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.