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Lecture #6, 10/27/2009Drift and Diffusion:Drift:electric charges will respond to a field byDrift:electric charges will respond to a field by moving in the direction towards the opposite pole (electrons move to the anode, holes move to the cathode)Diffusion: Random thermal motion of particles (l t hl d t t t )(electrons, holes, dopant atoms, etc.) cause a redistribution of concentration with time to reduce the concentration gradient. The diffusion of particles creates an effective particle current towards the points of lower concentration.DiffusionJ = -D (dc/dx) ….. Fick’s lawDiffusion coefficientParticle currentConcentration gradientTo obtain reasonably fast diffusion processes, Si wafers are typically heated to 1000 degrees C in the presence of a gas source such as phosphene arsene boron etcsource, such as phosphene, arsene, boron, etc...Typical diffusion furnace (a large tube furnace) for doping of semiconductors. The surface concentration is constantly replenished with a gas flow Contamination with otherreplenished with a gas flow. Contamination with other elements, such as carbon, should be carefully avoided.Diffusing into Si to define a resistorpn junctionp-n junctionDiffusion profile after diffusing p-impurities into a n-doped sample Note the position of the pn junction isdoped sample. Note the position of the p-n junction is where both p and n dopant densities are equalOxidation of SiliconThis process is a tremendously important fabrication step which is used throughout gCMOS fabrication for making masks and electrostatic gatesGate oxide thicknesses have to be controlled often to within less than 1 nm to make smallless than 1 nm to make small MOSFETsMuch thicker oxide layers are yoften used to isolate devices or as diffusion masks. These have to be about 1 micron thickto be about 1 micron thickDiffusion coefficients of elements in some common semiconductorsTwo common solutions to the diffusion equation (Fick’s second law)For fixed number of particles diffusing from a square concentration profile:from a square concentration profile:S = Number of initial particles (atoms, electrons)DDiffiiD = Diffusivityx = distance from initial boundaryt = elapsed timeFor an infinite supply of diffusing particles from a surface:erf(x) distribution Gaussian distributionExamples of diffusion profiles after diffusion from (a) a constant source (erfc distribution) and (b) a finite source (Gaussian)What is the surface concentration in a furnaceconcentration in a furnace diffusion process? Depending on the rate limiting step, it often is the solid solubility of the material in the semiconductor (in this case Si) Note that the(in this case Si). Note that the solubility is strongly dependent on the temperatureThe erf(x) function is very () yimportant for furnace diffusion problems where there is a surface sourcethere is a surface source which is replenished continuouslyHere are examples of some erf(x) andHere are examples of some erf(x) and complementary erfc(x) values tabulated for various values of x.Drift and Diffusion of carriersDrift DiffusionIn an electric field, hfFor electronshlthe current of charged carriers is controlled by both For holesydrift and diffusion. The sum of these two componentstwo components will determine how much current will flflowDerivation of the Einstein Relationship:Relationship:This relationship can be used to determine the mobility from the ydiffusion coefficient and vice versa.The Haynes-Schockley ExperimentThis experiment was pperformed to measure both the mobility (drift) and diffusion coefficient for charges withincharges within semiconductors A light pulse is generated on one side of the sample andone side of the sample, and the carrier density is detected as a function of time on the other.The diffusion widened pulse is then measured electrically Total numberBackground holeTotal number of carriers leftBackground hole densityDiffusivity Time delayCarrier recombination rateWhen we put p- and n-doped semiconductors in contact, the Fermi level of bothp-doped n-dopedpnpnthe Fermi level of both semiconductors has to become equal. Individual pieces pnpnThis results in the bending of of semiconductorgthe conduction and valence bands (band bending) and the establishment of an internalestablishment of an internal electrostatic potential VoWhen no current is flowing throught the junction, J(drift)+J(diffusion) = 0Finally, the total diode current is the sum of the hole and electron currents across the p-n junction and is given by:and is given by:Area of junction diffusivity diffusion length Applied voltage Temperature (K)L=DL=DCan also be substituted for LEnd of Lecture #6,


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CALTECH APH 9A - Lecture notes

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