Network for Computational Nanotechnology (NCN)
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1 Network for Computational Nanotechnology (NCN) Purdue, Norfolk State, Northwestern, MIT, Molecular Foundry, UC Berkeley, Univ. of Illinois, UTEP Abhijeet Paul, Gerhard Klimeck, Ben Haley, SungGeun Kim, and Lynn Zentner
2 Table of Contents Introduction 3» Origin of bands (electrons in vacuum, electrons in crystal)» Solution to electron in periodic potential 6» Basics of energy-bands, band-gap, and effective mass 8 The Periodic Potential Lab (What is it? What does it do?) 9» Detailed description of inputs 10» Explanation for outputs 15 What Happens When You Just Hit Simulate? 16 Suggested Exercises Using the Tool 20 Final Comments About the Tool 21 References 22 2
3 Introduction: Origin of Bands (electron in vacuum) Schrödinger Equation Free electron kinetic energy Hamiltonian k = Momentum vector E = Kinectic energy m = Effective mass 3
4 Introduction: Origin of Bands (solution of Schrödinger equation) Eigen Energy E-k relationship E = Bk 2 E k EigenVector Plane wave φ(k) = Aexp(-ik.R) Continuous energy band k = Momentum vector E = Kinectic energy 4
5 Introduction: Origin of Bands (electron in crystal) Schrödinger Equation E-k E GAP Electron Hamiltonian in a periodic crystal GAP Atoms Vpp GAP k Discontinuous energy bands Eigen vectors are no longer simple plane waves. Energy bands become discontinuous, thereby producing BAND-GAPS 5
6 How to Solve for Electron in 1D-crystal? Approximate crystal potential The solution can be obtained as a periodic potential problem. Atoms Original crystal potential V (ev) a A Vmax Vmin x 6
7 How to Solve for Electron in 1D-crystal? Single electron periodic potential Schrödinger Eqn k = Bloch wave number Periodic Potential 7
8 Energy bands, Bandgap, and Effective Mass E-k relationship in vacuum E E-k relationship in periodic potential E Band Gap k -π/a k π/a first BRILLOUIN ZONE Effective mass: energy band curvature. More details on effective mass can be found here: 8
9 The Periodic Potential Lab: What is it?» A MATLAB based tool» Tool developed at Purdue University Part of the teaching tools on nanohub.org (ABACUS) What does it do?» Solves single electron Schrödinger equation in different types of periodic potentials» Provides a variety of information for an electron in periodic potential Energy bands and electron wave functions Effective masses and band-gaps Developers:» Abhijeet Paul / Purdue University» Gerhard Klimeck / Purdue University 9
![Inputs [1]: Types of Periodic](/docs-images/78/77638374/images/10-7.jpg "Potentials Step well Parabolic well")
10 Inputs [1]: Types of Periodic Potentials Step well Parabolic well Four types of periodic potentials are available in the tool. Triangular Well Coulombic Well All images from Periodic Potential Lab on nanohub.org 10
![Inputs [2]: Details of Step Well Well Geometry = ΔE Vmin Epar Vmax + ΔE = NE](/docs-images/78/77638374/images/11-5.jpg "Egrid = linspace(vmin,vmax+δe,ne) Energy Details of Well: Vmax: Maximum energy")
11 Inputs [2]: Details of Step Well Well Geometry = ΔE Vmin Epar Vmax + ΔE = NE Egrid = linspace(vmin,vmax+δe,ne) Energy Details of Well: Vmax: Maximum energy barrier height in ev Vmin: Mininum energy level in the well in ev Energy of particle above the barrier provides the total energy range of the particle (ΔE) in ev Energy sampling points show how many points will be used for the energy grid (NE). Well Geometry Description: W:Total width of the well in angstroms (Ǻ) a: width of the barrier in Ǻ m o : mass of the travelling particle in terms of vacuum electronic mass (m = m o x 9.1e-31kg) All images from Periodic Potential Lab on nanohub.org 11
![Inputs [3]: Details of](/docs-images/78/77638374/images/12-6.jpg "Triangular Well Well")
12 Inputs [3]: Details of Triangular Well Well Geometry Well Geometry description and well Energy Details are the same as the step well description. All images from Periodic Potential Lab on nanohub.org 12
![Inputs [4]: Details of Parabolic Well Well](/docs-images/78/77638374/images/13-2.jpg "energy details are the same as the step well")
13 Inputs [4]: Details of Parabolic Well Well energy details are the same as the step well description Well Geometry Well Geometry Description: W:Total width of the well in angstroms (Ǻ). a: width of the barrier in Ǻ. [ a = W/2 ] m o : mass of the travelling particle in terms of vacuum electronic mass (m = m o x 9.1e-31kg) All images from Periodic Potential Lab on nanohub.org 13
![Inputs [5]: Details of Coloumbic Well](/docs-images/78/77638374/images/14-1.jpg "Well Geometry Energy details of well:")
14 Inputs [5]: Details of Coloumbic Well Well Geometry Energy details of well: -Vmax: Depth of well in ev. Emin: Lowest energy of particle approaching the the barrier in ev Energy of particle above the barrier provides the total energy range of the particle (ΔE) in ev Energy sampling points shows how many points will be used for energy grid (NE) Well geometry description is the same as the step well description All images from Periodic Potential Lab on nanohub.org 14
15 Explanation of Outputs Outputs All images from Periodic Potential Lab on nanohub.org [Energy functional]: provides information about the allowed energy states [Allowed bands]: energy bands where particle can stay [Band information]: band edges and effective band mass for electron [Parameter-Summary]: parameters provided by user as input [Effective mass information]: effective mass at band edges [Reduced/expanded dispersion relations]: in the expanded & reduced Brillouin zone [Periodic EK vs. free electron EK]: EK in crystal compared with free electron EK [Reduced EK vs. Eff. mass EK]: EK in crystal compared with parabolic EK at the band edges [Eigen energy and wave function]: provides the eigen energies and wave function on top of each eigen energies [Wave function probability plot]: modulus square of each wavefunction for different energy levels [Above %50 region of wavefunction(min/max)]: region where the wavefunction probability is more than 0.5 for minimum / maximum energy of each bands [1D DOS plot]: one dimensional density of states plot 15
![What Happens If You Just Hit SIMULATE? Default Inputs Default Outputs [1] Potential type [2] Energy details [3] Well geometry All images from Periodic Potential Lab on nanohub.](/docs-images/78/77638374/images/16-3.jpg "org [Energy functional]: provides information about the allowed energy states [Allowed bands]: energy bands where particle can stay [Band information]: band edges and effective band mass for electron")
16 What Happens If You Just Hit SIMULATE? Default Inputs Default Outputs [1] Potential type [2] Energy details [3] Well geometry All images from Periodic Potential Lab on nanohub.org [Energy functional]: provides information about the allowed energy states [Allowed bands]: energy bands where particle can stay [Band information]: band edges and effective band mass for electron [Parameter-Summary]: parameters provided by user as input [Effective mass information]: effective mass at band edges [Reduced/expanded dispersion relations]: in the expanded & reduced Brillouin zone [Periodic EK vs. free electron EK]: EK in crystal compared with free electron EK [Reduced EK vs. Eff. mass EK]: EK in crystal compared with parabolic EK at the band edges [Eigen energy and wave function]: provides the eigen energies and wave function on top of each eigen energy [Wave function probability plot]: modulus square of each wavefunction for different energy levels [Above %50 region of wavefunction(min/max)]: Region where the wavefunction probability is more than 0.5 for minimum / maximum energy of each bands [1D DOS plot]: one dimensional density of states plot 16
17 What Happens If You Just Hit SIMULATE? cont d Description of a few outputs Shows the graphical representation of real solutions of Kronig-Penney model Green shows allowed energy bands and red shows the energy band gaps Reduced bandstructure in crystal compared with effective mass bandstructure 17 Comparison of electron crystal E-K with free electron E-K All images from Periodic Potential Lab on nanohub.org.
18 What Happens If You Just Hit SIMULATE? cont d Wavefunctions/eigen energies for maximum/minimum energies of each band max min 18
19 What Happens If You Just Hit SIMULATE? cont d For minimum energy eigenvalues in each energy band The regions where the probability is larger than
20 Suggested Exercises Study the effect of well width (W) and particle mass variation in other types of wells. Results should be similar to what is given in this document. Study the effect of barrier height (Vmax-Vmin) and barrier width (a) for different wells. What happens to the following?:» Number of bands» Lowest Energy band» Band masse (effective mass) 20
21 Final Comments about the Tool Limitations of the Tool: Presently cannot treat any arbitrary periodic potential problem Does not provide the energy and wave-function plot together, which can be very useful. Opportunities: Use this tool to learn about electronic bandstructure in 1D periodic potential wells. Feel free to post (on tool webpage) about:» the bugs» new features you want (submit via wishlist) Contact the developers to collaborate on work using this tool. 21
22 References [1] URLs on Kronig Penney and 1D Periodic Potential :» Applet to explain Kronig-Penney model Books and notes:» Physics of Semiconductor Devices, S. M. Sze. New York: Wiley, 1969, ISBN ; 2nd ed., 1981, ISBN ; 3rd ed., with Kwok K. Ng, 2006, ISBN » Semiconductor Device Fundamentals, Robert Pierret, Addison-Wesley. ISBN-10: ISBN-13: Effective mass information:» Effective mass values in semiconductors (database) SVA/NSM/Semicond/ 22
23 References [2] Resource on nanohub.org» Teaching material on Kronig-Penney model, Elaborate description on Kronig-Penney model, Exercise on Kronig-Penney model» Link for the tool :» Always check the tool web-page for latest features, releases, and bug-fixes at : 23
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