1 Crystal Defects The crystal lattices represent an idealized, simplified system that can be used to understand many of the important principles governing the behavior of solids. Real crystals contain large numbers of defects, ranging from variable amounts of impurities to missing or misplaced atoms or ions. These defects occur for three main reasons: 1) It is impossible to obtain any substance in 100% pure form. 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, such as a hammer blow, 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 (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 defects – present for thermodynamic reasons • Extrinsic defects – not required by thermodynamics, can be controlled by purification or synthetic conditions • Chemical defects (foreign atom, mixed crystals, nonstoichiometry) • Geometrical defects (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, external surfaces Volume defects (3D) spaces of foreign matter – pores, inclusions 5 Classes of Crystal Defects 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 compounds where metal ions are able to assume multiple oxidation states Frenkel: ions moved to interstitial positions, vacancies, found in open structures (wurtzite, sphalerite, etc) with low coordination numbers, open structure provides room for interstital sites to be occupied 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 conc. 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 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 This 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 13 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 and LixWO3 Covalent compounds - held to together by very strong covalent bonds which are difficult to break, do not show a wide range of composition Ionic compounds - do not show a wide range because a large amount of energy is required to remove / add ions What oxidation states? 14 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 nonstoichiometric compounds can be controlled by varying their composition. 15 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 16 Dislocations Line imperfections in a 3D lattice • Edge • Screw • Mixed 17 Edge Dislocation • Extra plane of atoms • Burgers vector – Deformation direction – For edge dislocations it is perpendicular to the dislocation line 18 Edge Dislocation 19 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 20 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 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 21 Deformation 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” 22 Slip • When dislocations move slip occurs – Direction of movement – same as the Burgers vector • Slip is easiest on close packed planes • Slip is easiest in the close packed direction • Affects – Ductility – Material Strength 23 Schmidt’s Law • In order for a dislocation to move in its slip system, a shear force acting in the slip direction must be produced by the applied force. Slip direction Normal to slip plane  Ao  F r Fr / A - Resolved Shear Stress in the slip direction  = F/Ao = unidirectional stress applied to the cylinder Fr = F cos() A = A0/cos()  =  cos() cos() 24 Surface and Grain Boundaries • The atoms at the boundary of a grain or on the surface are not surrounded by other atoms – lower 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 25 Grain Boundaries High resolution STEM image from a grain boundary in gold at the atomic level, imaged on an FEI Titan STEM 80-300. 26 Low Angle Tilt Boundary Low Angle Tilt Boundary = Array of Edge dislocations sin b D  D = dislocation spacing b = Burgers vector  = misorientation angle 27 Low Angle Twist Boundary Low Angle Twist Boundary = a Screw dislocation 28 Stacking Faults Low Angle Twist Boundary = a Screw dislocation 29 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 30 Hall-Petch Equation y = 0 + K d –1/2 y = yield strength (stress at which the material permanently deforms) d = average grain diameter 0 = stress K = unpinning constant 31 Control of the Slip Process • Strain hardening • Solid Solution strengthening • Grain Size strengthening 32 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 33 100% 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 34 Diffusion Couple Experiments La2O3 CoO LaCoO3 Experimental conditions: T = 1370 – 1673 K pO2 = 40 Pa – 50 kPa 35 Diffusion CaTiO3-NdAlO3 diffusion couple fired at 1350 °C/ 6 h 36 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 37 Typical diffusion coefficients for ions (atoms) in a solid at room temperature 10-13 cm2 s-1 In solid state ionic conductors (e.g. Ag-ions in -AgI) the values are greater by orders of magnitude ( 10-6 cm2 s-1) Diffusion - Fick’s First Law 38 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 39 Mechanisms of Diffusion • Along Defects = Vacancy (or Substitutional) mechanism – Point Defects – Line Defects • Through Interstitial Spaces = Interstitial mechanism • Along Grain Boundaries • On the Surface 40 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 41 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 42 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 43 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 44 Interstitial Atoms 45 Interstitial Atoms 46 Interstitial Atoms 47 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 highest energy is for volume diffusion – Vacancy – Interstitial • Grain Boundary diffusion requires less energy • Surface Diffusion requires the least 48 Activation Energy Energy barrier for diffusion Initial state Final stateIntermediate state E Activation energy 49 Energy Barrier for Diffusion 50 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 51 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) 52 surface grain boundaries volume Ag in Ag C in Fe Diffusion Coefficients 53 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) 54 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 55 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 56 Second order differential equations are nontrivial. Consider diffusion in from a surface where the concentration of diffusing species is always constant. This solution applies to gas diffusion into a solid as in carburization of steels or 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 57       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 Fick's Second Law of Diffusion 58       Dt2 x erf-1= C-C C-C os ox