iffusion iffusion is a process of mass transport that involves the movement of one atomic species into another. It occurs by ranom atomic jumps from one position to another an takes place in the gaseous, liqui, an soli state for all classes of materials. water aing ye partial mixing time homogenization What is iffusion? iffusion is material transport by atomic motion. Inhomogeneous materials can become homogeneous by iffusion. For an active iffusion to occur, the temperature shoul be high enough to overcome energy barriers to atomic motion.
iffusion Mechanisms. There are two main mechanisms of iffusion of atoms in a crystalline lattice: the vacancy or substitutional mechanism the interstitial mechanism Atoms move from concentrate regions to less concentrate regions. Vacancy iffusion. To jump from lattice site to lattice site, atoms nee energy to break bons with neighbors, an to cause the necessary lattice istortions uring jump. This energy comes from the thermal energy of atomic vibrations (E av ~ kt). Materials flow (the atom) is opposite the vacancy flow irection.
Interstitial iffusion: Interstitial iffusion is generally faster than vacancy iffusion because boning of interstitials to the surrouning atoms is normally weaker an there are many more interstitial sites than vacancy sites to jump to. Requires small impurity atoms (e.g. C, H, O) to fit into interstices in host. Generation of Point efects Point efects are cause by: 1. Thermal energy X efect n efect n site * E C exp[ ( efect )] kt ln[ Xefect ] lnceefect/ kt Ln[X] E efect /k 1/T
Example If, at 400 o C, the concentration of vacancies in aluminum is.3 x 10-5, what is the excess concentration of vacancies if the aluminum is quenche from o C to room temperature? What is the number of vacancies in one cubic μm of quenche aluminum? Given, E s 0.6 ev ; k 86. x 10-6 ev/k, ; r Al 0.143 nm
iffusion Flux: The flux of iffusing atoms, J, is use to quantify how fast iffusion occurs. The flux is efine as either in number of atoms iffusing through unit area an per unit time (e.g., atoms/m -secon) or in terms of the mass flux - mass of atoms iffusing through unit area per unit time, (e.g., kg/m -secon). J M At J 1 δm A δt (Kg m - s -1 ); where M is the mass of atoms iffusing through the area A uring time t. in Area A out δc δx Flux is proportional to the concentration graient an the iffusion coefficient, (m /s), by Fick s first law: J δc δx Steay-State iffusion Flux oes not change with time Concentration profile concentration graient is maintaine constant. Concentration is expresse in terms of mass of iffusing species per unit volume of soli (kg/m 3 ) Negative sign inicates irection of graient It is the riving force [m /s (kg/m 3 )/m] kg/(m s)
δc δx c x B B c x A A J C C1 C or J x Δx Fick s First Law of iffusion Where J: the number of atom iffusing own the concentration graient per secon per unit area, unit: atoms/cm s C: the concentration of molecules (or the number of iffuse molecules per unit volume), unit: atoms/cm 3 x: atomic jump istance : iffusion coefficient, unit: cm /s [ ] cm J i units? s C x g cm 4, ( i x, y, z ) J i [ units ] g s cm
Example: (Fick s 1 st Law) : A thin plate of BCC Fe, T1000K Oxiizing ensity of Fe: ρ 7.9g/cm CO/CO 3 atmosphere Fe 8.9 10-7 cm /s at 1000K Calculate: the number of carbon atoms transport to t0.1cm back surface per secon through an area of 1cm carbon concentration: C 1 0.wt%; C 0% Solution: wt% ρ The concentration of carbon (atoms/cm 3 Fe ): C N A C C J 1 0.% 7.9g / cm 1.01g / mol 7.9 10 0 C x 6.9 10 15 0 3 atoms / cm 6.03 10 C1 C 8.7 10 t atoms / cm s 3 3 7 atoms / mol cm A 0 7.9 10 atoms / cm / s 0.1cm C 3
iffusivity -- the proportionality constant between flux an concentration graient epens on: Type of boning iffusion mechanism. Substitutional vs interstitial. Temperature. Type of crystal structure of the host lattice. Interstitial iffusion easier in BCC than in FCC. Type of crystal imperfections. (a) iffusion takes place faster along grain bounaries than elsewhere in a crystal. (b) iffusion is faster along islocation lines than through bulk crystal. (c) Excess vacancies will enhance iffusion. Concentration of iffusing species.
