Fick's First Law 1.1 2003/06/20 2003/06/20 00:00:00.001 GMT-5 Bill Wilson wlw@rice.edu Elizabeth Gregory lizzardg@rice.edu Bill Wilson wlw@rice.edu Jeffrey Silverman jsilv@rice.edu interstitial diffusion substitutional diffusion Arrhenius plot Fick's First Law of Diffusion substitutional diffusion This section explains Fick's First Law of Diffusion. We talked about diffusion in the context of diodes, and described Fick's First Law of Diffusion for some particle concentration N x t : Fick's First Law of Diffusion Flux D x N x t D is the diffusion coefficient and has units of cm^2/sec. In a semiconductor, impurities move about either interstitially, which means they travel around in-between the lattice sites (). Or, they move by substitutional diffusion, which means they hop from lattice site to lattice site (). Substitutional diffusion is only possible if the lattice has a number of vacancies, or empty lattice sites, scattered throughout the crystal, so that there are places into which the impurity can move. Moving interstitially requires energy to get over the potential barrier of the regions between the lattice sites. Energy is required to form the vacancies for substitutional diffusion. Thus, for either form of diffusion, the diffusion coefficient D, is a strong function of temperature.

Interstitial diffusion

Substitutional diffusion

To a very good degree of accuracy, one can describe the temperature dependence of the diffusion coefficient with an activation energy EA , such that: D T D o E a k T The activation energy EA and coefficient Do are obtained from a plot of the natural log of D vs. 1 k T , called an Arrhenius plot (). The slope gives EA and the projection to infinite T ( 1 T 0 ) gives Do .

Arrhenius plot of diffusion constant

The continuity equation holds for motion of impurities just like it does for anything else, so the divergence of the flux, F must equal the negative of the time rate of change of the concentration of the impurities, or, in one dimension: x Flux x N xt