Advanced Materials, 7.5 credits

Size: px

Start display at page:

Download "Advanced Materials, 7.5 credits"

Christopher Ramsey
6 years ago
Views:

1 Advanced Materials, 7.5 credits Teacher: Ludvig Edman Room A117, bottom floor, Fysikhuset Phone: Instructor (labs + project + exercises): Andreas Sandström Room C202, second floor, Fysikhuset andreas.sandstrom@physics.umu.se Phone: Homepages:

2 General information Course material: Home page: Handouts -- including lab instructions, problem sets, reading assignments, schedules, course status, project information, etc. -- are all available at: Literature on the web: Accessible also from home via proxy server: Textbook: Ashcroft/Mermin Solid State Physics or equivalent text book in solid-state physics such as Hook & Hall, Kittel

3 General information Course requirements Problems: Lecture-based problems assigned before each meeting (see home page). All must be solved, and solutions handed in before 27/5. Exception: Randomly picked person that solve all his/her assigned problems on black board (exception only valid for the set of problems assigned that specific meeting) Laboratory projects: Supervisor: Andreas Sandström Thermoelectric effects & Superconductivity Oral report at the end of each lab; otherwise written report Project: Supervisor: Andreas Sandström Single-person project Deadline to decide topic: 15/4 Deadline for written report (max 10 pages, template on home page): 10/5 Oral presentation (~15 mins): 25/5 Written exam: 2/6 9.00

4 General information Passing grade without taking exam Opportunity: A passing grade can be obtained without taking the final exam, provided that the following criteria are fulfilled: 1. All solutions to the problems handed in (or approved) before 27/5 2. Laboratory projects approved before 27/5 3. Project report handed in before 10/5 4. Active participation during lectures; important: only a maximum of two lectures can be missed!

5 Introduction to course Background: D-level course on advanced materials with energy relevance Alternative and renewable energy sources, energy storage and energy conversion materials and devices, and other future energy-relevant materials New types of advanced materials with unknown capacity

6 Introduction to course Examples of topics that will be covered Semiconductors (both conventional inorganic and emerging organic): Solar cells, efficient and/or flexible lighting and displays, small and large-scale electronic devices Superconductors: Very strong electromagnets, zero-loss power lines Thermoelectric materials: New routes to efficient power generation and vibration-free refrigerators Nanomaterials:??? Biomaterials: How to mimic (learn from) nature! Energy-storage devices and materials: Supercapacitors, batteries, hydrogen-rich materials, lithium compounds Efficient and/or green energy-conversion devices: Solar cells, hydrogen fuel cells, solid-state lighting, thermoelectric devices

7 1a. Energy and Materials What is Energy? Capacity of a system to do work Forms of energy? Kinetic: Potential: Thermal: Electrical: Chemical: Radiant: Nuclear: Sound: object movement object position heat content of object moving charges or static charge distributions stored in chemical bonds stored in electromagnetic waves stored in the atomic nucleus stored in compression waves

8 1a. Energy and Materials Examples of transformation of energy: (Manifestations of 1 st law of TD) Water at the top of a dam (potential energy) Fast flowing water driving a turbine at the bottom of the dam (kinetic energy) Generator produces AC current (electric energy) Electric radiator (heat energy) Two battery electrodes containing two different redox active materials (chemical energy) Electrode reactions produce electric potential difference, which in turn produces electric current in outer circuit (electrical energy) Current powers a lamp (radiation (and thermal) energy)

9 1a. Energy and Materials Renewable or not? Water power initially driven by ongoing fusion reactions within the sun (solar power): Renewable energy source (that is naturally and constantly replenished) Burning of fossil fuels (natural gas, oil and coal) which are remains of organisms that lived millions of years ago. This is a transformation of chemical energy stored in hydrocarbon bonds created by ancient solar power: Non-renewable energy source (that is depleted over time) Electrode materials in a rechargeable battery need to be regenerated by external (renewable or non-renewable) energy source: Energy conversion device

10 1a. Energy and Materials So which types of energy are we using? 83 % fossil fuels: 41 % oil, 20 % coal, 22 % natural gas (Sweden: 32 % (27 %, 4 %, 1 %) 11 % nuclear power (Sweden: 30 %) 2 % hydroelectric (Sweden: 22 %) ~>4 % other renewable: dominated by biomass, but also contributions from solar heat, wind (growing 30 %/year), geothermal, solar photovoltaic (large installations popular in Germany and Spain, small installations common in Kenya), etc. (Sweden: 16 %)

11 1a. Energy and Materials Which energy should we use? Non-renewable sources will eventually run out (1 trillion barrels of highquality oil have been used; 2-4 of lower quality remain) [1 barrel 160 l] Green-house effect related to burning of non-renewable fossil fuels Net oil producers in unstable regions (Saudi Arabia, Iran, Iraq, ) New population rich countries (e.g., China, India) expected to start consuming large quantities of energy Undesirable to expand on nuclear power production to unstable regions Renewable is desirable (e.g., George W. Bush: U.S. should convert to a hydrogen society ) But how can it be done?

12 1a. Energy and Materials Development of new and advanced materials for use in a number of hot renewable and/or energyefficient areas; just a few examples: Solar cells (organic & inorganic) Thermoelectric power generation Hydrogen storage and utilization (example: fuel cells) Thin and efficient batteries Organic, inorganic and bio-based electronics Solid-state lighting Nano-science and nanotechnology

1b. Lattices, crystals, and quantized vibrations The static lattice Bravais lattice (BL): Periodic structure in which each lattice point looks on identical lattice-point environment; often shown by

13 1b. Lattices, crystals, and quantized vibrations The static lattice Bravais lattice (BL): Periodic structure in which each lattice point looks on identical lattice-point environment; often shown by unit cells (14 different BLs identified by Bravais 1845) 3-D BL in vector notation: R = n 1 a 1 + n 2 a 2 + n 3 a 3 Examples of BLs: SC: BCC: FCC: Trigonal R: -Po Li, Na, K, Fe, W Al, Au, Cu, Pb, Ne (below 24 K) 5K

): Diamond 2 interpenetrating (IP) FCC BLs (C, Si, ) HCP 2 IP hexagonal BLs (Mg, Zn,

14 1b. Lattices, crystals, and quantized vibrations Bravais lattices with base more structures (230!): Diamond 2 interpenetrating (IP) FCC BLs (C, Si, ) HCP 2 IP hexagonal BLs (Mg, Zn, low-t α-co, ) Ideal case: c/a = (8/3) 1/2 NaCl Appear to be SC BL but 2 different atoms 2 IP FCC BLs, and also base different than diamond (many alkaline salts, CaO, ),

15 1b. Lattices, crystals, and quantized vibrations But what holds a crystal together? Chemical or physical bonds! What is the difference between different types of bonds?

1b. Lattices, crystals, and quantized vibrations Valence electron distribution differs in different bonds Molecular crystals: Very small change in valence electron density (ED) compared to

16 1b. Lattices, crystals, and quantized vibrations Valence electron distribution differs in different bonds Molecular crystals: Very small change in valence electron density (ED) compared to constituent atoms/molecules and consequently very little ED between ion cores (Ne, Ar, Kr,...) Ionic crystals: Valence electron(s) transferred from one constituent atom to other strong electrostatic forces (Na + Cl -, K + Cl -,...) Covalent crystals: Large change in ED; valence electrons localized in pairs in certain directions ( chemical bonds ) (diamond, Si,...) Metallic crystals: ED uniformly spread

17 1b. Lattices, crystals, and quantized vibrations But why does all these different bonds hold a solid together? Electric attraction between positive nuclei and negative electrons (via electronic structure) Molecular/ionic crystals: Valence electrons unperturbed & identical to isolated molecules/ions Bonding: Fluctuating dipoles in molecular crystals (E ~ -1/r 6 short range) and inter-ionic interactions in ionic crystals (E ~ -1/r long range) Repulsive energy: core-core repulsion of filled atomic shells (Pauli exclusion principle) Metals & Covalent crystals : Valence electrons strongly perturbed & delocalized Bonding mechanism (simplified): Valence electron delocalized Δx increases; uncertainty principle: Δx*Δp x ħ Δp x decreases max p x decreases E k (= p 2 /2m e ) for valence electron decreases It lowers its energy and gets stuck in its new configuration (unless sufficient energy provided to lift it out if its confined state)

1b. Lattices, crystals, and quantized vibrations Back to structures: Each specific structure (SC, FCC, etc.) can be described by a unit cell! (But why even bother with unit cells?

18 1b. Lattices, crystals, and quantized vibrations Back to structures: Each specific structure (SC, FCC, etc.) can be described by a unit cell! (But why even bother with unit cells? Very important concept for the understanding of lattice vibrations specific heat & heat transport - and electronic properties electric and thermal conductivity & color, etc.!) Primitive unit cells contain only one lattice point Wigner-Seitz primitive unit cell most common choice: Contains all space closest its one lattice point Basis for construction of first Brillouin zone (BZ) in k-space Simple cubic BL: a 1 = ax, a 2 = ay, a 3 = az First BZ in k-space: More on the usefulness of BZ:s soon b 1 = (2π/a)x, b 2 = (2π/a)y, b 3 = (2π/a)z

19 1b. Lattices, crystals, and quantized vibrations The dynamic lattice (advanced part!) So far static lattice, but what about lattice dynamics? C v = (dq/dt) V 0 & Thermal expansion! Some sort of vibrational motion in the lattice takes place. Difficult to analyze, but consider a simple case: i) Harmonic approximation of lattice potential: An atom vibrating around an equilibrium position, and restoring force proportional to equilibrium displacement (x): F = - U = - Kx ( U = U 0 + ½Kx 2 ) ii) Linear chain (1D Bravais lattice) of N atoms separated by a with periodic boundary condition for x: x(a) = x[(n+1)a] iii) Only nearest-neighbor interactions in 1D chain: F n = F n n+1 + F n n-1 iv) Assume wave solution for displacement: x(na,t) e i(kna-ωt) + math

20 1b. Lattices, crystals, and quantized vibrations Dispersion relation in 1D: ω(k) = (4K/m) 0.5 sin(½ak) (for discrete values of k = 2 n/an between N/2,..., N/2 in 1st BZ) Discussion of parameters: Atoms vibrating with a frequency, ω, around their equilibrium position Distance between atoms that are vibrating in phase is: λ = 2π/k

21 1b. Lattices, crystals, and quantized vibrations Dispersion relation in 1D: ω(k) = (4K/m) 0.5 sin(½ak) (for discrete values of k = 2 n/an between N/2,..., N/2 in 1st BZ) Important things to note: i) Each discrete k (=2π/λ) connected to a specific ω: a normal mode (NM) ii) Each NM can be excited to different energy ~(amplitude of disturbance) 2 iii) Small k (long λ, sin k = k) phase velocity (ω/k) = group velocity(dω/dk)~(k/m) 0.5 : sound-wave region (strong and light diamond: m/s; heavy Pb: m/s) iv) Large k (small λ a) linearity breaks down ( discrete media effect): dω/dk 0

very small value In 3D-Bravais lattice, each branch splits up into 3: one longitudinal and two transversal Note band gaps between different

22 1b. Lattices, crystals, and quantized vibrations 1D-Bravais lattice with a two-atom basis: 2 branches of ω(k)-relation: Acoustic branch (sound waves at small k) & optical branch (can interact with EM radiation with appropriate ω = E/ħ and k = p/ħ at small k) Note: p photon = ħk photon = very small value In 3D-Bravais lattice, each branch splits up into 3: one longitudinal and two transversal Note band gaps between different branches production of photonic crystals, which can be built from many periodic structures including tennis balls, but more common structures include

photonic crystal fiber) and in nature (e.g.

23 1b. Photonic crystals Periodic nanostructures designed to influence motion of photons (in similar way that periodicity of SC crystals affects motion of electrons, lecture 2). Photonic crystals occur in synthetic forms (e.g. photonic crystal fiber) and in nature (e.g. opal) No allowed propagation of EM modes in photonic band gap high-reflecting mirrors & low-loss wave-guiding Since basic physical phenomenon based on diffraction periodicity of photonic crystal structure ~ 0.5* ~ 300 nm for applications in visible part of EM spectrum Difficult fabrication

1b. Lattices, crystals, and quantized vibrations But how can lattice dynamics, i.e. dispersion relations, be measured in a real crystal (periodic structure)?

24 1b. Lattices, crystals, and quantized vibrations But how can lattice dynamics, i.e. dispersion relations, be measured in a real crystal (periodic structure)? Two ways: Neutron or photon inelastic scattering Interaction with material excites or extinguishes one (or more) phonons; energy and (crystal) momentum must be conserved dispersion relations in various directions given by change in energy and momentum between incoming and outgoing particles Note: fundamental difference between E-k relationships of neutrons and photons. Photons can in principle only excite phonons close to k = 0 due to their low k-values (momentum)

25 1b. Lattices, crystals, and quantized vibrations Ok, we understand the dynamics of the lattice; can we also predict the elusive C v? Classical treatment: 1. A continuum of NM energy levels (given by square of amplitude) 2. E NM (average) = k B T 3. 3N different NMs E th = 3N*k B T C v = (1/V)*(dE th /dt) = 3Nk B /V (Dulong-Petit law, independent of T) Poor agreement at low T!

26 1b. Lattices, crystals, and quantized vibrations Quantum treatment: NM energies (and amplitudes) quantized: E NM = (n+½)ħω n = # excitations of a NM = # phonons (can be considered quasiparticles; remember waveparticle duality in quantum physics!) C v : ~ T 3 (low T) = 3Nk B /V (high T: all NMs excited) Intermediate T: Debye model (acoustic dispersion relations approximated by 3 linear branches) + Einstein approximation (optical branches with constant ω E ) employed Expression for C v in good agreement with experiment

27 U 1b. Lattices, crystals, and quantized vibrations Last problem: Anharmonic effects (= more complicated interaction between atoms than harmonic approximation: U = U 0 + ½Kx 2 + mx 3 + ) must be considered to explain: harmonic Anharmonic 1. Certain equilibrium properties, e.g. thermal expansion. When atoms start to vibrate more at higher T their equilibrium position shifts and the crystal expands 2. Certain transport properties; for instance, a perfect harmonic crystal should have infinite thermal conductivity ( ), which is obviously not observed experimentally: j heat = *ΔT/Δx -1,0-0,5 0,0 0,5 since very small temperature gradient would produce infinite heat current! x

28 Next lecture Friday 26/3, at in N335: Electrons in solids

Lecture 11 - Phonons II - Thermal Prop. Continued

Lecture 11 - Phonons II - Thermal Prop. Continued Phonons II - hermal Properties - Continued (Kittel Ch. 5) Low High Outline Anharmonicity Crucial for hermal expansion other changes with pressure temperature Gruneisen Constant hermal Heat ransport Phonon