General Description of Activated Process: 1
Diffusion: 2
Diffusion: Temperature Plays a significant role in diffusion Temperature is not the driving force. Remember: DRIVING FORCE GRADIENT of a FIELD VARIALE Remember: Driving force for diffusion is a difference in the chemical potential. µ phase1 µ phase2 i.e. µ 0 or µ 0 HOWEVER: Temperature increases the activity of a diffusing species. Question: What is the probability that an atom will diffuse? o Atoms are oltzons o Use Maxwell-oltzmann Statistics! βε Ne f ( ε) βε e g( ε) dε N( ε) dε f( ε) g( ε) dε No. Impurities with E > E βε Ne g( ε) dε Total No. of Impurities βε e g( ε) dε E A Ae 3 A
Diffusion: 4
Diffusion: Frequency of Jump is proportional to the probability Coefficient of Diffusion is proportional to Jump Frequency ν ν f ( ε) 0 D ν Aν e 0 E Aν e 0 E Diffusivity or Diffusion Coefficient (Arrhenius Rate Equation): D De o E A E act is the activation energy for diffusion k b T is the thermal energy D o, the pre-exponential factor, contains a number of physical constants and properties including:» entropy of formation of the defect» attempt frequency for jumps into available neighboring sites» lattice constant» crystal structure dependence 5
Diffusion: Diffusivity or Diffusion Coefficient: D De o E A 6
Diffusion: Diffusivity or Diffusion Coefficient: Question: How does one obtain: o Activation energy o Pre-exponential ln D E m k A D De o E act Linearize Diffusivity: Eact ln D ln De o EA + ln D o Has form of Equation of a Line! y mx + b Thus: 1 EA y ln D; x ; m ; b lnd T k o est way to calculate E : y ln D ln D m x 1 1 T T So : E k slope A 2 1 2 1 ln D ln D k 1 1 T T 2 1 2 1 A 1 T 7
KINETICS: Diffusion and "Chemical Reactions: Thermally Activated Processes We will use Si as an example of a system with various diffusion mechanisms. Two types of diffusion mechanisms: o Direct diffusion mechanisms: diffusion without the aid of point defects. Interstitial diffusion o Indirect diffusion mechanisms: diffusion with the aid of point defects As + V AV As + I AI As + I Ai Vacancy mechanism Interstitialcy mechanism Kick-out mechanism A A + V Dissociative (Frank-Turnbull) mechanism s i 8
KINETICS: Diffusion and "Chemical Reactions: Thermally Activated Processes Chemical reaction causes a change in concentration, C I, of interstitials s reac I k A + I AI C t k f r kc k CC r AI f I A s k r & k f : forward & reverse Coefficients of reaction k ke o E A For the equation that describes the total kinetics, that is, the total change in the concentration of C, or C total : C C total diff C + t t t rctn 9
Extra notes for those that are interested Arrhenius behavior is observed in many areas of science Conduction in solids 10
Extra notes for those that are interested Other Examples of Arrhenius Rate ehavior: Much of Kinetics shows this behavior o Carrier concentration and conduction in semiconductors and insulators o Mass Transport o Defect Formation o Rates of Chemical Reactions (Coefficient of Reaction Rate) o Creep Rate o Dislocation motion 11
Carrier Concentration in Semiconductors: Intrinsic Intrinsic E e- e - E c E e- e - E c e - e - E v e - e - e - E v E e- e - E c E e- n-type e - e - e - e - e - E d E c e - E d n-type E v 12
Carrier Concentration: n ne o E G 2 Intrinsic carrier concentration as a function of 1/T (Arrhenius plot). 13
Carrier Conductivity in insulators: σ n σ E σ e A 14
SiO 2 Growth Kinetics Models: Deal-Grove Model A & /A C1 C A 1 b e (oxidant diffusion) 2 E e E 2 b (interface reaction rate) Plots of & /A using values in table. 15
Chemical Vapor Deposition (CVD) J 1 J 2 J rxn J 2 kc s s J SL J 1 D D gas ( Cgas Cwaf. surf. ) gas δ g S C x ( gas waf. surf. ) h C C 16