Question bank — Ion implantation and diffusion

False. The minus sign means flux flows down the gradient (high to low concentration), whichever spatial direction that happens to be — it has nothing to do with the direction of increasing $x$.

False. Depth scales as $\sqrt{Dt}$, so you must quadruple the time to double the depth — diffusion is a slow $\sqrt{t}$ crawler, per the diffusion equation.

False. Ions must travel a bit before losing all their energy, so the Gaussian peaks at the buried projected range $x=R_p$; the surface concentration is lower.

False. Stopping depth is statistical — each ion suffers a different random sequence of collisions, so depths spread out with standard deviation $\Delta R_p$ (the straggle). "Same energy" fixes only the average depth $R_p$.

False. Most implanted atoms land in interstitial (non-lattice) sites and the crystal is damaged, so they carry no free charge until you anneal to activate them and repair the lattice — see Doping and PN Junctions.

False. $D=D_0{e}^{-E_a/kT}$ is exponential (an Arrhenius law); a $100{}^{\circ }\text{C}$ change can move $D$ by an order of magnitude. Temperature is the master knob.

True. ${\int }_0^{\infty }\frac{Q}{\sqrt{\pi Dt}}{e}^{-x^2/4Dt}dx=Q$ for all $t$; the source is fixed (limited), so the peak drops as the profile spreads while the area stays constant.

False. The surface is held at fixed $N_s$ by an unlimited supply, so more atoms keep entering: $Q(t)=\frac{2}{\sqrt{\pi }}{N}_s\sqrt{Dt}$ grows with time.

False. It is just conservation of atoms (continuity, $\partial N/\partial t=-\partial J/\partial x$) with Fick's first law substituted in — no new physics, only bookkeeping of atoms in a slab.

False. The implant itself is done cold (near room temperature) because energy comes from acceleration, not heat. Only the later anneal is hot — a separate, controllable step.

The error is the linear assumption. Since $x_j\propto \sqrt{Dt}$, reaching $2x_j$ needs $4\times$ the time — quadratic cost in depth.

The error is ignoring penetration. Ions plough in before stopping, so the maximum $N_p$ is buried at $x=R_p$; treating the surface as the peak misplaces the whole profile.

The error is confusing area with height. $Q$ is atoms per area (the integral of $N$); the peak height is $N_p=Q/(\sqrt{2\pi }\Delta R_p)$ — you must divide by the Gaussian's effective width.

The error is reading "small" as "zero." $\mathrm{erfc}$ has an infinitely long tail; concentration is tiny but nonzero at every depth, which is exactly why junctions form where $N(x)$ crosses the background doping.

The error is forgetting that anneal is hot, so diffusion happens too. The straggle grows from $\Delta R_p$ to $\sqrt{\Delta R_p^2+2Dt}$ — the profile broadens.

The error is over-generalising. Pulling $D$ outside the derivative required $D$ constant; concentration-dependent $D$ gives ${\partial }_{t}N={\partial }_x(D(N){\partial }_{x}N)$, which cannot be simplified that way.

The error is reversing the regimes. At high energy electronic stopping (drag from electrons) dominates; nuclear stopping — and the damage it causes — dominates at low energy near the end of the ion's path.

Because $D$ has units cm²/s, the only length you can build from $D$ and $t$ is $\sqrt{Dt}$. So all profiles must scale with $x/\sqrt{Dt}$ — that is why depth grows like $\sqrt{t}$, per Fick's Laws.

An atom must clear an energy barrier $E_a$ to hop between lattice sites; the fraction of atoms with at least that much thermal energy is $e^{-E_a/kT}$, so the hopping rate — and hence $D$ — carries that exponential.

Dose $Q$ is just delivered charge over area, $Q=It/(qA)$ — you count electrons in the beam, an electrical measurement you can meter precisely. Diffusion dose depends on temperature, time, and surface chemistry, all harder to pin down.

Each ion's final depth is the sum of many small independent random collision losses; by the central-limit tendency, a sum of many random contributions clusters into a bell curve about the mean $R_p$ with spread $\Delta R_p$.

The prefactor $Q/\sqrt{\pi Dt}$ and the width $\sqrt{4Dt}$ are tuned against each other: a taller-narrower bell (small $t$) and a shorter-wider bell (large $t$) enclose the same area, because the standard bell integral ${\int }_{-\infty }^{\infty }{e}^{-x^2/(4Dt)}dx=\sqrt{4\pi Dt}$ exactly cancels the prefactor's $\sqrt{Dt}$. So the normalization is not a coincidence — it is built in so the fixed dose is conserved.

Implanted ions travel mostly straight down along the beam (anisotropic), so they scatter little sideways. Thermal diffusion is isotropic — atoms wander equally in all directions, spreading under mask edges. This sharpness matters for tight MOSFET geometries.

Dopants only enter where the wafer is exposed; the patterned oxide/photoresist blocks ions or slows diffusion, so the mask defines where the doped regions land — the whole point of controlled placement.

A fixed dose $Q$ (fixed area under the curve) spread over a wider Gaussian must be shorter; since area $=N_p\Delta R_p\sqrt{2\pi }=Q$, a larger width forces a smaller peak.

It collapses to a step: $N=N_s$ at $x=0$ and $N\to 0$ for any $x>0$, because the argument $x/2\sqrt{Dt}\to \infty$ everywhere except the surface — no time has passed to spread anything.

Exactly $N_s$, since $\mathrm{erfc}(0)=1$. That is precisely the boundary condition the solution was built to satisfy — the surface is pinned.

It falls to zero like $1/\sqrt{t}$ (from the $Q/\sqrt{\pi Dt}$ prefactor) while the width grows — the fixed dose keeps spreading thinner forever, area conserved.

Zero at that instant, because $J=-D\partial N/\partial x$. No gradient means no net hopping preference — even though atoms are still jiggling, equal numbers cross each way.

The peak sits at the surface. The wafer only occupies $x\ge 0$, so the mathematical Gaussian's $x<0$ tail is unphysical: those ions would have to be outside the solid. In practice the surface acts as a reflecting boundary — an ion cannot stop in vacuum, so that missing tail is folded back into $x\ge 0$ (mirror image), or one simply notes that only the $x\ge 0$ half is real. If you naively kept a fixed peak $N_p$ and integrated only over $x\ge 0$, you would count half the dose; the reflecting-boundary fix restores the full dose by piling the reflected half onto the near-surface region.

The $2Dt$ term is negligible, so the width stays essentially $\Delta R_p$ — but then dopants also don't activate. This is the trade-off: enough heat to activate inevitably broadens the profile.

$D\to D_0$ (the barrier factor $\to 1$), the maximum attempt-limited rate. Physically the silicon melts first, so this is a mathematical ceiling, not a usable regime.

The one with the smaller $\Delta R_p$: it packs the dose into a narrower, taller Gaussian, giving a steeper concentration fall-off and a more sharply defined junction depth.