iffusion coefficient epens on the temperature ln e o ln RT o - R T is the iffusivity or iffusion Coefficient (m / sec ) o is the prexponential factor or iffusion constant (m / sec ) is the activation energy for iffusion (joules / mole ) R is the gas constant ( joules / (mole eg) ) T is the absolute temperature ( K in Kelvin ) /R
Non Steay State iffusion iffusion flux an the concentration graient at some particular point in a soli vary with time, with a net accumulation of epletion of the iffusing species resulting Fick s secon law apples (when is inepenent of composition)
Fick s n Law x A Fick s n Law: C high J in J out C low C t V ( J in J out ) A VA x The rate of change of the number of atoms in the slice V The rate that atoms entering the slice the rate that atoms leaving the slice C t ( J in x J out ) A V C x J x C x C t C x
C t x C In wors: The rate of change of composition at position x with time, t, is equal to the rate of change of the prouct of the iffusivity,, times the rate of change of the concentration graient, C x /x, with respect to istance, x. Solutions to the E are possible when physically meaningful bounary conitions are specifie Particularly important solution semi-infinite soli in which surface concentrations are constant, iffusing species is usually a gas, an the partial pressure is maintaine at a constant value Secon orer ifferential equations are nontrivial an ifficult to solve. Consier iffusion in from a surface where the concentration of iffusing species at the surface is always constant. This solution applies to gas iffusion into a soli as in carburization of steels or oping of semiconuctors. Bounary Conitions For t 0, C C o at 0 < x For t > 0 C C S at x 0 an C C o at x
where C C - C - C C S surface concentration 1- C o initial uniform bulk concentration x s o o erf x t C x concentration of element at istance x from surface at time t x istance from surface iffusivity of iffusing species in host lattice t time erf error function erf (x/ t) is the Gaussian error function this is like a continuous probability ensity function from 0 to x/ t
The equation below emonstrates the relationship between concentration, position, an time C x being a function of the imensionless parameter x/ t may be etermine at any time an position if the parametes C o, C x, an are known C C x s - C - C o o 1 - erf x t Special Case esire to achieve some specific concentration of solute, C 1 in an alloy, then C C x s - C - C o o constant x t constant
Example The carburization of a steel gear at a temperature of 1000 o C in gaseous CO/CO mixture, took 10hours. How long will take to carburize the steel gear to attain similar concentration conitions at 100 o C? For C in γ iron 0. exp{ - 34000 / T} cm /s
Example: (Fick s n Law) etermine the time it takes to obtain a carbon concentration of 0.4% at epth 0.01cm beneath the surface of an iron bar at 1000 o C. The initial concentration of carbon in the iron bar is 0.0% an the surface concentration is maintaine at 0.40%. The Fe has FCC structure an the iffusion coefficient is 14,000 J / mol RT 10-5 m /s exp( ). Known: T1000 o C, epth x 0.01cm, C X 0.4% C O 0.%, C S 0.4% 10-5 m /s 14,000 exp( J / mol ) R 8.314 J/K RT Fin: time t?
Solution: 14,000 173K 10-5 m /s exp ( ) 8.314 173 173K.98 10-11 m /s.98 10-7 cm /s C C X S C C O O 0.4 0. 0.4 0. 0.04 0. erf(z) 0.8, where z erf(z) 0.8 0. 1 erf x t x t erf ( z) erf ( z1) erf ( z ) erf ( z ) 1 z z z1 z 1 0.8 0.797 0.809 0.7970 z 0.90 0.95 0.90
x z 0.906 0.906 t (0.01/1.81) t [x / ( 0.906)] / 104s 1.73min. 7.98 10 C0 + Cs C( xeff, t), C0 + Cs C C 0 C C C C s 0 s 0 t 1.73min. Effective penetration istance: x eff (for 50% of concentration) C0 ( Cs C0) / 0.5 Cs C0 C C0 0.5 1 erf C C Fick s n eff Law: ( ) s 0 x t erf (0.5) 0.5 x eff t
Effective penetration istance In general, for most iffusion problems x eff γ t where γ: a geometry-epenent parameter γ 1 for a flat plate γ for cyliners
Thermal iffusion of Impurities into Silicon The ability to moify the properties of a semiconuctor through the aition of controlle amounts of impurity atoms is an important aspect of silicon evice an IC manufacture. There are two principal methos which are use to introuce impurities into silicon, thermal iffusion an ion implantation. We will iscuss the basic equations escribing the impurity profiles below the surface of the wafer using the thermal iffusion metho. Thermal iffusion is a high temperature process where the opant atoms are eposite on to or near the surface of the wafer from the gas phase. Wafers can be batch-processe in furnaces. The impurity profile or istribution is etermine mainly by the iffusion temperature an time, an ecreases monotonically from the surface. The maximum concentration of a particular iffusing impurity is always foun at the surface.
The impurity concentration C(x,t) as a function of epth below the wafer surface, x, an iffusion time, t is etermine from Fick's iffusion law; is the iffusion coefficient an varies markely from one impurity to another; some impurities iffuse quickly through silicon (fast iffusants), while others move more slowly (slow iffusants). of impurities in silicon. epens on the temperature of iffusion an can be expresse in the generalize form as (T) o exp(-e A / k B T) where o is the iffusion coefficient extrapolate to infinite temperature an E A is an activation energy (usually quote in ev). Thus, a plot of log (T) (µm / hr) vs 1/T (K -1 ) will give a straight line with slope E A.
iffusion: Smaller atoms iffuse more reaily than big ones, an iffusion is faster in open lattices or in open irections Self-iffusion coefficients for Ag epen on the iffusion path. In general the iffusivity if greater through less restrictive structural regions grain bounaries, islocation cores, external surfaces.
Example (A)For an ASTM grain size of 6, approximately how many grains woul there be per square inch at a magnification of 100? (B)The iffusion coefficients for copper in aluminum at an o C are 4.8x10-14 an 5.3x10-13 m s -1, respectively. etermine the approximate time at o C that will prouce the same iffusion results (in terms of concentration of Cu at some specific point in Al) as a 10 hour heat treatment at o C. (C) For the problem (B) compute the activation energy for the iffusion of Cu in Al.
(A) This problem asks that we compute the number of grains per square inch for an ASTM grain size of 6 at a magnification of 100x. All we nee o is solve for the parameter N in the equation below, inasmuch as n 6. Thus N n1 6 1 3 grains/in (B) Fick s secon law, as it is esire to achieve some specific concentration conitions. t cons tan t t t t t ( 13 1 5.3x10 m. s )( 10hours) ( 14 1 4.8x10 m s ) 110.4hours
(C) Using the equation an RT RT RT RT RT o RT o RT o RT o e e e e e e e ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] 1 1 13 1 14 1 1. 134.7 773 1 873 1. 10 ln 5.3. 10 4.8 ln.. 8.31 1 1 ln ln 1 1 ln ln ln mol kj K K s m x s m x K mol J T T R T T R RT RT