Ion Implantation and Synthesis of Materials - Studium

9.4 Diffusion Mechanisms 119δ Cxt ( , )δt=2δ CD2δx(9.4)9.3.1 Diffusion CoefficientThe diffusion coefficient, D, is a strong function of temperature. The temperaturedependence arises because some energy (typically a few electron volts, eV) isrequired for an atom to jump from one atomic position to another. This energy isoften called the activation energy, E A . The diffusion coefficient can be writtenAD D exp ⎛ E ⎞= 0 ⎜ −kT ⎟⎝ ⎠(9.5)where D 0 is a pre-exponential parameter with typical values between 10 −1 and10 2 cm 2 s −1 and values of E A between 1 and 5 eV, depending on the diffusingspecies in Si.9.3.2 Diffusion of Doping Atoms into SiThe diffusion of impurities into Si wafers typically is done in two steps. In the firststep, dopants are implanted into the substrate to a relatively shallow depth of a fewthousand angstroms. After the impurities have been introduced into the Sisubstrate, they are diffused deeper into the substrate to provide a suitable impuritydistribution in the substrate. The solid solubility and diffusion of dopant atoms inSi are given in the top and bottom, respectively, of Fig. 9.10.The dopants introduced by the ion implantation step can be redistributed deeperin the substrate to lower the concentrations by a drive-in step. In the drive-in step,the total amount of dopant atoms, Q, remains fixed. The concentration profile dueto the drive-in diffusion is given by2Q ⎛ x ⎞Cxt ( , ) = exp⎜−⎟πDt⎝ 4Dt⎠(9.6)9.4 Diffusion MechanismsFor diffusion in a crystalline solid, consider an impurity atom located between thehost atoms, Fig. 9.11a. This impurity atom, called an interstitial, can jump fromone interstitial site to the next vacant interstitial site. Occasionally, an impurityatom located in a lattice site that is normally occupied by a host atom will jump to