Advanced Prop. of Materials: What else can we do?

Transcription

1 1.021, 3.021, , : Introduction to Modeling and Simulation : Spring 2011 Part II Quantum Mechanical Methods : Lecture 6 Advanced Prop. of Materials: What else can we do? 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 Lesson outline Review some stuff Optical properties Magnetic properties Transport properties Vibrational properties Courtesy of Elsevier, Inc., Used with permission.

4 The Saga of Length and Time Scales 10-3 Macro scale h = 0 Length (m) 10-6 Here be dragons Nano scale h = Time (s) Image by MIT OpenCourseWare.

5 Size vs.accuracy Size/duration Classical empirical methods Pair potentials Force fields Shell models Quantum empirical methods Tight-binding Embedded atom Quantum self-consistent methods Density functional theory Hartree-Fock Quantum many-body methods Quantum Monte Carlo MP2, CCSD(T), Cl GW, BSE Accuracy Image by MIT OpenCourseWare.

6 Review: inverse lattice Schrödinger certain quantum equation symmetry number hydrogen atom spherical symmetry [H, L 2 ] = HL 2 L 2 H = 0 [H, L z ] = 0 ψ n,l,m (r) periodic translational solid symmetry ψ n, ( [H, T ] = 0 k r)

7 Review: inverse lattice kz some G ky kx ψ k (r) ψ k+ G (r) E k = E k+ G

8 Review:The band structure 6 Silicon E (ev) 0 Γ 15 Γ' 25 X 1 E C E V S 1 k is a continuous variable -10 L Λ Γ 1 Figure by MIT OpenCourseWare. Γ X U,K k Σ Γ Image by MIT OpenCourseWare.

9 Review: inverse lattice R ψ 0 (r + R ) = ψ 0 (r) periodic over unit cell ψ (r + 2R ) = ψ (r) G/2 G/2 periodic over larger domain R R

10 Review: inverse lattice choose certain k-mesh e.g. 8x8x8 N=512 number of k-points (N) unit cells in the periodic domain (N)

11 Review: inverse lattice Distribute all electrons over the lowest states. N k-points You have (electrons per unit cell)*n electrons to distribute!

12 Let s Do A Few Simulations

13 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

14 The Fermi energy 6 E (ev) 0 unoccupied Γ 15 E Γ' C 25 X 1 E V Fermi energy each band can hold: -10 L Λ S 1 occupied Γ 1 Figure by MIT OpenCourseWare. Γ X U,K k Σ Γ 2N electrons and you have (electrons per unit cell)*n or two electrons and you have (electrons per unit cell) Image by MIT OpenCourseWare.

15 E (ev) silicon 6 0 Electrical properties 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 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.

16 Electrical properties diamond: insulator

17 Electron Transport E dv ee 1 E-field = v = 0 dt m τ At equilibrium v = eτe m m, e j = nev = ne 2 τ m E σe Electric current Electrical conductivity σ = ne2 τ m

18 Electron Transport Calculating σ from band structure σ = e 2 τ Fermi function dk f 4π 3 v(k)v(k) E 1 v(k) = k E(k) Curvature of band structure source unknown. All rights reserved. This content is excluded from our Creative Commons license. For more information, see

19 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.

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

21 Simple optical properties

22 Magnetism S N Origin of magnetism: electron spin An electron has a magnetic moment of μb, Bohr magneton. µ = µ B (n n ) Spin up n Spin down n

23 Magnetization spin-polarized calculation: separate density for electrons with spin up down Integrated difference between up and down density gives the magnetization.

24 Magnetism In real systems, the density of states needs to be considered. 3 bcc Fe DOS (states/ev) E - E F (ev) EF µ = µ B de[g (E) g (E)]

25 Quantum Molecular Dynamics and let us, as nature directs, begin first with first principles. Aristotle (Poetics, I) F=ma Use Hellmann-Feynman!

26 Carbon Nanotube Growth Carbon nanotube growth:

27 Silicon Nanocluster Growth Silicon nanocluster growth: source unknown. All rights reserved. This content is excluded from our Creative Commons license. For more information, see

28 Water Henry Cavendish was the first to describe correctly the composition of water (2 H + 1 O), in He reported his findings in terms of phlogiston (later the gas he made was proven to be hydrogen) and dephlogisticated air (later this was proven to be oxygen). Cavendish was a pretty neat guy. A University dropout, he also compared the conductivities of electrolytes and expressed a version of Ohm's law. His last major work was the first measurement of Newton's gravitational constant, with the mass and density of the Earth.The accuracy of this experiment was not improved for a century.

29 Water Which of the following is the correct picture for H2O? Martin Chaplin. License: CC-BY-SA-ND. This content is excluded from our Creative Commons license. For more information, see Cool water site:

30 Classical or Quantum? Please see the second table at More than 50 classical potentials in use today for water. Which one is best?

31 Mg++ in Water classical! quantum! Martin Chaplin. License: CC-BY-SA-ND. This content is excluded from our Creative Commons license. For more information, see Important Differences!

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

33 Vibrational properties animated phonons on the web leeviz/phonon/phonon.html sound in solids determined by acoustical phonons (shock waves) some optical properties related to optical phonons heat capacity and transport related to phonons

34 Summary of properties structural properties electrical properties optical properties magnetic properties vibrational properties

35 Literature Charles Kittel, Introduction to Solid State Physics Ashcroft and Mermin, Solid State Physics wikipedia, phonons, lattice vibrations,...

36 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: