Quantum Modeling of Solids: Basic Properties

Transcription

1 1.021, 3.021, , : Introduction to Modeling and Simulation : Spring 2011 Part II Quantum Mechanical Methods : Lecture 5 Quantum Modeling of Solids: Basic Properties Jeffrey C. Grossman Department of Materials Science and Engineering Massachusetts Institute of Technology

2 Part II Outline theory & practice example applications 1. It s A Quantum World:The Theory of Quantum Mechanics 2. Quantum Mechanics: Practice Makes Perfect 3. From Many-Body to Single-Particle; Quantum Modeling of Molecules 4. From Atoms to Solids 5. Quantum Modeling of Solids: Basic Properties 6. Advanced Prop. of Materials:What else can we do? 7. Nanotechnology 8. Solar Photovoltaics: Converting Photons into Electrons 9. Thermoelectrics: Converting Heat into Electricity 10. Solar Fuels: Pushing Electrons up a Hill 11. Hydrogen Storage: the Strength of Weak Interactions 12. Review

3 Motivation? mechanical properties? electrical properties? optical properties

4 Lesson outline 6 Review structural properties Calc. the band structure E (ev) 0 Γ 15 Γ' 25 X 1 E C E V Calc. the DOS S 1 Metal/insulator Magnetization -10 L Λ Γ 1 Figure by MIT OpenCourseWare. Γ X U,K k Σ Γ Image by MIT OpenCourseWare. Let s take a walk through memory lane for a moment...

5 In the Beginning... There were some strange observations by some very smart people. e - + _ Image by MIT OpenCourseWare. Image courtesy NASA.

6 In the Beginning... The weirdness just kept going. Image removed due to copyright restrictions. See the image here:

7 It Became Clear......that matter behaved like waves (and vice versa). And that we had to lose our classical concepts of absolute position and momentum. And instead consider a particle as a wave, whose square is the probability of finding it. "(r,t) = Aexp[i(k r #$t)] But how would we describe the behavior of this wave?

8 Then, F=ma for Quantum Mechanics V M m v Image by MIT OpenCourseWare. 2 2m 2 + V (r, t) ψ(r, t) = i ψ(r, t) t

9 It Was Wonderful It explained many things. hydrogen s 3p 3d 2s 2p Courtesy of David Manthey. Used with permission. Source: It gave us atomic orbitals. 1s It predicted the energy levels in hydrogen.

10 It Was Wonderful It gave us the means to understand much of chemistry. BUT...

11 Nature Does > 1 electron! It was impossible to solve for more than a single electron. Enter (stage left) computational quantum mechanics! But...

12 We Don t Have The Age of the Universe Which is how long it would take currently to solve the Schrodinger equation exactly on a computer. So...we looked at this guy s back. Screenshot of article removed due to copyright restrictions; see the article online: And started making some approximations.

13 The Two Paths Ψ is a wave function of all positions & time. 2 2m 2 + V (r, t) ψ(r, t) = i ψ(r, t) t Chemists (mostly) Physicists (mostly) Ψ = something simpler - H = something simpler mean field methods

14 Working with the Density E[n] = T [n] + V ii + V ie [n] + V ee [n] kinetic ion-ion ion-electron electron-electron n=# Ψ(N 3n ) ρ(n 3 ) electrons , ion potential Hartree potential exchange-correlation potential

15 Review:Why DFT? 100,000 Number of Atoms 10, QMC Linear scaling DFT DFT MP2 10 Exact treatment CCSD(T) Year Image by MIT OpenCourseWare.

16 Review: Self-consistent cycle Kohn-Sham equations Iterations to selfconsistency n(r) = φ i (r) 2 i scf loop 3.021j Introduction to Modeling and Simulation - N. Marzari (MIT, May 2003)

17 Review: Crystal symmetries A crystal is built up of a unit cell and periodic replicas thereof. lattice unit cell Image of M. C. Escher's "Mobius with Birds" removed due to copyright restrictions.

