1.021, 3.021, 10.333, 22.00 : Introduction to Modeling and Simulation : Spring 2012 Part II – Quantum Mechanical Methods : Lecture 9 Some Review & Introduction to Solar PV Jeffrey C. Grossman Department of Materials Science and Engineering Massachusetts Institute of Technology � Part II Topics 1. 2. 3. 4. 5. 6. 7. 8. It’s a Quantum World:The Theory of Quantum Mechanics 10. 11. Application of Quantum Modeling of Solids: Solar Cells Part II Quantum Mechanics: Practice Makes Perfect From Many-Body to Single-Particle; Quantum Modeling of Molecules Application of Quantum Modeling of Molecules: Solar Thermal Fuels Application of Quantum Modeling of Molecules: Hydrogen Storage From Atoms to Solids Quantum Modeling of Solids: Basic Properties Advanced Prop. of Materials:What else can we do? Application of Quantum Modeling of Solids: Nanotechnology 2 Lesson outline • Discussion of PSET • Review for the Quiz • Introduction to Solar PV 3 Motivation: ab-initio modeling! ? mechanical properties ? electrical properties ? optical properties 4 Vision without Acton is a Dream Acton without Vision is a Ni htmare Japanese proverb 5 Why quantum mechanics? Problems in classical physics that led to quantum mechanics: • “classical atom” • quantization of properties • wave aspect of matter • (black-body radiation), ... 6 Wave aspect of matter light matter wave character particle character _ _ _ _ _ _ _ _ _ _ _ _ _ Image by MIT OpenCourseWare. Image in public domain. See Wikimedia Commons. 7 Wave aspect of matter particle:E and momentum pk wave k h k p k = nk k= λ |k k| de Broglie: free particle can be described a as h planewave i(� k·� r −ωt) ψ(k r , t) = Ae λ= with . mv 8 Interpretation of a wavefunction ψ(k r , t) wave function (complex) |ψ|2 = ψψ ∗ interpretation as probability to find particle! � ∞ −∞ �(r, t) �(r, t) 2 Image by MIT OpenCourseWare. 9 ψψ ∗ dV = 1 Schrödinger equation H time independent: in f˙(t) f (t) = ψ(k r , t) = ψ(k r ) · f (t) Hψ(k r) ψ(k r) Hψ(k r ) = Eψ(k r) = const. = E ψ(k r , t) = ψ(k r) · e time independent Schrödinger equation stationary Schrödinger equation �� − �i Et The hydrogen atom stationary Schrödinger equation Hψ = Eψ � � e v l o s just � − � T + V ψ = Eψ � n r ) = Eψ(k r) − ∇2 + V ψ(k 2m n2 2m 2 2 ∇ − e2 4π�0 r �� � ψ(k r ) = Eψ(k r) The hydrogen atom quantum numbers n 1 2 2 2 l 0 0 1 1 ml F(φ) P(θ) 0 1 2π 1 2 a03/2 0 1 2π 1 2 -r / 2a 1 r 0 2e 2 2a03/2 a0 0 1 2π 6 cos θ 2 1 r -r / 2a0 e 3/2 a 2 6 a0 0 1 ±iφ e 2π 3 sin θ 2 1 r -r / 2a0 e 3/2 a 2 6 a0 0 ±1 R(r) 2 e -r/a0 Image by MIT OpenCourseWare. �2 The hydrogen atom Energies: © R. Nave. All rights reserved. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/. �3 Atomic units 1 eV = -19 1.6021765 J 1 Rydberg = 13.605692 eV = 2.1798719-18 J 1 Hartree = 2 Rydberg 1 Bohr =5.2917721-11 m Energies in Ry Distances in Bohr Atomic units (a.u.): Also in use: -10 1Å=10 m, nm= �4 -9 10 m Everything is spinning ... Stern–Gerlach experiment (1922) k F = −∇E k = ∇m k ·B Image courtesy of Teresa Knott. �5 Everything is spinning ... new quantum number: spin quantum number for electrons: spin quantum number can ONLY be up down �6 Pauli’s exclusion principle Two electrons in a system cannot have the same quantum numbers! hydrogen ... 3s 2s quantum numbers: main n: 1,2,3 ... orbital l: 0,1,...,n-1 magnetic m: -l,...,l spin: up, down 1s �7 ... ... 