1. Network for Computational Nanotechnology (NCN) Purdue, Norfolk State, Northwestern, MIT, Molecular Foundry, UC Berkeley, Univ. of Illinois, UTEP Periodic Potential Lab First-Time User Guide Periodic Potential Lab Abhijeet Paul, Gerhard Klimeck, Ben Haley, SungGeun Kim, and Lynn Zentner NCN@Purdue

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) E-k relationship E k Continuous energy band φ(k) = Aexp(-ik.R) EigenVector k = Momentum vector E = Kinectic energy Eigen Energy Plane wave E = Bk 2 4

5. Introduction: Origin of Bands (electron in crystal) Atoms Vpp Schrödinger Equation GAP E k Discontinuous energy bands Discontinuous energy bands E-k Eigen vectors are no longer simple plane waves. Energy bands become discontinuous, thereby producing BAND-GAPS Eigen vectors are no longer simple plane waves. Energy bands become discontinuous, thereby producing BAND-GAPS Electron Hamiltonian in a periodic crystal Electron Hamiltonian in a periodic crystal 5

6. How to Solve for Electron in 1D-crystal ? Atoms Original crystal potential Approximate crystal potential ∞ ∞ A a V min V max V (eV) x The solution can be obtained as a periodic potential problem. KRONIG-PENNEY The KRONIG-PENNEY Model 6

7. How to Solve for Electron in 1D-crystal ? Single electron periodic potential Schrödinger Eqn Periodic Potential The electron eigen vector follows this relation due to the periodic potential. This is known as BLOCH THEOREM the BLOCH THEOREM. k = Bloch wave number 7

8. Energy bands, Bandgap, and Effective Mass E-k relationship in periodic potential E-k relationship in periodic potential More details on effective mass can be found here: http://en.wikipedia.org/wiki/Effective_mass_(solid-state_physics) first BRILLOUIN ZONE E k Band Gap π/a -π/a E k E-k relationship in vacuum E-k relationship in vacuum Effective mass: energy band curvature. 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)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

10. Inputs [1]: Types of Periodic Potentials Four types of periodic potentials are available in the tool. Parabolic well Step well Coulombic Well Triangular Well All images from Periodic Potential Lab on nanoHUB.org 10

11. Inputs [2]: Details of Step Well Well Geometry 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) 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 = ΔE Vmin ≤ Epar ≤ Vmax + ΔE Egrid = linspace(Vmin,Vmax+ΔE,NE) = 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). 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). 11

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

13. Inputs [4]: Details of Parabolic Well All images from Periodic Potential Lab on nanoHUB.org Well energy details are the same as the step well description 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) 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) Well Geometry 13

14. Inputs [5]: Details of Coloumbic Well Well Geometry Well geometry description is the same as the step well description All images from Periodic Potential Lab on nanoHUB.org 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) 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) 14

15. Explanation of Outputs [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 Outputs All images from Periodic Potential Lab on nanoHUB.org 15

16. What Happens If You Just Hit SIMULATE ? Default Outputs Default Inputs [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 Comparison of electron crystal E-K with free electron E-K Reduced bandstructure in crystal compared with effective mass bandstructure All images from Periodic Potential Lab on nanoHUB.org. 17

18. What Happens If You Just Hit SIMULATE? cont’d… max min Wavefunctions/eigen energies for maximum/minimum energies of each band 18

19. What Happens If You Just Hit SIMULATE? cont’d… The regions where the probability is larger than 0.5. For minimum energy eigenvalues in each energy band 19

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:tool webpage »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 : »http://en.wikipedia.org/wiki/Particle_in_a_one- dimensional_lattice_(periodic_potential) »http://lamp.tu-graz.ac.at/~hadley/ss1/KronigPenney/KronigPenney.php »Applet to explain Kronig-Penney modelApplet to explain Kronig-Penney model Books and notes: »Physics of Semiconductor Devices, S. M. Sze. New York: Wiley, 1969, ISBN 0-471-84290-7; 2nd ed., 1981, ISBN 0-471-05661-8; 3rd ed., with Kwok K. Ng, 2006, ISBN 0-471-14323-5.ISBN 0-471-84290-7ISBN 0-471-05661-8ISBN 0-471-14323-5 »Semiconductor Device Fundamentals, Robert Pierret, Addison-Wesley. ISBN- 10: 0201543931 ISBN-13: 9780201543933 Effective mass information: »http://en.wikipedia.org/wiki/Effective_mass_(solid-state_physics) »Effective mass values in semiconductors (database) http://www.ioffe.rssi.ru/SVA/NSM/Semicond/ 22

23. References [2] Resource on nanoHUB.org »Teaching material on Kronig-Penney model, http://nanohub.org/resources/4959 http://nanohub.org/resources/4959 »Elaborate description on Kronig-Penney model, http://nanohub.org/resources/4962 Exercise on Kronig-Penney model »http://nanohub.org/resources/4851 Link for the tool : » http://nanohub.org/tools/kronig_penney Always check the tool web-page for latest features, releases, and bug-fixes at : http://nanohub.org/tools/kronig_penney 23