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Device Modeling from Atomistic First Principles: theory of the nonequilibrium vertex correction

Device Modeling from Atomistic First Principles: theory of the nonequilibrium vertex correction. Eric Zhu 1 , Leo Liu 1 , Hong Guo 1,2 1 Nanoacademic Technologies Inc. Brossard, QC J4Z 1A7, Canada 2 Dept. of Physics, McGill Univ., Montreal, Quebec, H3A 2T8 Canada.

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Device Modeling from Atomistic First Principles: theory of the nonequilibrium vertex correction

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  1. 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 NEGF5, Jyvaskyla, Finland

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

  3. … 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 ? NEGF5, Jyvaskyla, Finland

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

  5. Ex 2: roughness scattering increases resistance of Cu interconnects With Daniel Gall of RPI. $: SRC NEGF5, Jyvaskyla, Finland

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

  7. New quantum mechanics Physics atomic simulations materials, chemistry, physics device modeling < 5nm (1000 atoms) Semi-empirical device modeling, 10,000 to 100,000 atoms device parameters TCAD Can we calculate realistic device parameters? engineering science Quantitative prediction of quantum transport from atomic first principles without any parameter NEGF5, Jyvaskyla, Finland

  8. NEGF-DFT Method - NEGF-DFT: non-equilibrium density matrix ‘DFT’: density functional theory NEGF: Keldysh nonequilibrium Green’s function `DFT’ in NEGF-DFT is not the usual ground state DFT: density matrix of NEGF-DFT is constructed at non-equilibrium. No variational solution. DFT NEGF-DFT 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). NEGF5, Jyvaskyla, Finland

  9. 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 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). NEGF-DFT/CPA-NVC NEGF5, Jyvaskyla, Finland

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

  11. 1 2 To build intuition, let’s solve a toy problem exactly 1D tight binding chain nearest neighbor coupling on-site energy NEGF5, Jyvaskyla, Finland

  12. 1 2 SL SR Self-energies for the leads: How to handle half-infinite chain ? Self energy: The problem is reduced to 3 sites plus self-energies NEGF5, Jyvaskyla, Finland

  13. L R Physical quantities and NEGF Express physical quantities in terms of NEGF: average over disorder configurations The problem is reduced to calculate disorder averaged NEGF NEGF5, Jyvaskyla, Finland

  14. 1 2 SL SR Disorder average can be done exactly for the 3-site model NEGF5, Jyvaskyla, Finland

  15. 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. NEGF5, Jyvaskyla, Finland

  16. 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) NEGF5, Jyvaskyla, Finland

  17. and are solved from CPA equation CPA: CPA picture: effective media Implementation: needs a method that does one atom at a time: LMTO, KKR, etc.. NEGF5, Jyvaskyla, Finland

  18. NEGF-DFT Average over random disorder: X Non-equilibrium density matrix: nonequilibrium vertex 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) NEGF5, Jyvaskyla, Finland

  19. X Essence of Nonequilibrium Vertex Correction (NVC) 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. NEGF5, Jyvaskyla, Finland

  20. NVC Equation: some complicated technical details Youqi Ke, Ke Xia and Hong Guo PRL 100, 166805 (2008). NEGF5, Jyvaskyla, Finland

  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. NEGF5, Jyvaskyla, Finland

  22. and are solved from NVC equation NVC solution for the 3-site toy model NEGF5, Jyvaskyla, Finland

  23. Comparison for the 3-site toy model: specular part diffusive part Excellent ! NEGF5, Jyvaskyla, Finland

  24. no NVC NVC exact 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. NEGF5, Jyvaskyla, Finland

  25. 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 NEGF5, Jyvaskyla, Finland

  26. The band gap problem of local functionals in DFT DFT calculation of band gaps: MBJ computation time is ~LDA NEGF5, Jyvaskyla, Finland

  27. Some relevant band gaps for transistors materials: NEGF5, Jyvaskyla, Finland

  28. 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. NEGF5, Jyvaskyla, Finland

  29. Experimental data Calculated data MBJ potential + CPA: works. Good agreement with experimental data. NEGF5, Jyvaskyla, Finland

  30. 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). NEGF5, Jyvaskyla, Finland

  31. 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 NEGF5, Jyvaskyla, Finland

  32. 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 NEGF5, Jyvaskyla, Finland

  33. Roadmap for locality-less computation of density matrix of large systems algorithms Large: ~20,000 atoms Do not depend on locality iterative method direct method no preconditioner with preconditioner gmres bicgstab qmr Jacobi SOR ILU MCS H-Matrix NEGF5, Jyvaskyla, Finland

  34. Roadmap (cont.) algorithms iterative method direct method nested dissection principal layer pardiso single wall double wall thin & long thick & short NEGF5, Jyvaskyla, Finland

  35. Lx = periodic Ly = 10 nm Lz = 5, 10 ,15, 20 ,25, 30nm 160 cores 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 Benchmark: 160 cores, 3GB / core, 12,800 atomic sites, <30 min/step NEGF5, Jyvaskyla, Finland

  36. Solving ~20,000 atomic spheres for open devices at nonequilibrium Open device structure of Si: parallel NEGF-DFT run on 480 cores. Summary: NEGF-DFT modeling has reached realistic device sizes! NEGF5, Jyvaskyla, Finland

  37. 1,024,000 Si atoms 10.9 nm ×10.9 nm × 173.8 nm time =2 days If using tight binding model: huge systems can be done Run on a single computing node with 12 cores and 36 GB memory • 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 NEGF5, Jyvaskyla, Finland

  38. 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 NEGF5, Jyvaskyla, Finland

  39. Commercial TCAD tool: 1328 pages of parameter and physics descriptions NEGF5, Jyvaskyla, Finland

  40. Hundreds of parameters are needed ! NEGF5, Jyvaskyla, Finland

  41. NEGF-DFT/CPA-NVC versus Sentaurus Atomic model: parameter-free Continuum model with external parameters Sentaurus: Drift-diffusion coupled with Poisson solver in real space grids NEGF-DFT/CPA-NVC NEGF5, Jyvaskyla, Finland

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

  43. Verification for MOSFET channels Green: Sentaurus. Red: Nanodsim NEGF5, Jyvaskyla, Finland

  44. 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 NEGF5, Jyvaskyla, Finland

  45. Full double gate FET simulation: LG gate Tox LS LD oxide TSi n+ p n+ sS sD Tox oxide gate NEGF5, Jyvaskyla, Finland

  46. 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) NEGF5, Jyvaskyla, Finland

  47. Example 1: localized doping Localized doping suppresses off-state source-to-drain tunneling and reduces performance variability. Acknowledgement: Dr. Jesse Maassan (ECE, Purdue) NEGF-DFT-CPA-NVC Jesse Maassan & H.G. preprint (2012). NEGF5, Jyvaskyla, Finland

  48. New idea: suppressing S-to-D off-state tunneling NEGF5, Jyvaskyla, Finland

  49. 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 Youqi Ke, Ke Xia and Hong Guo, PRL 105, 236801 (2010) D.P. Liu, X.F. Han and Hong Guo, PRB 85, 245436 (2012). NEGF-DFT-CPA-NVC NEGF5, Jyvaskyla, Finland

  50. 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. NEGF5, Jyvaskyla, Finland

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