3p 3d 2p ... Periodic table of elements This image is in the public domain. Source: Wikimedia Commons. �8 e­ Next? Helium? r12 e­ + Hψ = Eψ ⇥ H1 + H2 + W ψ(k r1 , k r2 ) = Eψ(k r1 , k r2 ) r2 ⇥ T1 + V1 + T2 + V2 + W ψ(k r1 , k r2 ) = Eψ(k r1 , k r2 ) − n2 2m ∇21 − e2 4π� 0 r1 − n2 2m ∇22 − e2 4π� 0 r2 + e2 4π� 0 r12 ⇥ ψ(k r1 , k r2 ) = Eψ(k r1 , k r2 ) problem! cannot be solved analytically �9 Solutions density functional theory quantum chemistry Moller-Plesset perturbation theory MP2 coupled cluster theory CCSD(T) 2� The Two Paths Ψ is a wave function of all positions & time. ⇥ ∂ 2 − r , t) ψ(k r , t) = in ψ(k ∇ + V (k r , t) 2m ∂t n2 Chemists (mostly) Ψ = something simpler - - - - - Physicists (mostly) H = something simpler “mean field” methods 2� Working with the Density E[n] = T [n] + Vii + Vie [n] + Vee [n] kinetic ion-ion Walter Kohn © unknown. All rights reserved. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/. n=# 1 10 100 1,000 ion-electron electron-electron Ψ(N3n) 8 109 1090 10900 ion potential 22 ρ(N3) 8 8 8 8 Hartree potential exchange-correlation potential Why DFT? 100,000 10,000 Number of Atoms e n Li c s ar 1000 g n ali T F D DFT MP2 100 QMC CCSD(T) 10 1 Exact treatment 2003 2007 2011 Year 2015 Image by MIT OpenCourseWare. 23 Density functional theory E[n] = T [n] + Vii + Vie [n] + Vee [n] kinetic ion-ion ion-electron electron density n(kr) = electron-electron |φi (k r )| 2 i Eground state = min E[n] φ Find the wave functions that minimize the energy using a functional derivative. 24 Self-consistent cycle Kohn-Sham equations sc f lo op |φi (k r )|2 n(k r) = i 25 Density functional theory Only one problem: vxc not known!!! approximations necessary local density approximation LDA general gradient approximation GGA 26 • structure • bulk modulus • shear modulus • elastic constants • vibrational properties • sound velocity binding energies reaction paths forces pressure stress ... 27 Convergence for molecules g i b x o b y m s a ? h W g u o en ? f c s e h t t i x e I t d h i g i D r e h t t a loop ? t n i po 28 as my basis b ig enoug h? Crystal symmetries S = simple BC = body centered FC = face centered © Sandeep Sangal/IITK. All rights reserved. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/. 29 The inverse lattice The real space lattice is described by three basis vectors: k = n1k a1 + n2k a2 + n3k a3 R The inverse lattice is described by three basis vectors: k = m1kb1 + m2kb2 + m3k G b3 iG·R e ψ(k r) = =1 iGj ·r cj e j automatically periodic in R! 3� 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-SA. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/. 3� The Brillouin zone Brillouin zone of the FCC lattice 32 Periodic potentials V (k r) k R k R k R © Prof. Dr. Helmut Föll. All rights reserved. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/fairuse. llattice vector k V (k r ) = V (k r + R) NEW quantum number k that N lives in the inverse lattice! Bloch’s theorem ψk (kr) = e ik·r uk (k r) k) uk(k r ) = uk (k r+R 33 Inverse lattice Results of Bloch’s theorem: ik·R k ψk (k r + R) = ψk (k r )e 2 2 k |ψk (k r + R)| = |ψk (k r )| if solution r) ⇤ k (⇤ with r) ⇤ ⇤ (⇤ k+G ⇥ E⇤k = E⇤k+G ⇤ 34 charge density is lattice periodic also solution Structural properties Etot finding the stress/pressure and the bulk modulus V V0 p=− ∂E ∂V σbulk = −V 35 ∂p ∂V =V 2 ∂ E ∂V 2 Calculating the band structure 1. Find the converged ground state 3-step procedure 2. 3. density and potential. For the converged potential calculate the energies at k-points along lines. Use some software to plot the band structure. 6 E (eV) Kohn-Sham equations G15 G'25 0 EC EV X1 S1 n(k r) = |φi (k r )| -10 2 G1 Figure by MIT OpenCourseWare. L i L G D X U,K S G k 36 Image by MIT OpenCourseWare. Metal/insulator silicon 6 E (eV) G15 G'25 0 EC EV X1 Fermi energy S1 Number of electron in unit cell: EVEN: MAYBE INSULATOR ODD: FOR SURE METAL -10 G1 Figure by MIT OpenCourseWare. L L G D X U,K Are any bands crossing the Fermi energy? YES: METAL NO: INSULATOR S G k Image by MIT OpenCourseWare. 