Some Review & Introduction to Solar PV

Transcription

1 1.021, 3.021, , : 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

2 Part II Topics 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. Application of Quantum Modeling of Molecules: Solar Thermal Fuels 5. Application of Quantum Modeling of Molecules: Hydrogen Storage 6. From Atoms to Solids 7. Quantum Modeling of Solids: Basic Properties 8. Advanced Prop. of Materials:What else can we do? 10. Application of Quantum Modeling of Solids: Solar Cells Part II 11. Application of Quantum Modeling of Solids: Nanotechnology 2

3 Lesson outline Discussion of PSET Review for the Quiz Introduction to Solar PV 3

4 Motivation: ab-initio modeling!? mechanical properties? electrical properties? optical properties 4

5 Vision without Acton is a Dream Acton without Vision is a Ni htmare Japanese proverb 5

6 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

7 Wave aspect of matter light matter wave character particle character _ Image by MIT OpenCourseWare. Image in public domain. See Wikimedia Commons. 7

8 Wave aspect of matter particle: E and momentum kp wave p k = nk = h k λ k de Broglie: free particle can be described a as planewave with. ψ(kr, t) =Ae i( k r ωt) 8 λ = h mv

9 Interpretation of a wavefunction ψ(kr, t) ψ 2 = ψψ wave function (complex) interpretation as probability to find particle! ψψ dv = 1 Ψ(r, t) Ψ(r, t) 2 Image by MIT OpenCourseWare. 9

10 Schrödinger equation H time independent: ψ(kr, t) =ψ(kr) f(t) f (t) Hψ(kr) in = = const. = E f(t) ψ(kr) Hψ(kr) =Eψ(kr) i Et ψ(kr, t) =ψ(kr) e time independent Schrödinger equation stationary Schrödinger equation

11 The hydrogen atom stationary Schrödinger equation Hψ = Eψ [ ] T + V ψ = Eψ just solve [ n 2 ] 2 + V ψ(kr) =Eψ(kr) 2m n 2 e 2 ] 2 ψ(kr) =Eψ(kr) 2m 4πɛ 0 r

12 The hydrogen atom quantum numbers n l m l F(φ) P(θ) R(r) π a 0 3/2 e -r/a π a 3/ r a e 0 -r / 2a π 6 2 cos θ 1 26a r a e 3/ r / 2a ± 1 1 2π e±iφ 3 2 sin θ 1 26a r a e 3/ r / 2a 0 Image by MIT OpenCourseWare. 2

13 The hydrogen atom Energies: R. Nave. All rights reserved. This content is excluded from our Creative Commons license. For more information, see 3

14 Atomic units 1 ev = J 1 Rydberg = ev = J 1 Hartree = 2 Rydberg 1 Bohr = m Atomic units (a.u.): Energies in Ry Distances in Bohr Also in use: 1Å=10-10 m, nm= 10-9 m 4

15 Everything is spinning... Stern Gerlach experiment (1922) kf = E = m k Bk Image courtesy of Teresa Knott. 5

16 Everything is spinning... new quantum number: spin quantum number for electrons: spin quantum number can ONLY be up down 6

17 Pauli s exclusion principle Two electrons in a system cannot have the same quantum numbers! quantum numbers: main n: 1,2,3... orbital l: 0,1,...,n-1 magnetic m: -l,...,l spin: up, down hydrogen s 3p 3d 2s 2p 1s 7

18 Periodic table of elements This image is in the public domain. Source: Wikimedia Commons. 8

19 e Next? Helium? r12 Hψ = Eψ + r2 e H 1 + H 2 + W ψ(kr 1,kr 2 )=Eψ(kr 1,kr 2 ) T 1 + V 1 + T 2 + V 2 + W ψ(kr 1,kr 2 )=Eψ(kr 1,kr 2 ) n e n e m 4πɛ 0 r 1 2m 4πɛ 0 r 2 e 2 4πɛ 0 r 12 ψ(kr 1,kr 2 )=Eψ(kr 1,kr 2 ) cannot be solved analytically problem! 9

20 Solutions density quantum chemistry functional theory Moller-Plesset perturbation theory MP2 coupled cluster theory CCSD(T) 2

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

22 Working with the Density E[n] =T [n] +V ii + V ie [n] +V ee [n] kinetic ion-electron ion-ion electron-electron Walter Kohn unknown. All rights reserved. This content is excluded from our Creative Commons license. For more information, see n=# Ψ(N 3n ) ρ(n 3 ) , ion potential Hartree potential exchange-correlation potential 22

23 Why DFT? 100,000 Number of Atoms 10, QMC Linear scaling DFT DFT MP2 10 Exact treatment CCSD(T) Year 23 Image by MIT OpenCourseWare.

