1 Crystal Defects The crystal lattices = an idealized, simplified system of geometrical points used to understand important principles governing the behavior of solids Real crystals - contain large numbers of defects, e.g., variable amounts of impurities, missing or misplaced atoms or ions These defects occur for three main reasons: 1) It is impossible to obtain any substance in 100% pure for, some impurities are always present 2) Even if a substance were 100% pure, forming a perfect crystal would require cooling the liquid phase infinitely slowly to allow all atoms, ions, or molecules to find their proper positions. Cooling at more realistic rates usually results in one or more components being trapped in the “wrong” place in a lattice or in areas where two lattices that grew separately intersect 3) Applying an external stress to a crystal can cause microscopic regions of the lattice to move with respect to the rest, thus resulting in imperfect alignment 2 Crystal Defects Perfect crystals - every atom of the same type in the correct equilibrium position in the cell (does not exist at T > 0 K) Real crystals - all crystals have some imperfections - defects most atoms are in ideal locations, a small number are out of place • Intrinsic – present for thermodynamic reasons • Extrinsic – not required by thermodynamics, can be controlled by purification or synthetic conditions • Chemical – foreign atom, mixed crystals, nonstoichiometry • Geometrical – vacancy, interstitials, dislocations, boundaries, surface Defects dominate the material properties: Mechanical, Chemical, Electrical, Diffusion Defects can be added intentionally 3 Crystal Defects Perfect crystal Real crystal Does not exist at T > 0 K 4 Classes of Crystal Defects Point defects (0D) places where an atom is missing or irregularly placed in the lattice structure – lattice vacancies, self-interstitial atoms, substitution impurity atoms, interstitial impurity atoms Linear defects (1D) groups of atoms in irregular positions – dislocations Planar defects (2D) interfaces between homogeneous regions of the material - grain boundaries, stacking faults, shear planes, external surfaces Volume defects (3D) spaces of foreign matter – pores, inclusions, mosaic, domains 5 Classes of Crystal Defects a - interstitial impurity atoms, b, g - dislocations, c - self-interstitial atom, d - vacancy, e,f - inclusions, h - substitution impurity atom 6 Point Defects Point defects an atom is missing or is in an irregular position in the lattice • self interstitial atoms • interstitial impurity atoms • substitutional impurity atoms • vacancies 7 Point Defects – Ionic Compounds perfect crystal lattice AB interstitial imputity cation vacancy anion vacancy substitution of a cation substitution of an anion BA antisite defect AB antisite defect 8 Point Defects – Ionic Compounds • Vacancy • Interstitial • Substitutional • Frenkel • Schottky Schottky: a pair of vacancies, missing cation/anion moved to the surface, equal numbers of vacancies at both A and B sites preserving charge balance, found in highly ionic compounds where cation and anion have comparable size and high coordination numbers (NaCl), metal ions are able to assume multiple oxidation states Frenkel: ions moved to interstitial positions, vacancies, found in open structures (wurtzite, sphalerite, AgX, etc.) with low coordination numbers, open structure provides room for interstital sites to be occupied, cations much smaller than anions, more covalent bonding 9 Vacancies There are naturally occurring vacancies in all crystals Equilibrium defects – thermal oscillations of atoms at T > 0 K The number of vacancies grows as the temperature increases The number of vacancies: • N = the total number of sites in a crystal • Nv = the number of vacancies • Ha = the activation energy for the formation of a vacancy • R = the gas constant Nv goes up exponentially with temperature T         RT H NN a V exp 10 Crystal Energies Point defects = equilibrium concentration Enthalpy H is positive but configurational entropy S is positive – defects = disorder Minimum on free energy G = equilibrium concentration of defects The concentration of vacancies grows as the temperature increases Extended defects = no equilibrium concentration Enthalpy is HIGHLY positive, configurational entropy cannot outweight No minimum on free energy G Metastable defects – dislocations, grain boundaries, surface Heating = minimize free energy: polycrystalline  single crystal grain growth Grains with high dislocation density consumed Atoms move across grain boundary STHG  11 Typical Point Defects in Crystals Alkali halides Schottky (cations and anions) Alkaline earth oxides Schottky (cations and anions) Silver halides Frenkel (cations) Alkaline earth fluorides Frenkel (anions) Typical activation energies for ion diffusion Na+ in NaCl  0,7 eV Cl- in NaCl  1 eV Schottky pair  2,3 eV (1 eV/molecule = 96.49 kJ/mol) 12 The addition of the dopant (an impurity) into a perfect crystal = point defects in the crystal NaCl heated in Na vapors Na is taken into the crystal and changes its compostion NaCl  Na1+ x Cl Na atoms occupy cation sites, an equivalent number of unoccupied anion sites, Na atoms ionize, Na+ ions occupy the cation sites, the electrons occupy the anion vacancies – F centers – color (Farbe) Such solid is now a non-stoichiometric compound as the ratio of atoms is no longer the simple integer Violet color of Fluorite (CaF2) = missing F anions replaced by e Extrinsic Defects Solid Solutions 13 Substitution (mixing, solution) of ions on specific sites Forsterite: Mg2SiO4 Can substitute Fe for Mg Fayalite: Fe2SiO4 Olivine - the substitution is very readily accomplished and any intermediate composition is possible Olivine: (Mg, Fe)2SiO4 Olivine is a solid-solution series in which any ratio of Mg/Fe is possible as long as they sum to two ions per formula unit required for electric neutrality 14 Non-Stoichiometric Compounds Non-stoichiometry can be caused by • introducing an impurity (doping) • the ability of an element to show multiple valencies Vanadium oxide varies from VO0.79 to VO1.29 other examples: TiOx, NixO, UOx, WOx, and LixWO3 Covalent compounds - held to together by very strong covalent bonds which are difficult to break, do not show a wide range of compositions Ionic compounds - do not show a wide range because a large amount of energy is required to remove / add ions What oxidation states? 15 Non-Stoichiometric Compounds Non-stoichiometric ionic crystals a multi-valent element - changes in the number of ions can be compensated for by changes in the charge on the ions, therefore maintaining charge balance but changing the stoichiometry Non-stoichiometric compounds have formulae with non-integer ratios and can exhibit a range of compositions The electronic, optical, magnetic and mechanical properties of non-stoichiometric compounds can be controlled by varying their composition 16 Non-Stoichiometric Compounds Non-stoichiometric superconductor YBCO YBa2Cu3O6.5 a multi-valent element = Cu YBa2Cu3O6.8−7.0 90 K superconductor YBa2Cu3O6.45−6.7 60 K superconductor YBa2Cu3O6.0−6.45 antiferromagnetic semiconductor Oxygen content Tcritical Kelvin 17 Non-Stoichiometric Compounds Magnéli phase WO2.9 (W20O58) WnO3n-1 the shear planes align along (102) where ‘n’ varies from 30 to 19 WO2.96 to WO2.933 WnO3n-2 the reduction below WO2.92, the defect planes along (103) for ‘n’ values between 25 and 15 W20O58 (WO2.9), W5O14 (WO2.8), W4O11 (WO2.75), W18O49 (WO2.72), W3O8 (WO2.67), W2O5 (WO2.5), 18 Dislocations Line imperfections in a 3D lattice • Edge • Screw • Mixed 19 Edge Dislocation • Extra plane of atoms • Burgers vector – Deformation direction – For edge dislocations it is perpendicular to the dislocation line 20 Screw Dislocation • A ramped step • Burgers vector – Direction of the displacement of the atoms – For a screw dislocation it is parallel to the line of the dislocation 21 Deformation When a shear force is applied to a material, the dislocations move Plastic deformation = the movement of dislocations (linear defects) The strength of the material depends on the force required to make the dislocation move, not the bonding energy 22 Deformation Millions of dislocations in a material - result of plastic forming operations (rolling, extruding,…) Any defect in the regular lattice structure (point, planar defects, other dislocations) disrupts the motion of dislocation - makes slip or plastic deformation more difficult Dislocation movement produces additional dislocations Dislocations collide – entangle – impede movement of other dislocations - the force needed to move the dislocation increases the material is strengthened Applying a force to the material increases the number of dislocations Called “strain hardening” or “cold work” 23 Surface and Grain Boundaries • The atoms at the boundary of a grain or on the surface are not surrounded by other atoms – lower coordination number (CN), weaker bonding • Grains line up imperfectly where the grain boundaries meet • Dislocations can not cross grain boundaries • Tilt and Twist boundaries • Low and High angle boundaries High resolution STEM image from a grain boundary in Au at the atomic level, imaged on an FEI Titan STEM 80-300 Crystal Shear Planes 24 Mosaic Crystals 25 Boundary of slightly mis-oriented volumes within a single crystal Lattices are close enough to provide continuity (so not separate crystals) Has short-range order, but not long-range order 26 Stacking Faults ABCABCABCABABCABC AAAAAABAAAAAAA ABABABABABCABABAB 27 Effect of Grain Size on Strength • Material with a small grain = a dislocation moves to the boundary and stops – slip stops • Material with a large grain = the dislocation can travel farther • Small grain size = more strength • Hall-Petch Equation y = 0 + K d –1/2 y = yield strength (stress at which the material permanently deforms) d = average grain diameter 0 = yield stress for bulk single crystal K = unpinning constant 28 Amorphous Structures • Cooling a material off too fast - it does not have a chance to crystallize • Forms a glass • Easy to make a ceramic glass • Hard to make a metallic glass • There are no slip planes, grain boundaries in a glass 29 100% Concentration Profiles 0 Cu Ni Interdiffusion: atoms migrate from regions of large to lower concentration Initial state (diffusion couple) After elapsed time 100% Concentration Profiles 0 Diffusion 30 Diffusion Couple Experiments La2O3 CoO LaCoO3 Experimental conditions: T = 1370 – 1673 K pO2 = 40 Pa – 50 kPa 31 Diffusion Couple Experiments CaTiO3-NdAlO3 diffusion couple fired at 1350 °C/ 6 h 32 Diffusion - Fick’s First Law J = diffusion flux [mol s1 m2] D = diffusion coefficient diffusivity [m2 s1] dc/dx = concentration gradient [mol m3 m1] A = area [m2] x diffusion flux Fick’s first law describes steady-state diffusion Velocity of diffusion of particles (ions, atoms ...) in a solid mass transport and concentration gradient for a given point in a solid 33 Conditions for diffusion: • an adjacent empty site • atom possesses sufficient energy to break bonds with its neighbors and migrate to adjacent site (activation energy) The higher the temperature, the higher is the probability that an atom will have sufficient energy Diffusion rates increase with temperature Mechanisms of Diffusion Diffusion = the mechanism by which matter is transported into or through matter Diffusion at the atomic level is a step-wise migration of atoms from lattice site to lattice site 34 Mechanisms of Diffusion • Along Defects = Vacancy (or Substitutional) mechanism – Point Defects – Line Defects • Through Interstitial Spaces = Interstitial mechanism • Along Grain Boundaries • On the Surface 35 Vacancy Mechanisms of Diffusion • Vacancies are holes in the matrix • Vacancies are always moving • An impurity can move into the vacancy • Diffuse through the material 36 Atoms can move from one site to another if there is sufficient energy present for the atoms to overcome a local activation energy barrier and if there are vacancies present for the atoms to move into The activation energy for diffusion is the sum of the energy required to form a vacancy and the energy to move the vacancy Vacancy Mechanisms of Diffusion 37 Interstitial Mechanisms of Diffusion • There are holes between the atoms in the matrix • If the atoms are small enough, they can diffuse through the interstitial holes • Fast diffusion 38 Interstitial