37 Simple optical properties 6 unoccupied G 15 E (eV) E=hv G'25 0 S1 EC EV X1 occupied -10 G1 Figure by MIT OpenCourseWare. L L G D k X U,K S G Image by MIT OpenCourseWare. shortest wave length visible 400 nm ; corresponds to photon with E=3.1 eV 38 Image in the public domain. Vibrational properties lattice vibrations are called: phonons force What is the frequency of this vibration? 39 iron Magnetization &control calculation = 'scf', pseudo_dir = '' / &system ibrav=3, celldm(1)=5.25, nat=1, ntyp=1, ecutwfc=25.0, occupations='smearing' smearing='gauss', degauss=0.05, nspin=1 starting_magnetization(1)=0.0 / &electrons conv_thr=1.0d-10 / ATOMIC_SPECIES Fe 55.847 iron.UPF ATOMIC_POSITIONS {crystal} Fe 0.0 0.0 0.0 K_POINTS {automatic} 444111 nspin=1: non spin-polarized nspin=2: spin-polarized starting magnetization for each atom perform three calculations and find lowest energy: non spin-polarized spin-polarized ferromagnetic 4� anti-ferromagnetic With the band structure and DOS we find: • • • • • • electrical conductivity (insulator/metal/semiconductor) thermal conductivity optical properties magnetization/polarization magnetic/electric properties ... 4� Convergence for solids h s e m k y m ? s h a g u W o n e e fin ? f c s e h t t i x e I t d h i g i D r e h t t a loop ? t n i po 42 as my basis b ig enoug h? Summary of properties structural properties electrical properties optical properties magnetic properties vibrational properties 43 Image of Airbus A380 on Wikimedia Commons. License: CC-BY-SA. Images of circuit board, phone, hard drive © sources unknown. All rights reserved. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/. Aerothermodynamic image of shuttle, courtesy NASA. It’s Not Only About Warming: Abundance of Affordable Energy Resources Can Uplift the World x4 x13 HUMAN WELL-BEING INCREASES WITH INCREASED PER-CAPITA ENERGY USE © AIP Publishing. All rights reserved. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/. 44 Courtesy:Vladimir Bulovic It’s Not Only About Warming: Abundance of Affordable Energy Resources Can Uplift the World North American Electrical Blackout 08/14/2003 … August 15 … just 24 hours into blackout Air Pollution was Reduced SO2 >90% O3 ~50% Light Scattering Particles ~70% “This clean air benefit was realized over much of eastern U.S.” Marufu et al., Geophysical Research Letters 2004 New York City Skyline by 4:13 pm 256 power plants were off-line Courtesy:Vladimir Bulovic Images courtesy of Vladimir Bulovic. All rights reserved. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/. 45 Map of the World Scaled to Energy Consumption by Country Newman, U. Michigan, 2006 Courtesy of M. E. J. Newman. • In 2002 the world burned energy at a rate of 13.5 TW • How fast will we burn energy in 2050? • (assume 9 billion people) • If we use energy like in U.S. we will need 102 TW • Conservative estimate: 28~35 TW Energy use estimates from Nocera, Daedalus 2006 & Lewis and Nocera, PNAS 2006 46 U.S. Energy Consumption Goal: consume half of our electricity through renewable sources by the year 2050. © unknown. All rights reserved. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/. 47 Energy from the Sun • Energy released by an earthquake of magnitude 8 (1017 J): • the sun delivers this in one second • Energy humans use annually (1020 J): • sun delivers this in one hour • Earth’s total resources of oil (3 trillion barrels, 1022 J): • the sun delivers this in two days 48 Courtesy of SOHO/EIT (ESA & NASA) consortium. MIT OpenCourseWare http://ocw.mit.edu 3.021J / 1.021J / 10.333J / 18.361J / 22.00J Introduction to Modelling and Simulation Spring 2012 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.