24 Density functional theory E[n] = T [n] + V ii + V ie [n] + V ee [n] kinetic ion-ion ion-electron electron-electron electron density n(kr) = φ i (kr) 2 E ground state =mine[n] φ i Find the wave functions that minimize the energy using a functional derivative. 24

25 Self-consistent cycle Kohn-Sham equations scf loop n(kr) = i φ i (kr) 2 25

26 Density functional theory Only one problem: vxc not known!!! approximations necessary local density approximation LDA general gradient approximation GGA 26

27 structure bulk modulus reaction shear modulus forces elastic constants stress binding energies paths pressure vibrational properties... sound velocity 27

28 Convergence for molecules Was my box big enough?? Did I exit the scf loop at the right point? as my basis big enough? 28

29 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 29

30 The inverse lattice The real space lattice is described by three basis vectors: kr = n 1 ka 1 + n 2 ka 2 + n 3 ka 3 The inverse lattice is described by three basis vectors: k k k k G = m 1 b 1 + m 2 b 2 + m 3 b 3 e ig R = 1 ψ(kr) = c j e ig j r j automatically periodic in R! 3

31 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 3

32 The Brillouin zone Brillouin zone of the FCC lattice 32

33 Periodic potentials V (kr) kr kr krk Prof. Dr. Helmut Föll. All rights reserved. This content is excluded from our Creative Commons license. For more information, see lattice vector V (kr) =V (kr + R) k NEW quantum number k that lives in the inverse lattice! Bloch s theorem ψ ( rk )=e ik r u (kr) k k u (kr) =u (kr + Rk ) k k 33

34 Inverse lattice Results of Bloch s theorem: ψ (kr + Rk ) =ψ (kr)e k k ik R charge density k k is lattice periodic ψ (kr + Rk ) 2 = ψ (kr) 2 if solution k ( r) k+ G ( r) also solution with E k = E k+ G 34

35 Structural properties finding the Etot stress/pressure and the bulk modulus V0 V E p 2 E p = σ bulk = V = V V V V 2 35

36 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(kr) = φ i (kr) 2-10 Γ 1 Figure by MIT OpenCourseWare. i L Λ Γ k X U,K Σ Γ 36 Image by MIT OpenCourseWare.

37 Metal/insulator silicon E (ev) 6 0 Γ 15 Γ' 25 X 1 E C E V Fermi energy Are any bands crossing the Fermi energy? YES: METAL NO: INSULATOR -10 S 1 Γ 1 Figure by MIT OpenCourseWare. Number of electron in unit cell: EVEN: MAYBE INSULATOR ODD: FOR SURE METAL L Λ Γ X U,K Σ Γ k Image by MIT OpenCourseWare. 37

38 Simple optical properties 6 E=hv E (ev) 0 unoccupied Γ 15 E Γ' C 25 X 1 E V S 1 occupied -10 Γ 1 Figure by MIT OpenCourseWare. L Λ Γ k X U,K Σ Γ shortest wave length visible 400 nm ; corresponds to photon with E=3.1 ev 38 Image by MIT OpenCourseWare. Image in the public domain.

39 Vibrational properties lattice vibrations are called: phonons force What is the frequency of this vibration? 39

40 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 iron.upf ATOMIC_POSITIONS {crystal} Fe K_POINTS {automatic} nspin=1: non spin-polarized nspin=2: spin-polarized starting magnetization for each atom perform three calculations non spin-polarized and find lowest energy: ferromagnetic spin-polarized anti-ferromagnetic 4

41 With the band structure and DOS we find: electrical conductivity (insulator/metal/semiconductor) thermal conductivity optical properties magnetization/polarization magnetic/electric properties... 4

42 Convergence for solids Was my k-mesh fine enough?? Did I exit the scf loop at the right point? as my basis big enough? 42

43 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 Aerothermodynamic image of shuttle, courtesy NASA.

44 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 44 Courtesy: Vladimir Bulovic

45 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 SO 2 >90% O 3 ~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 45

46 Map of the World Scaled to Energy Energy use estimates from Nocera, Daedalus 2006 & Lewis and Nocera, PNAS 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

47 U.S. Energy Consumption Goal: consume half of our electricity through renewable sources by the year unknown. All rights reserved. This content is excluded from our Creative Commons license. For more information, see 47

48 Energy from the Sun Energy released by an earthquake of magnitude 8 (10 17 J): the sun delivers this in one second Energy humans use annually (10 20 J): sun delivers this in one hour Earth s total resources of oil (3 trillion barrels, J): Courtesy of SOHO/EIT (ESA & NASA) consortium. the sun delivers this in two days 48

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