Atoms • An atom must be small to fit into the interstitial voids • H and He can diffuse rapidly through metals by moving through the interstitial voids • Interstitial atoms like hydrogen, helium, carbon, nitrogen, etc. must squeeze through openings between interstitial sites to diffuse around in a crystal • The activation energy for diffusion is the energy required for these atoms to squeeze through the small openings between the host lattice atoms • Interstitial C is used to strengthen Fe = steel, it distorts the matrix • The ratio of r/R is 0.57 – needs an octahedral hole • Octahedral and tetrahedral holes in both FCC and BCC – however the holes in BCC are not regular polyhedra • The solubility of C in FCC-Fe is much higher than in BCC-Fe 39 Interstitial Atoms 40 Interstitial Atoms 41 Interstitial Atoms 42 Activation Energy • All the diffusion mechanisms require a certain minimum energy to occur - the activation energy • The higher the activation energy, the harder it is for diffusion to occur • The order of energy for diffusion types: Volume (Vacancy, Interstitial)  Grain Boundary  Surface The activation energy = Energy barrier for diffusion 43 Activation Energy Energy barrier for diffusion Initial state Final stateIntermediate state E Activation energy 44 Diffusion in Perovskites ABX3 A cation diffusion B cation diffusion B B B B BBO OO O O O O O AA EA = 379 Activation energies (kJ mol-1) The A cation diffusion is easier EA = 1420 EA = 746 (100)Cubic plane 45 Diffusion Rate D = the diffusivity, which is proportional to the diffusion rate D = D for T  Q = the activation energy R = the gas constant T = the absolute temperature D is a function of temperature Thus the flux (J) is also a function of temperature High activation energy corresponds to low diffusion rates The logarithmic representation of D verus 1/T is linear, the slope corresponds to the activation energy and the intercept to D         RT Q DD exp Diffusion coefficients show an exponential temperature dependence (Arrhenius type) Diffusion coefficients for impurities in Si 46 surface grain boundaries volume Ag in Ag C in Fe Diffusion Coefficients         RT Q DD exp ln D = ln D  Q/RT 47 Velocity of diffusion of particles (ions, atoms ...) in a solid - mass transport and concentration gradient for a given point in a solid Ji = -Di  ci/ x [ mol cm-2 s-1] (const. T) Ji: flow of diffusion (mol s-1 cm-2); Di: diffusion coefficient (cm2 s-1) ci/ x: concentration gradient (mol cm-3 cm-1) (i.e. change of concentration along a line in the solid) dx dc D Adt dn J  Diffusion Knowledge of D allows an estimation of the average diffusion length for the migrating particles: = 2Dt (: average square of diffusion area; t: time) 48 Diffusion Diffusion FASTER for: • open crystal structures • lower melting T materials • materials w/secondary bonding • smaller diffusing atoms • lower density materials Diffusion SLOWER for: • close-packed structures • higher melting T materials • materials w/covalent bonding • larger diffusing atoms • higher density materials Non-Steady-State Diffusion 49 Fick's Second Law of Diffusion       xd Cd D xd d = td Cd xx The rate of change of composition at position x with time, t, is equal to the rate of change of the product of the diffusivity, D, times the rate of change of the concentration gradient, dCx/dx, with respect to distance, x Fick's Second Law of Diffusion 50 Second order differential equations are nontrivial Diffusion in from a surface where the concentration of diffusing species is always constant, e.g. : – gas diffusion into a solid as in carburization of steels – doping of semiconductors Boundary Conditions For t = 0, C = Co at 0  x For t > 0 C = Cs at x = 0 C = Co at x =  Fick's Second Law of Diffusion 51       Dt2 x erf-1= C-C C-C os ox Cs = surface concentration Co = initial uniform bulk concentration Cx = concentration of element at distance x from surface at time t x = distance from surface D = diffusivity of diffusing species in host lattice t = time erf = error function The solution to Fick's second law is the relationship between the concentration Cx at a distance x below the surface at time t