Physics of disordered materials. Gunnar A. Niklasson Solid State Physics Department of Engineering Sciences Uppsala University

Transcription

1 Physics of disordered materials Gunnar A. Niklasson Solid State Physics Department of Engineering Sciences Uppsala University

2 Course plan Familiarity with the basic description of disordered structures and associated physical concepts Giving background for following research literature. Follows the layout of an introductory course based on classic books, for example, C. Kittel: Introduction to Solid State physics. Outline Structural models Structural characterization Atomic vibrations (phonons) Electronic structure Electrical properties Optical properties

3 Resources Lecture notes Reference books in the Solid State Physics library, house 4, level 3. Examination: Home assignments, 3 sets. Contact: Gunnar Niklasson Room 4412, phone: gunnar.niklasson@angstrom.uu.se Homepage:

4 Motivation for the course The traditional field of disordered materials concerned glasses, amorphous materials and also liquids (see books by Cusack, Elliott) From the 1970 s and 80 s much interest was directed to fractal structures (B. Mandelbrot) In this course we combine the subjects of disordered solid materials and random fractal structures Next we look at the concept of disorder from various perspectives

5 Crystalline materials Introductory solid state physics physics of crystalline materials. Example: Si crystal used in semiconductor technology Effects of disorder important for many applications. Doping, nanoparticles, thin films Fundamental research Source: Wikimedia, M. Lincetto

6 Crystalline vs. amorphous solids

7 Weak disorder No disorder: Chain, flat surface, crystal Weak disorder: Folded chain, surface steps, defects, vacancies, dislocations, grain boundaries Perturbations of the perfect order Schematic surface Source: www1.columbia.edu

8 Strong disorder Homogeneous materials: Local density similar to the average density. Ex: Glass Inhomogeneous materials: Large fluctuations of the local density, density depends on length scale, pores of different size Ex: Aggregates, porous materials Ni particles

9 Fractals Structure repeats itself over a range of length scales. Physics: Disordered fractals Fractal dimension (B. Mandelbrot 1975) Lower cutoff: Atom, molecule or particle size Upper cutoff: Correlation length, size of structure Ex: Metallic particle aggregate, rough surface (ordered - von Koch curve) Source: Wikipedia

10 Types of disorder Topological (structural) No translational symmetry. Orientational (magnetic spins, dipoles) Substitutional (chemical, compositional) Vibrational (atoms vibrate around their equilibrium positions even at T=0. Source: Elliott PoAM

11 Ordering rule A property, p 1, of one object in a set is related to p i of object i by an ordering rule. Perfect order: p i of all objects can be derived from knowing one of them, by applying the ordering rule. Crystalline order: Translational symmetry is the ordering rule and position is the property. Not only used for structural order but also order referred to different physical properties.

12 Order parameter Atoms A and B placed on the sites of a crystal lattice with concentrations c A and c B =1-c A. Crystal: All A-atoms are in A-sites (r=1) and all B- atoms in B-sites. Complete disorder: Atoms placed randomly on the sites. Fraction c A of A-atoms in A-sites (r=c A ). Order parameter r ξ = 1 c c A A

13 Entropy and disorder S = k B ln W W number of microscopic states of the system compatible with the macrostate. Vacancies in a crystal Many arrangements possible: W increases Configurational entropy S>0 Mixing of two types of atoms or particles at constant volume W increases Entropy of mixing >0

14 Types of disordered solid materials Amorphous solids Many thin films a-semiconductors a-metals Glasses (supercooled liquids) Vitreous silica chalcogenide glasses Composites Random alloys Metal-insulator mixtures Particle dispersions Porous materials Often exhibit fractal structure at intermediate length scales

15 Glass transition Rapid cooling of a liquid: Supercooled liquid Not enough time to go into the lower energy crystalline state at the melting temperature T m As the temperature decreases below T m the liquid becomes more viscous Forms a glass (a quenched amorphous structure) in a region around a glass transition temperature T g T g depends on cooling rate and thermal history

16 Glass transition temperature Gradual change in volume at T g Steep change in some thermodynamic properties (heat capacity, thermal expansion, isothermal compressibility) From liquid-like to solid-like values Glass transition determined by thermal methods (Ex: DSC)

17 Viscosity A solid has viscosity η~ Pa s Ratio of shear stress to velocity gradient in a fluid ( resistance to flow ) η is proportional to the structural relaxation time, τ Limit above is arbitrary corresponding to a relaxation time of one day Structural relaxation becomes very slow below T g Most simple liquids: Pa s Glass transition corresponds to Pa s, depending on definition.

18 Viscosity Strong glass-formers: Often covalent bonds, ex. SiO 2. Arrhenius law: η =η exp( A/ 0 T Fragile glass formers: Often ionic and organic materials. Vogel- Tammann-Fulcher law: η =η ) exp( B /( T 0)) 0 T

19 Glass transition T g for some materials: SiO K B 2 O K Se 313 K PMMA 378 K Nylon K Polyethylene 253 K Poly(ethylene oxide) 218 K Poly (propylene oxide) 211 K Physics still not known in detail Many theoretical ideas Thermodynamics, free volume theory, mode coupling theory Connection to different relaxations, low frequency vibrations Subject stimulates both theoretical thinking and novel experimental methods

20 Origin of disorder Physical constraints: Atoms and particles may not penetrate each other Chemical bonds Interactions between atoms and molecules Attractive forces between particles; v.d. Waals, dipolar Randomness Disorder Physical Laws

21 Example: DLA aggregate Aggregation of particles that move by a random walk Which physical law gives rise to aggregation? Particles stick to the growing aggregate due to short-range attractive forces Such structures common in physics and nature

22 Fractional Brownian motion Displacement of a random walker as a function of time Trail exhibits fractal scaling with Hurst exponent H H=0.5 is the normal random walk (Brownian motion) Different rules of the random walk lead to trails with fractal scaling having different fractal dimensions D=2-H. D=1: A straight line. D=2: space filling Ex: Rough surfaces

23 Randomness vs. design Fractional Brownian motion Displacement vs. time (R.F. Voss) H=0.2 Bach concerto for two violins in D minor, movement 2, Pitch vs. time H=0.18