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Network for Computational Nanotechnology (NCN) UC Berkeley, Univ.Of Illinois, Norfolk State, Northwestern, Purdue, UTEP

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Network for Computational Nanotechnology (NCN) UC Berkeley, Univ.Of Illinois, Norfolk State, Northwestern, Purdue, UTEP Network for Computational Nanotechnology (NCN) UC Berkeley, Univ.of Illinois, Norfolk State, Northwestern, Purdue, UTEP Abhijeet Paul, Ben Haley, and Gerhard Klimeck NCN @Purdue University West Lafayette, IN 47906, USA Abhijeet Paul Table of Contents I • Introduction » Origin of bands (electrons in vacuum and in crystal) 5 » Energy bands, bandgap, and effective mass 6 » Different types of device geometries 7 » How is band structure calculated? 8 » Assembly of the device Hamiltonian 9 » Self-consistent E(k) calculation procedure 10 • Band Structure at a Glance 11 » Features of the Band Structure Lab 12 » A complete description of the inputs 14 • What Happens When You Just Hit Simulate? 20 Abhijeet Paul 2 Table of Contents II • Some Default Simulations » Circular silicon nanowire E(k) 23 » Silicon ultra-thin-body (UTB) E(k) 25 » Silicon nanowire self-consistent simulation 27 • Bulk Strain Sweep Simulation 29 • Case Study 31 • Suggested Exercises Using the Tool 34 • Final Words about the Tool 35 • References 36 • Appendices 38 » Appendix A: job submission policy for Band Structure Lab » Appendix B: information about high symmetry points in a Brillouin zone » Appendix C: explanation for different job types used in the tool Abhijeet Paul 3 Origin of Bands: Electrons in Vacuum Schrödinger Equation Eigen Energy E(k) relationship E = Bk2 E Free electron kinetic energy Hamiltonian k Plane Waves Continuous energy band φ(k) = Aexp(-ikr) (Eigen vectors) • Single electron (in vacuum) Schrödinger Equation provides the solution: Plane waves as eigen vectors k = Momentum vector E(k) = Bk2 as eigen energy E = Kinetic energy • Eigen energy can take continuous values for every value of k • E(k) relationship produces continuous energy bands Abhijeet Paul 4 Origin of Bands: Electrons in Crystal Schrödinger equation E(k) relationship E GAP Electron Hamiltonian in a periodic crystal GAP GAP k Discontinuous Periodic potential energy bands due to crystal (Vpp) Atoms • An electron traveling in a crystal sees an extra crystal potential, Vpp. • Eigen vectors are no longer simple plane waves. • Eigen energies cannot take all the values. • Energy bands become discontinuous, thereby producing the BAND-GAPS. Abhijeet Paul 5 Energy Bands, Bandgap, and Effective Mass Continuous bands Energy bands E Lattice constant = E -π/a ≤ k ≤ π/a. This is called Bandgap first BRILLOUIN ZONE. E(k) relation in this zone is called reduced E(k) relation k k E(k) relationship E Vacuum electron in periodic potential E(k) relationship Electron mass in vacuum = 9.1e-31kg Band Gap • Similar E(k) relationship • Now free electron mass is replaced by -π/a k π/a effective mass (m*) • Effective mass provides the energy band curvature Abhijeet Paul 6 Description of Geometries Y Bulk X Nanowire 1D Periodic Y X (3D periodic) Z Z • Nanowires have 3 cross-sectional shapes: circular, triangular, Y UTB (2D periodic) rectangular. X • The semiconductor is represented atomistically for the E(k) calculation. Semiconductor • The oxide is treated as Z continuum material for self- Oxide consistent simulations. • X -> transport direction • Y,Z -> confinement directions Abhijeet Paul 7 How Band Structure is Calculated [4] Dispersion (E(k)) Bulk [2] Assemble device relation 3D periodicity Hamiltonian [H] E Band Gap Quantum well 2D periodicity -π/a k π/a Y X In the Band Structure Z Lab, the device Hamiltonian E1 Device is assembled using the axis semi-empirical E2 tight-binding method. Quantum wire E3 1D periodicity confinement [3] Diagonalized H provides [1] Select crystal eigen-energies dimensionality periodicity Abhijeet Paul 8 Assembly of the Device Hamiltonian Device Hamiltonian • Device Hamiltonian is assembled Anion using semi-empirical tight-binding [TB] • Each atom is represented using an onsite block [Hon_a or Hon_c]. • Coupling with nearest Cation neighbor is taken in coupling blocks [Vac, Vca] • Size of these blocks depends on the basis set and spin-orbit coupling • Basis sets are made of orthogonal atomic orbitals like s,p,d,etc. Anion Onsite Cation-Anion block [Hon_a] Coupling block [Vca] • The Band Structure Lab uses sp3d5s* basis set with 10 Cation Onsite Anion-Cation basis functions block [Hon_c] Coupling block [Vac] Abhijeet Paul 9 Self-consistent E(k) Calculation Procedure Electronic structure 20 band sp3d5s* model with spin orbit coupling Appropriate for treating atomic level disorder Strain treatment at atomic level Structural, material & potential variations treated easily Top of the barrier ballistic transport Self consistent iteration scheme Abhijeet Paul 10 Band Structure Lab at a Glance • What is the Band Structure Lab and what does it do?: » A C++ based code to perform electronic structure calculation » A tool powered by OMEN-BSLAB, C/C++ MPI based parallel code » Solves single electron Schrödinger equation in different types of semiconductor crystals using the semi-empirical tight-binding method: For pure crystals with and without strain For gates semiconductor systems with applied external biases for nanowires and ultra thin bodies (UTB) » Provides various information on an electron in a periodic potential Energy bands Effective masses and band-gaps This tool was developed at Purdue University and is part of the teaching tools on nanoHUB.org (AQME) . Abhijeet Paul 11 Features of the Band Structure Lab • Calculation of energy dispersion(E(k)) for semiconductor materials: » In bulk (3D), Ultra Thin Bodies [UTB] (2D), and Nanowires (1D) » With and without strain in the system, it can handle: Hydrostatic strain (equal strain in all directions) Biaxial strain (equal strain on a plane) Uniaxial strain (strain along any arbitrary axis) Arbitrary strain (all directions have different strains) • Provides following information » Effective masses in bulk, nanowires, and UTBs » 3D dispersion for bulks in 1st Brillouin zone » Bandgaps and bandedges • Self-consistent simulations: » Provides charge and potential profile in nanowire FETs and in UTB DGMOS for the applied gate bias » Change in E(k) relation due to applied bias Screen shot from http://nanohub.org Abhijeet Paul 12 Computational Aspects of Band Structure Lab • This tool has 3 levels of parallelism, namely: » Parallel over all the gate biases » Parallel over the kz point calculations for each Vg point » Parallel over the kx point calculation for each Kz point • Runs on multiple CPUs and on various clusters to provide a faster turn-around time for simulations • Tool has internal job submission method, depending on the kind of job the user wants to run • User can override these internal settings, but this should be done with care. See Appendix [A] for additional information on the job submission policy. Abhijeet Paul 13 Inputs [1]: Device Structure Types of geometries and related parameters are selected on this page [1] Geometry [2] Device Information [2.3] Device Directions: [a] Transport direction (X) [100],[110],[111] [2.1] Job type: [b] Confinement direction(Z) Bulk: Band structure calculation [c] 3rd orthogonal direction(Y) nanowire & UTB: determined automatically [1] Band structure calculation [2] Band structure calculation under an applied bias [3] Material 4 Material 5 types of geometry Types: [periodicity]: [2.2] Device Dimension: Depending on job type, select: [a] Silicon [1] Bulk [3D] [a] Dimension of NW or UTB [b] Gallium [2] Circular nanowire [1D] semiconductor core in nm Arsenide [3] Rectangular nanowire [1D] [b] Thickness of oxide in nm (This is [c] Indium [4] Triangular nanowire [1D] available for self-consistent E(k) Arsenide [5] Ultra thin body [2D] calculation.) [d] Germanium Screen shot from http://nanohub.org 14 Abhijeet Paul Inputs [2]: Electronic Structure Properties used to obtain the electronic dispersion are set on this page. [1] Tight Binding Model This is the basis, set model used for calculating the band structure. Presently, the sp3d5d* model is supported by the tool. [2] Spin Orbit (SO) Coupling [3] Dangling Bond Energy • This produces the effect of • This is the energy barrier set at the electron spin on band structure. external boundary of the structure. • Should be always “ON” for valence • This value is utilized to remove the bands. spurious states in the bandgap. Default • Produces negligible effect on value of 30 eV is good. conduction bands. • Smaller value means lower barrier • With SO on calculations are slower and larger value means higher barrier. due to larger matrix sizes. • Usually there is no need to change this value. Screen shot from http://nanohub.org Abhijeet Paul 15 Inputs [3.a]: Analysis - Bulk This page provides options for the kind of simulations that can be run, depending on the selected geometry. Two types of simulations: bulk Strain sweep analysis: dispersion and strain sweep • Provide the initial Effect of strain on E(k) and final % strain value • Provide number Bulk dispersion [E(k)] calculation: Select the % strain value of points for strain (eps_xx, eps_yy, eps_zz) along sweep the 3 axes Explore bands [1] Along std. symmetry directions* [2] Along some symmetry directions* Only 3 models available Strain Models for strain sweep analysis [1] Bi-axial Show 3D E(k) [2] Uniaxial Produces energy isosurface plots. [3] Hydrostatic User can set the kx, ky, kz region, as well * See Appendix [B] as the energy limit. for the bands [4] Arbitrary Abhijeet Paul 16 Inputs [3.b] Analysis - UTB Self-consistent E(k) calculation options calculation options Select Type of Band Select Type of DGMOS CB or VB N-type or P-type Depending on source-drain doping: Direction along which E(k) to be Select the calculated* number of Bias selection: sub-bands. • Set gate bias. • Set drain bias. Select the number of sub-bands. Select the • Set gate work function. number of k • Set semiconductor points. (Higher k electron affinity. points are good • Set device temperature. Select the number of k points. for a P-type • Set DIBL. simulation, but they increase the Select backgate simulation time. Select the strain type and values. configuration. (Strain detail is the same as bulk) Select the strain Set source/drain doping.
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