4.4 Fick's second law

Taking the derivative with respect to $x$ of Fick's first law

In order to get a solution to the diffusion equation, we must first assume some boundary conditions. We will deal with asemi-infinite wafer, and assume that

We also have to decide something about initial conditions. We will make the assumption that we have at time $t=0$ and $x=0$ some surface concentration of impurities which we will call $Q_0$ ( $\frac{\mathrm{impurities}}{\mathrm{cm}^2}$ ). This is the situation we would have if we introduce the impurities using a relatively shallow implant step. Analternative surface boundary condition would be one where the concentration of impurities remains at some fixed value. This iswhat happens when there are impurities in the gas flow over the wafer during the time that they are in the diffusion oven. This iscalled an infinite source diffusion .

The first condition is called a limited source diffusion and that is what we shall consider further here. It is not too hard to show that with this initialcondition, the solution to the diffusion equation is:

Note that $N(x, t)$ is a function of distance into the wafer, and time $t$ . The time is, of course, the time of the diffusion process. $D$ , the diffusion constant, depends on the temperature at which the diffusion takes place. is a plot of $D$ for three of the most commonly used dopants in silicon. Phosphorus and boron are the most common acceptor anddonor respectively. Arsenic is sometimes used because it is significantly bigger in diameter than either P or B and thus,moves around less after an implant.

Suppose we do a relatively shallow implant of boron into our p-type wafer, and deposit a $Q_0$ of $5E13$ phosphorus $\frac{\mathrm{atoms}}{\mathrm{cm}^2}$ . We then perform an anneal diffusion at 1100°C for 60 minutes. At 1100°C, $D$ for phosphorus seems to be about $2E-13\frac{\mathrm{cm}^2}{\sec }$ . We will make a plot of $N(x)$ for various times. If you do this at home, be sure to put time in seconds, not minutes, hours, or fortnights. Looking at , is pretty easy to see how the impurities move into the semiconductor, and how theconcentration at the surface, $N(0, t)$ , decreases as more and more of the impurities moves deeper into the wafer.