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Physics of disorderedmaterials
Gunnar A. NiklassonSolid State Physics
Department of Engineering SciencesUppsala University
Crystalline materials
• Introductory solid statephysics – physics of crystalline materials.
• Example: Si crystal used in semiconductor technology
• Effects of disorder importantfor many applications.
• Doping, nanoparticles, thinfilms
• Fundamental research
Source: Wikimedia, M. Lincetto
Crystalline vs. amorphous solids
www.ces-world.com
Course plan• Familiarity with the basic
description of disorderedstructures and associatedphysical concepts
• Giving neededbackground for followingresearch literature.
• Follows the layout of an introductory course basedon for example the book: C. Kittel: Introduction to Solid State physics
Outline• Structural models• Structural
characterization• Atomic vibrations
(phonons)• Electronic structure• Electrical properties• Optical properties• Magnetic properties
(P. Nordblad)
Resources• Lecture notes• Reference books in the Solid State Physics
library, house 4, level 3.• Examination: Home assignments, 4 sets.• Contact: Gunnar Niklasson
Room 4412, phone: 3101. [email protected]
• Homepage: http://www.teknik.uu.se/ftf/education/Disordered_materials/Phys_disord_mtrl.html
Weak disorder• No disorder: Chain,
ordered surface, crystal
• Weak disorder: Surface steps, defects, vacancies, dislocations, grain boundaries
• Perturbations of the perfect order
Schematic surface
Source: www1.columbia.edu
Strong disorder• Homogeneous materials:
Local density similar to the average density.Ex: Glass
• Inhomogeneous materials: Large fluctuations of the local density, densitydepends on length scale, pores of different sizeEx: Aggregates, porousmaterials
Ni particles
Types of disorder
• Topological (structural)No translationalsymmetry.
• Orientational (magneticspins, dipoles)
• Substitutional (chemical, compositional)
• Vibrational (atoms vibrate around theirequilibrium positions even at T=0.
Source: Elliott PoAM
Inhomogeneous disorder• Structure repeats itself over
a range of length scales.• Lower cutoff: Atom,
molecule or particle size• Upper cutoff: Correlation
length, size of structure• Can often be described by
fractal geometry(B. Mandelbrot, 1975)
• Ex: Metallic particleaggregate, rough surface(von Koch curve)
Source: Wikipedia
Ordering rule
• A property, p1, of one object in a set is related to pi of object ”i” by an ordering rule.
• Perfect order: pi of all objects can be derivedfrom 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.
Order parameter• Atoms A and B placed on
the sites of a crystallattice with concentrationscA and cB=1-cA.
• Crystal: All A-atoms are in A-sites (r=1) and all B-atoms in B-sites.
• Complete disorder: Atoms placed randomlyon the sites. Fraction cAof A-atoms in A-sites(r=cA).
• Order parameter
A
A
ccr
−−
=1
ξ
Entropy and disorder
• Vacancies in a crystalMany arrangementspossible: W increases
• Configurational entropyΔS>0
• Mixing of two types of atoms or particles at constant volumeW increases
• Entropy of mixing >0
S = kB ln W
W – number of microscopic states of the system compatible with the macrostate.
Types of disordered solid materials
• Amorphous solidsMany thin filmsa-semiconductorsa-metals
• Glasses (supercooledliquids)Vitreous silicachalcogenide glasses
• CompositesRandom alloysMetal-insulator mixturesParticle dispersionsPorous materials
• Often exhibit fractalstructure at intermediatelength scales
Glass transition
• Rapid cooling of a liquid: Supercooled liquid• Not enough time to go into the lower energy
crystalline state at the melting temperature Tm
• As the temperature decreases below Tm the liquid becomes more viscous
• Forms a glass (a quenched amorphousstructure) in a region around a glass transitiontemperature Tg
• Tg depends on cooling rate and thermal history
Glass transition temperature• Gradual change in
volume at Tg• Steep change in some
thermodynamicproperties (heat capacity, thermalexpansion, isothermalcompressibility)
• From liquid-like to solid-like values
• Determined by thermalmethods (Ex: DSC)
Viscosity
• A solid has viscosity η~1013.6 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 Tg• Most simple liquids: 10-3 – 1 Pa s• Glass transition corresponds to 109 – 1012 Pa s,
depending on definition.
Viscosity• Strong glass-formers:
Often covalent bonds, ex. SiO2. Arrhenius law:
• Fragile glass formers: Often ionic and organicmaterials. Vogel-Tammann-Fulcher law:
)/exp(0 TAηη =
))/(exp( 00 TTB −=ηη
Glass transition• Tg for some materials:
SiO2 1453 KB2O3 530 KSe 313 KPMMA 378 KNylon-6 323 KPolyethylene 253 KPoly(ethyleneoxide) 218 KPoly (propyleneoxide) 211 K
• Physics still not known in detail
• Many theoretical ideas• Thermodynamics, free
volume theory, mode coupling theory…
• Connection to different relaxations, lowfrequency vibrations
• Subject stimulates boththeoretical thinking and novel experimental methods
Origin of disorder
• Physical constraints:• Atoms and particles
may not penetrateeach other
• Chemical bonds• Interactions between
atoms and molecules• Attractive forces
between particles; v.d. Waals, dipolar…
Randomness Physical Laws
Disorder
Example: DLA• Aggregation of particles
that move by a randomwalk
• Which physical law gives rise to aggregation?
• Particles stick to the growing aggregate due to short-range attractiveforces
• Such structures commonin physics and nature