18 Review: Crystal symmetries 4 Lattice Types 7 Crystal Classes Bravais Lattice Triclinic Monoclinic Orthorhombic Tetragonal Trigonal Parameters a 1 = a 2 = a 3 α 12 = α 23 = α 31 a 1 = a 2 = a 3 α 23 = α 31 = 90 0 α 12 = 90 0 a 1 = a 2 = a 3 α 12 = α 23 = α 31 = 90 0 a 1 = a 2 = a 3 α 12 = α 23 = α 31 = 90 0 a 1 = a 2 = a 3 α 12 = α 23 = α 31 < Simple (P) Volume Centered (I) Base Centered (C) Face Centered (F) Bravais The most common Bravais lattices are the cubic ones (simple, bodycentered, and facecentered) plus the hexagonal closepacked arrangement....why? Cubic a 1 = a 2 = a 3 α 12 = α 23 = α 31 = 90 0 Hexagonal a 1 = a 2 = a 3 α 12 = α 23 = α 31 = 90 0 a 3 a 2 a 1 Image by MIT OpenCourseWare.

19 Reciprocal Lattice & Brillouin Zone Associated with each real space lattice, there exists something we call a reciprocal lattice. The reciprocal lattice is the set of wave-vectors which are commensurate with the real space lattice. It is defined by a set of vectors a*, b*, and c* such that a* is perpendicular to b and c of the Bravais lattice, and the product a* x a is 1.

20 Reciprocal Lattice & Brillouin Zone It is defined by a set of vectors a*, b*, and c* such that a* is perpendicular to b and c of the Bravais lattice, and the product a* x a is 1. a * b " c In particular: = a # b " c Brillouin a * b " c = a # b " c Image from Wikimedia Commons, source unknown. All rights reserved. This content is excluded from our Creative Commons license. For more information, see

21 Review:The inverse lattice real space lattice (BCC) inverse lattice (FCC) z a a2 a 1 x a 3 y Image by MIT OpenCourseWare.

22 The Brillouin zone inverse lattice The Brillouin zone is a special unit cell of the inverse lattice. Image by Gang65 on Wikimedia Commons. License: CC-BY. This content is excluded from our Creative Commons license. For more information, see

23 The Brillouin zone Brillouin zone of the FCC lattice

24 Bloch s Theorem (Take 2) Reciprocal lattice vectors have special properties of particular value for calculations of solids. Remember that we write the reciprocal lattice vector: G = 2"na * + 2"mb * + 2"oc * We added the 2 simply for convenience, and the n, m, o, are integers. Now consider the behavior of the function exp(igr).

25 Bloch s Theorem (Take 2) exp(ig" r) = exp[i(2#na * + 2#mb * + 2#oc * )" ($a + %b + &c)] = exp[i(2#n$ + 2#m% + 2#o& )] = cos(2#n$ + 2#m% + 2#o& ) + isin(2#n$ + 2#m% + 2#o& ) As r is varied, lattice vector coefficients (α,β,γ) change between 0 and 1 and the function exp(ig r) changes too. However, since n, m, and o are integral, exp(ig r) will always vary with the periodicity of the real-space lattice. i G j r e G R = 1 ψ(r) = c j e i j automatically periodic in R!

26 Bloch s Theorem (Take 2) The periodicity of the lattice in a solid means that the values of a function (e.g., density) will be identical at equivalent points on the lattice. The wavefunction, on the other hand, is periodic but only when multiplied by a phase factor. NEW quantum number k that lives in the inverse lattice! This is known as Bloch s theorem. ψ k (r) = e i k r u k (r) u k (r) = u k (r + R )

27 Periodic potentials Results of the Bloch theorem: R ψ k (r + R ) = ψ k (r)e i k 2 2 charge density ψ k (r + R ) = ψ k (r) is lattice periodic if if solution ψ (r) G k ψ k+ (r) also solution with E k = E k+ G G

28 Periodic potentials Schrödinger certain quantum equation symmetry number hydrogen atom periodic solid spherical symmetry [H, L 2 ] = HL 2 L 2 H = 0 [H, L z ] = 0 translational symmetry [H, T ] = 0 ψ n,l,m (r) ψ n, ( k r)

29 The band structure Different wave functions can satisfy the Bloch theorem for the same k: eigenfunctions and eigenvalues labelled with k and the index n energy bands

30 The band structure Atom Molecule Solid Energy Antibonding p Conduction band from antibonding p orbitals p Antibonding s Conduction band from antibonding s orbitals s Bonding p Valence band from p bonding orbitals Bonding s Valence band from s bonding orbitals k Image by MIT OpenCourseWare.

31 The band structure 6 Silicon E (ev) 0 Γ 15 Γ' 25 X 1 E C E V S 1 energy levels in the Brillouin zone -10 L Λ Γ 1 Figure by MIT OpenCourseWare. Γ X U,K k Σ Γ Image by MIT OpenCourseWare.

32 The band structure 6 Silicon E (ev) 0 unoccupied Γ 15 E Γ' C 25 X 1 E V S 1 occupied energy levels in the Brillouin zone -10 L Λ Γ 1 Figure by MIT OpenCourseWare. Γ X U,K k Σ Γ Image by MIT OpenCourseWare.

33 The Fermi energy E (ev) 6 0 unoccupied Γ 15 E Γ' C 25 X 1 E V gap: also visible in the DOS Fermi energy -10 S 1 occupied Γ 1 Figure by MIT OpenCourseWare. one band can hold two electrons (spin up and down) L Λ Γ k X U,K Σ Γ Image by MIT OpenCourseWare.

34 The electron density electron density of silicon

35 Structural properties Forces on the atoms can be calculated with the Hellmann Feynman theorem: For λ=atomic position, we get the force on that atom. Forces automatically in most codes.

36 Structural properties Etot finding the equilibrium lattice constant mass density alat a mu= Kg

37 Structural properties finding the stress/pressure and the bulk modulus Etot V0 V E p 2 E p = σ bulk = V = V V V V 2

38 Calculating the band structure 1. Find the converged ground state density and potential. 3-step procedure 2. For the converged potential calculate the energies at k-points along lines. 3. Use some software to plot the band structure. 6 Kohn-Sham equations E (ev) 0 Γ 15 Γ' 25 X 1 E C E V S 1 n(r) = φ i (r) 2-10 Γ 1 Figure by MIT OpenCourseWare. i L Λ Γ X U,K k Σ Γ Image by MIT OpenCourseWare.

39 Calculating the DOS 1. Find the converged ground state density and potential. 3-step procedure 2. For the converged potential calculate energies at a VERY dense k-mesh. 3. Use some software to plot the DOS. Kohn-Sham equations n(r) = φ i (r) 2 i Image removed due to copyright restrictions. Please see Fig. 3 in Yu, R., and X. F. Zhang. "Platinum Nitride with Fluorite Structure." Applied Physics Letters 86 (2005):

40 silicon E (ev) 6 0 Γ 15 Γ' 25 Metal/insulator X 1 E C E V Fermi energy Are any bands crossing the Fermi energy? YES: METAL NO: INSULATOR -10 S 1 Number of electrons in unit cell: EVEN: MAYBE INSULATOR ODD: FOR SURE METAL Γ 1 Figure by MIT OpenCourseWare. L Λ Γ X U,K Σ Γ k Image by MIT OpenCourseWare.

41 Metal/insulator diamond: insulator

42 Metal/insulator BaBiO3: metal Courtesy of Elsevier, Inc., Used with permission.

43 Simple optical properties 6 E=hv E (ev) 0 unoccupied Γ 15 E Γ' C 25 X 1 E V gap photon has almost no momentum: only vertical transitions possible energy conversation and momentum conversation apply -10 L Λ S 1 occupied Γ 1 Figure by MIT OpenCourseWare. Γ X U,K k Σ Γ Image by MIT OpenCourseWare.

44 Silicon Solar Cells Have to Be Thick ($$$) It s all in the band- structure! Please see graph at

45 Literature Charles Kittel, Introduction to Solid State Physics Ashcroft and Mermin, Solid State Physics wikipedia, solid state physics, condensed matter physics,...

46 MIT OpenCourseWare J / 1.021J / J / J / 22.00J Introduction to Modeling and Simulation Spring 2011 For information about citing these materials or our Terms of use, visit: