ECE 6450 - Dr. Alan DoolittleGeorgia TechLecture 3DiffusionReading: Chapter 3ECE 6450 - Dr. Alan DoolittleGeorgia TechImpurity Diffusion:Pfann patented the idea of using diffusions in Si and Ge in 1952. Diffusions are most commonly used for:1.) Bases, emitters, and resistors in bipolar technology2.) Form source and drain regions and dope polysilicongate/interconnect lines in MOS technology.When to use it and when not to use it:1.) Use when damage from Ion Implantation leads to unacceptable decreases in Minority carrier lifetime, electrical junctions need to be very deep, or a cheap easy solution is needed2.) Do not use it for ultra-shallow junctions, majority carrier devices (use ion implantation instead) or total impurity “dose” is critical (ex. MOSFET channel)Diffusions sources include:1.) Chemical source in a vapor at high temperature.2.) Doped oxide source (either deposited at high temperature oras a “Spin on polymer).3.) Diffusion/annealing from an Ion implanted source.iECE 6450 - Dr. Alan DoolittleGeorgia TechImpurity Diffusion:Traditional Tube FurnaceBaffles used to “mix”gasesWafers Held in a quartz boatHot FurnaceECE 6450 - Dr. Alan DoolittleGeorgia TechFick’s first law states that “impurities” flow (with flux J) toward a decrease in concentration,(1)The diffusion coefficient, D, also called diffusivity, or diffusion constant, characterizes a particular impurity’s resistance to flow when exposed to an impurity gradient.We do not measure impurity gradients or impurity fluxes. These quantities are difficult to obtain.Thus, using the law of conservation of matter,(2)This second law simply states that the total change in flux leaving a volume equals the time rate of change in the concentration in the volume.xtxCDJ∂∂−=),(xJttxC∂∂−=∂∂ ),(Impurity Diffusion:ECE 6450 - Dr. Alan DoolittleGeorgia TechPlugging (1) into (2), one can rewrite Fick’s first law as Fick’ssecond law,(3)In certain special cases, D is independent of x,(4)We will examine various solutions of this differential equation later.More generally in 3D:(5)Note that generally, D=f( T , x , and even C). ⎟⎠⎞⎜⎝⎛∂∂∂∂=∂∂xtxCDxttxC ),(),(22),(),(xtxCDttxC∂∂=∂∂()CDtC∇•∇=∂∂Impurity Diffusion:ECE 6450 - Dr. Alan DoolittleGeorgia TechWhat distinguishes one impurity from another is it’s diffusivity.Consider the general case where an atom can exist as both a substitutional or interstitial impurity. We can define a few terms:[NS] = Solubility of the impurity in a substitutional site[NI] = Solubility of the impurity in a interstitial siteγs= Substitutional jump frequency (~1013Hz, depends on Temperature, other factors)γi= Interstitial jump frequencyImpurity Diffusion:Diffusion CoefficientECE 6450 - Dr. Alan DoolittleGeorgia Tech[][][]≡+issNNNFraction of time the impurity spends at the Substitutional Lattice sites[][][]≡+isiNNNFraction of time the impurity spends at the Interstitial Lattice sites[][][][][][]isIIissseffectiveNNNNNN+++=γγγthen the effective jump frequency can be defined as,Impurity Diffusion:Diffusion CoefficientECE 6450 - Dr. Alan DoolittleGeorgia TechktEoaeDD−=Note: Dois assumed constant but in fact has a slight dependence on temperature through the γeffectiveterm.Impurity Diffusion:Diffusion CoefficientSiforeVEwhereedDmotionEeffectivealInterstiti2.11.0~,kT2motion−∝⎟⎠⎞⎜⎝⎛−γwhere d is the distance for a jump.For a substitutional impurity to move, it must first “create” a vacancy-interstitial pair. Thus, often it’s motion is limited by the energy required to create the defectFor an interstitial, no defect must be created before the impurity can diffuse. Thus, the diffusivity is,()SiforeVEwhereedDcreationdefectEEeffectiveonalsubstituticreationdefect54~,kT2motion−∝⎟⎠⎞⎜⎝⎛+−γGenerally,ECE 6450 - Dr. Alan DoolittleGeorgia TechImpurity Diffusion:Diffusion MechanismsThe diffusivity is dependant on the point defect reactions that take place. Several mechanisms exist: Interstitial, vacancy, interstitialcy, and dissociativemechanisms. Note: the kickoutmechanism is a subset of the full interstitialcy mechanism but does not require the presence of self-interstitials.ECE 6450 - Dr. Alan DoolittleGeorgia Tech++++−−−−⎥⎦⎤⎢⎣⎡+⎥⎦⎤⎢⎣⎡+⎥⎦⎤⎢⎣⎡+⎥⎦⎤⎢⎣⎡+⎥⎦⎤⎢⎣⎡+⎥⎦⎤⎢⎣⎡+⎥⎦⎤⎢⎣⎡+⎥⎦⎤⎢⎣⎡+=443221144332211DnpnpDnpDnpDnnDnnDnnDnnDDiiiiiiiConsider “simple” vacancy diffusion:The diffusivity can be considered a superposition of all the diffusivity of all individual vacancy species (different charges):Rarely are third and forth order terms important.Impurity Diffusion:Diffusion CoefficientECE 6450 - Dr. Alan DoolittleGeorgia TechArsenic in Si is an important case where multiple diffusivity terms must be considered. At high concentrations of As, n>>ni. Thus, the first 2 terms must be used. The doping has enhanced the diffusion. At lower concentrations, n<ni, and only the intrinsic diffusivity term needs to be considered. See example 3.1 in Campbell. This concentration dependent diffusivity leads to a steep drop in the concentration when the concentration becomes lower than ni. (Traffic jam analogy).Impurity Diffusion:Diffusion CoefficientECE 6450 - Dr. Alan DoolittleGeorgia TechAt high concentrations, doubly charged vacancies, V2-, combine with ionized phosphorous, P+to form a complex, (PV)-These dominate over singly charged vacancies. Thus,22⎥⎦⎤⎢⎣⎡+=−iinnDDDAt P-levels resulting in a fermi energy less than 0.11 eV, the (PV)-pair disassociates according to, This results in an excess of V-resulting in a “kink region”. This excess concentration leads to an enhanced diffusivity in the tail region.−+−−+⇔+= VPeunstablePVPV )()()(0In reality, many diffusions are dominated by more than just vacancy reactions. Other defect complexes form. Consider P in Si.Impurity Diffusion:Diffusion CoefficientECE 6450 - Dr. Alan DoolittleGeorgia TechExpected diffusion profile IF phosphorous was governed by simple intrinsic diffusion process.ECE 6450 - Dr. Alan DoolittleGeorgia Tech0,2),( >⎟⎟⎠⎞⎜⎜⎝⎛= tDtzerfcCtzCS() ( ) ()DttCdztzCtQT,02,0π==∫∞sBjCCerfcDtx12−=Many useful solutions exist in “The mathematics of Diffusion”, 2nd Ed., John C. Frank (1975).Three important cases exist:1.) Infinite Source (non-depleting) Assumption (often termed