( 7) ( 9) ( 8) Applying Thermo: an Example of Kinetics - Diffusion. Applying Thermo: an Example of Kinetics - Diffusion. dw = F dr = dr (6) r

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1 Fundamental Phyic of Force and Energy/Work: Energy and Work: o In general: o The work i given by: dw = F dr (5) (One can argue that Eqn. 4 and 5 are really one in the ame.) o Work or Energy are calar potential (e.g., voltage). o Force i a Vector field. o If a potential i contant, there i no FIELD! o Work i done by a Force! o Combining uation 4 and 5, we have: Y dw = F dr = dr (6) r o Key Point: If the field potential i not changing, then no work would be done. Enough fundamental! Let apply thi phyic and thermo! 1 Thermo & Phyic a applied to Diffuion: Conider a force/force acting on an atom producing atomic motion. The applied force i given by the previou uation: F The motion of the atom will be often interrupted by other atom and colliion occur. Thu: the velocity of the diffuing atom over a time period larger than the time between colliion i an average velocity. The velocity i proportional to the applied force and can be written a: v= μ The contant of proportionality i called the mobility. Conider: o Flux of atom,, diffuing at an average velocity, v, through a homogeneou ditribution of B atom. o The flux of atom through B i ual to the product of: Number of atom per unit volume (i.e., concentration, C ) verage velocity of the atom, v. o Thi i given by: = Y mob F ( 7) ( 8) = C v ( 9) 1

2 By combining the lat two uation, we obtain: = C μ F mob ( 10) Subtituting the Force Field Eqn. 7 (field gradient): = C μ Y mob ( 11) o Thi lat uation i a general form of Fick 1 t Law of Diffuion. o That i, the flux of atom through a homogeneou ditribution of B atom i due to the gradient of ome potential field. o NOTE: Thi potential field can be any of the one hown in the previou table. Becaue the gradient of a potential field follow the uperpoition principle, the more general form of Fick 1 t Law i: = C μ Y, ( 1) mob i i Thermo & Phyic a applied to Diffuion (cont.) Derivation of Fick 1 t Law: In the following lide, Fick 1 t law, in which the concentration gradient i obtained from the chemical potential, i explicitly derived. Key point: Diffuion in olid i baed on thi delineation. The chemical potential of atom i given by the thermodynamic relation: Ε μ = N TPN,, ( 13) where Ε i a free energy of the atom in B. Example of Free Energy, E: o Gibb, G = G(T, P, N) o Helmhotz, F = F(T, V, N) o Enthalpy, H = H(S, P, N) o Omega potential or Grand Potential, Ω = Ω(T, V, μ) o where the um indicate the uperpoition of potential field gradient. 3 4

3 Subtituting n. 13 into n. 11, the flux with repect to the chemical potential gradient i obtained: = C μ μ mob ( 14) uming one-dimenional diffuion, uation 14 implifie to: μ = C μ mob ( 15) The chemical potential of the atom may be written a a function of the chemical activity (i.e., a ) of in a ditribution of B: o where a i the activity of among B, k i Boltzmann contant and μ Α i the chemical potential of in the pure tate. The a may be decribed a the amount that the chemical potential of deviate from the ideal or pure tate (i.e., ideality). The ideality can be interpreted a the abence of - interaction upon adding an extra atom to the ytem. Thu, the enthalpy change, ΔH, of the ytem i zero. φ μ = μ + kt ln a 16 5 The mathematical decription of the activity i given by: a = γ C ( 17) o where γ i the activity coefficient and C i the concentration of the atom. Cae I: (Henry Law) a range of C much maller than the concentration of B atom (C B ), γ become contant. o The chance of interaction between and B atom i mall ince i o dilute in B. o The primary interaction of i with B. o Thi phenomenon i known a Henry law. o Mathematically, a C approache zero, the activity coefficient of, γ, i given by: a γ = contant 18 C Cae II: (Raoult law) C >> C B (i.e., B rather than atom follow Henry law), activity coefficient of i 1. o The atom have a mall probability of interacting with B atom. o The primary interaction of atom i with other atom. o Hence, the olution of i effectively pure. 6 3

4 In thi cae, and the activity coefficient i given by: a γ = 1 19 C o Thi condition i known a Raoult law. o Raoult law i predominant for mot diffuion procee in Si ince C Si >> C dopant For either Henry or Raoult law, Fick 1 t and nd law may till be derived from the chemical potential. Thi eventually can be een by combining uation 15, 16, and 17 into the following form: = C μ mob φ ( μ kt lnγ C) + ( 0) o Since the chemical potential of a pure ubtance i contant, it derivative i zero. ofurthermore, k and T are contant. o Under thee circumtance and taking the derivative of the natural logarithm, uation 0 become: = C μ mob kt γ C ( γ C ) ( 1) o For Henry and Raoult law, γ i a contant or one, repectively. o In either cae, γ may be taken out of the differential. 7 Since the ratio of both γ and C factor to 1, then: = μmobkt Eintein relation tate that the diffuivity, D, of an atom i proportional to it mobility where the contant of proportionality i kt. Mathematically & in term of atom, thi i written a: D = ktμ mob ( 3) ο μ mob unit = quare of the ditance per unit time per unit energy. o kt unit = energy. o Thu, D unit = quare of the ditance per unit time. Invoking Eintein relation with repect to uation 3, Fick 1 t law i obtained: = D ( 4) Note: It i the concentration gradient that drive the flux of atom from one area to another. Fick 1 t law = Steady State Diffuion C f ( t) C, at every point, doe not change wrt time. Fick 1 t law 8 4

5 NON-STEDY STTE DIFFUSION It ha been found that a the concentration of atom in B change with time, the concentration change with poition. Known a Fick nd law, it ha the following form: = ( 5) ume that D i concentration independent. For diffuion in S/C, the concentration of dopant atom i very mall, thu aumption may be comfortably made. Subtituting uation 4 into uation 5, Fick nd law, i given by: Diffuion: Solving Fick nd Law Solve for C(x,t) Infinite Solution Need Boundary Condition Two Primary Boundary Condition: o Fixed Surface Concentration (Infinite Source) Solution: Complimentary Error Function o Reditribution of a contant total number of diffuing atom (Finite Source) Solution: Gauian Function = D C = D C 6 Fick nd law

6 Diffuion: Solving Fick nd Law = D 1 t of Primary Boundary Condition: o Fixed Surface Concentration (Infinite Source) Solution: Complimentary Error Function Boundary Condition: For t = 0, C = Co at 0 x For t > 0, C = C at x= 0 C = C at x= o C Diffuion: Solving Fick nd Law o Fixed Surface Concentration (Infinite Source) Solution: Complimentary Error Function C( x, t) = C ( C Co) erf Dt = Co + ( C Co) erfc Dt C( x, t) = C ( C Co) erf Dt = Co + ( C Co) erfc Dt = 1 erfc z erf z z = 1 e 0 π y dy

7 Diffuion: Solving Fick nd Law = D nd of Primary Boundary Condition: C o Reditribution of a contant total number of diffuing atom (Finite Source) Solution: Gauian Function Boundary Condition: C( x, t) = C( 0, t) exp Dt = C( 0, t) exp 4Dt 13 Diffuion: Thermally ctivated Procee Temperature Play a ignificant role in diffuion Temperature i not the driving force. Remember: DRIVING FORCE = GRDIENT of a FIELD VRIBLE Remember: Driving force for diffuion i a difference in the chemical potential, μ, n. 13. μ i.e. 0 phae1 μphae Δμ or μ 0 HOWEVER: Temperature increae the activity of a diffuing pecie. Diffuivity or Diffuion Coefficient: D= D e Eact kt B o E act i the activation energy for diffuion k b T i the thermal energy D o, the pre-exponential factor, contain a number of phyical contant and propertie including: o entropy of formation of the defect o attempt fruency for jump into available neighboring ite o lattice contant o crytal tructure dependence 14 7

8 Diffuion: Thermally ctivated Procee Diffuivity or Diffuion Coefficient: D= D e Eact kt B o Diffuion Mechanim = Procee = Reaction We will ue Si a an example of a ytem with variou diffuion mechanim. Two type of diffuion mechanim: o Direct diffuion mechanim: diffuion without the aid of point defect. Intertitial diffuion o Indirect diffuion mechanim: diffuion with the aid of point defect + V V Vacancy mechanim + I I Intertitialcy mechanim + I + V i i Kick-out mechanim Diociative (Frank-Turnbull) mechanim k f + I I C t reac I kr = kc k CC r I f I k r & k f : forward & revere Coefficient of reaction 15 total diff = + rctn 16 8

9 Diffuion Mechanim = Procee = Reaction Indirect diffuion mechanim: diffuion with the aid of point defect 1.0 B P 0.8 Ga, l 0.6 I cceptor (p-type dopant) Donor (n-type dopant) Fit to Donor Sb C D C D D D D eff, XI XI XV XV eff eff X = + = XI + XV CX C X eff CXI DXI CI CXV DXV CV eff CI eff CV X = + = XI + XV CX C I CX C V CI CV eff eff eff X D XI CI DXV CV CI CV = + = eff, eff eff eff eff I + V X D XI + DXV CI DX I + DXV CV CI C V D D D D D f.1 intertitial v radiu.o 5/ r/r I (atomic radiu) Equilibrium Nonuilibrium 17 9