band gap, large size and accuracy, NEGF-DFT method can begin to predict device characteristics parameter-free for realistic nanoFET structures. Other details were included into NEGF-DFT as well: electron-phonon, collinear and non-collinear spin, spin-orbit, photon, high frequency, transient, etc. Endless application possibilities… Integration with industry TCAD tools possible. Further reduction of computation time underway … 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

NEGF5, Jyvaskyla, Finland 2017/4/21 Thank you ! Acknowledgements to Canadian funding: NSERC, CIFAR, FQRNT, IRAP, McGill University. We are grateful to Hong Kong government which funded AoE at HKU where the Sentaurus benchmark was done. 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

Today’s Circuit simulation: 2017/4/21 Today’s Circuit simulation: How do we know device parameters? Slide courtesy from Munsun Chan (HKUST) 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

Device parameters: measured in foundry 2017/4/21 Device parameters: measured in foundry capacitance transconductances diodes Geometry scaling Several hundred parameters are needed. 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

Today: the Moore’s law of fitting parameters 2017/4/21 Today: the Moore’s law of fitting parameters Too many parameters, very difficult to go on Slide courtesy from Munsun Chan (HKUST) 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

Why there are so many fitting parameters ? 2017/4/21 Why there are so many fitting parameters ? Model is beautiful, real device is ugly. Ex: parameters are added to take into account the fringing fields and shape irregularity. More and more parameters are added to account for changes of device due to whatever reason. Slide courtesy from Munsun Chan (HKUST) 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

Computational nanoelectronics should have: 2017/4/21 Computational nanoelectronics should have: discrete materials – atomic first principles quantum transport in discrete materials – quantum equations non-equilibrium quantum transport conditions - distributions reasonably accurate results: band gap, disorder, phonon - methods thousands and up to 70,000 atoms – large systems wide scope of materials and device structures – complexity reasonably fast computation – computer science, applied math … Red – must have Black – nice to have Present status: certain level of every item has been developed, more at hands-on sessions. 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

Can we simulate a FET from atomic first principles? 2017/4/21 Can we simulate a FET from atomic first principles? Picture from Taur and Ning, Fundamentals of Modern VLSI Devices ~100nm ~10nm 2011: L=22nm 2013 (?): L=14nm To model, needs ~10,000 to 70,000 Si atoms in the channel region. Can we calculate? DFT: ~1000 atoms. 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

Wide range of research are carried out by NEGF-DFT 2017/4/21 Wide range of research are carried out by NEGF-DFT Leakage current is MOSFET; Resistivity of Cu interconnects; Conductance, I-V curves of transport junctions; Computation of capacitance, diodes, inductance, current density; TMR, spin currents, and spin injection in magnetic tunnel junctions; Transport in nanowires, rods, films, clusters, nanotubes; Resistance of surface, interface, grain boundary; STM image simulations; Strongly correlated issues in transport at the large U limit; Transport in semiconductor devices; Transport through molecules; …. a progressing field... 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

Direct method: principal layer (PL) 2017/4/21 Direct method: principal layer (PL) After attempting many mathematical methods for solving the large matrix problem in NEGF, the principal layer approach for the LMTO implementation of NEGF-DFT wins for problems of roughly N < 20,000 atoms. 10 nm 6 nm CHANNEL SOURCE DRAIN Computation: O(N) along transport, O(M2) along transverse direction 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial

Verification on p-i-p MOSFET channels 2017/4/21 Verification on p-i-p MOSFET channels Red – NEGF-DFT Green – Sentaurus with Fermi statistics Black – Sentaurus with Boltzman statistics 2012-08-29 NEGF5, Jyvaskyla, Finland HP corporate template tutorial