Ion implantation and diffusion

Big picture: To make a transistor, we must put dopant atoms (like Boron or Phosphorus) into precise regions of a silicon wafer at precise concentrations and depths. Two tools do this: diffusion (thermal, atoms wander in) and ion implantation (shoot ions in like darts). This note derives both from first principles.

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1. WHY do we dope at all?

Two knobs matter:

• Dose $Q$ = number of dopant atoms per unit area (atoms/cm²).
• Profile $N(x)$ = concentration vs depth (atoms/cm³).

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2. Diffusion — atoms wander down a concentration gradient

WHY the flux law (Fick's 1st law)

Imagine a plane at depth $x$. More atoms sit on the high-concentration side, so more of them hop across the plane that way than back. The net flux is proportional to how steep the concentration is:

$J = -D\,\frac{\partial N}{\partial x}$

• Why the minus sign? Flux goes down the gradient (from high to low). If $N$ decreases with $x$, $\partial N/\partial x<0$, so $J>0$ (flow into the wafer). ✔
• $D$ = diffusion coefficient (cm²/s), strongly temperature dependent: $D = D_0\,e^{-E_a/kT}$ Why Arrhenius? An atom must clear an energy barrier $E_a$ to hop; the fraction with enough thermal energy is the Boltzmann factor $e^{-E_a/kT}$.

WHY the diffusion equation (Fick's 2nd law)

Take a thin slab between $x$ and $x+dx$. Whatever flows in minus flows out must accumulate:

$\frac{\partial N}{\partial t} = -\frac{\partial J}{\partial x}$

Why? This is just conservation of atoms (continuity). Substituting Fick's 1st law (with constant $D$):

$\boxed{\dfrac{\partial N}{\partial t} = D\,\dfrac{\partial^2 N}{\partial x^2}}$

HOW we solve it — two classic profiles

Case A: Constant surface concentration (predeposition). Surface held at fixed $N_s$ (unlimited dopant supply). Solution:

$N(x,t) = N_s\,\operatorname{erfc}\!\left(\frac{x}{2\sqrt{Dt}}\right)$

Case B: Fixed dose, drive-in (limited source). A dose $Q$ already sits at the surface, then we heat with source removed. Solution is a Gaussian:

$N(x,t)=\frac{Q}{\sqrt{\pi D t}}\;e^{-x^2/(4Dt)}$

Why Gaussian? It's the fundamental solution of the diffusion equation for a point/sheet source, and it automatically conserves the total dose $\int_0^\infty N\,dx = Q$.

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3. Ion Implantation — shoot the dopants in

HOW ions stop

Ions lose energy two ways:

1. Nuclear stopping — collisions with lattice nuclei (dominant at low energy, causes damage).
2. Electronic stopping — drag from electrons (dominant at high energy).

The spread of stopping depths (statistics of many random collisions) makes the profile approximately Gaussian:

$\boxed{N(x)=N_p\,\exp\!\left[-\frac{(x-R_p)^2}{2\,\Delta R_p^{2}}\right]}$

Why the peak is buried, not at surface: ions must travel a bit before losing all energy, so the max concentration $N_p$ sits at $x=R_p$, inside the wafer.

Relating peak to dose — total dose = area under the curve:

\;\Rightarrow\; \boxed{N_p=\frac{Q}{\sqrt{2\pi}\,\Delta R_p}}$$ *Why this step?* A Gaussian $N_p e^{-(x-R_p)^2/2\sigma^2}$ integrates to $N_p\sigma\sqrt{2\pi}$; setting it equal to the known dose pins down $N_p$. ### After implantation: **annealing** Implantation smashes the crystal (damage) and leaves dopants in interstitial (non-electrical) sites. We heat (~900–1000 °C) to: - **Repair** the lattice, and - **Activate** dopants (move them onto lattice sites). This anneal *also diffuses* the profile: the Gaussian's straggle grows to $\sqrt{\Delta R_p^2 + 2Dt}$. --- ## 4. Diffusion vs Implantation — the 80/20 table | | Diffusion | Ion implantation | |---|---|---| | Depth control | poor (√t, thermal) | ==excellent (set by energy)== | | Dose control | moderate | ==precise (count charge)== | | Temperature | high (~1000 °C) | ==low (room temp)== | | Lateral spread | large (isotropic) | small (anisotropic) | | Crystal damage | none | yes → needs anneal | | Profile shape | erfc / Gaussian at surface | ==Gaussian peaked at $R_p$== | --- ## 5. Worked Examples > [!example] E1 — Diffusion depth scaling > A drive-in at $T$ gives junction depth $x_j$ after 30 min. How long for **double** the depth (same $T$)? > *Why:* junction depth $\propto\sqrt{Dt}$ (from $L=2\sqrt{Dt}$). > Doubling depth ⇒ $\sqrt{t_2}=2\sqrt{t_1}\Rightarrow t_2=4t_1=120$ min. > **Lesson:** depth is a *slow* function of time — a huge cost for small gains. > [!example] E2 — Implant peak concentration > Implant boron, dose $Q=1\times10^{14}$ cm⁻², straggle $\Delta R_p=0.05\,\mu\text{m}=5\times10^{-6}$ cm. > *Why:* use $N_p=Q/(\sqrt{2\pi}\,\Delta R_p)$. > $N_p=\dfrac{10^{14}}{2.5066\times5\times10^{-6}}\approx 7.98\times10^{18}\ \text{cm}^{-3}.$ > *Why this step?* Peak concentration must pack the whole dose into a Gaussian of width $\Delta R_p$. > [!example] E3 — Implant dose from beam > Beam current $I=10\ \mu$A over area $A=200\ \text{cm}^2$ for $t=100$ s. $q=1.6\times10^{-19}$ C. > *Why:* $Q=It/(qA)$. > $Q=\dfrac{(10^{-5})(100)}{(1.6\times10^{-19})(200)}=\dfrac{10^{-3}}{3.2\times10^{-17}}\approx 3.1\times10^{13}\ \text{cm}^{-2}.$ > *Why this step?* Every incoming ion carries charge $q$; total charge / area / $q$ = atoms/area. --- ## 6. Common Mistakes (Steel-manned) > [!mistake] "Depth is proportional to diffusion time." > **Why it feels right:** more time → deeper, seems linear. > **Fix:** the solution depends on $x/\sqrt{Dt}$, so depth $\propto\sqrt{t}$. Quadrupling time only doubles depth. > [!mistake] "The implant peak is at the surface." > **Why it feels right:** you fire ions *at* the surface, so surface = most ions? > **Fix:** ions penetrate before stopping. The Gaussian peaks at $x=R_p$ *inside* the wafer; surface concentration is lower. > [!mistake] "Implantation alone makes a working junction." > **Why it feels right:** the dopants are physically there. > **Fix:** most sit in interstitial/damaged sites and are electrically **inactive** + the lattice is wrecked. You *must* anneal to activate and repair. > [!mistake] "$D$ barely changes with temperature." > **Fix:** $D=D_0e^{-E_a/kT}$ — exponential. A 100 °C change can shift $D$ by an order of magnitude. Temperature is the master knob for diffusion. --- ## 7. Flashcards #flashcards/hardware Fick's 1st law and meaning of its minus sign ::: $J=-D\,\partial N/\partial x$; flux flows *down* the gradient (high → low), hence minus. Fick's 2nd law (diffusion equation) ::: $\partial N/\partial t = D\,\partial^2 N/\partial x^2$, from continuity + Fick's 1st law. Temperature dependence of D ::: $D=D_0 e^{-E_a/kT}$ (Arrhenius); atoms must clear barrier $E_a$ to hop. Constant-source diffusion profile ::: $N(x,t)=N_s\,\mathrm{erfc}\big(x/2\sqrt{Dt}\big)$. Limited-source (drive-in) profile ::: $N(x,t)=\dfrac{Q}{\sqrt{\pi Dt}}e^{-x^2/4Dt}$ (Gaussian, conserves dose). Diffusion length ::: $L=2\sqrt{Dt}$; depth scales as $\sqrt{t}$. Implant profile shape ::: Gaussian peaked at range $R_p$ with width $\Delta R_p$. Peak implant concentration formula ::: $N_p=Q/(\sqrt{2\pi}\,\Delta R_p)$, from integrating the Gaussian = dose. Dose from beam current ::: $Q=It/(qA)$. Why annealing after implantation ::: repair lattice damage + move dopants onto lattice sites to activate them (also diffuses profile). Two ion stopping mechanisms ::: nuclear stopping (nuclei collisions, low-E, damage) and electronic stopping (electron drag, high-E). Implant vs diffusion for depth control ::: implant far better — depth set by energy, not slow √t thermal spread. --- > [!recall]- Feynman: explain to a 12-year-old > Imagine silicon is a Lego wall and we want to sprinkle special colored bricks (dopants) inside it. > **Diffusion** = pile the colored bricks on top and heat the wall so bricks slowly wiggle inward — but heating makes them wiggle *sideways too*, and they only creep in slowly (double the time ≈ only 1.4× deeper). > **Implantation** = load the colored bricks into a *toy gun* and shoot them so they lodge at a chosen depth. Super precise, but the wall cracks a bit where they hit — so afterward you gently warm it (annealing) to smooth the cracks and lock the bricks into place. > [!mnemonic] > **"DIFFUSE = √time, IMPLANT = √π".** > Diffusion depth grows like **√(Dt)**; implant peak divides dose by **√(2π)·ΔRp**. And remember **A.I.D.** for implant order: **A**ccelerate → **I**mplant → **D**amage-anneal. ## Connections - [[Semiconductor Fabrication]] - [[Doping and PN Junctions]] - [[Thermal Oxidation]] (also uses high-T furnaces; oxide masks implants) - [[Photolithography]] (defines *where* dopants go) - [[Fick's Laws]] / [[Diffusion Equation]] - [[MOSFET Structure]] (source/drain formed by implant + anneal) - [[Arrhenius Equation]] ## 🖼️ Concept Map ```mermaid flowchart TD DOPE[Need doping] -->|creates| NP[n-type and p-type regions] NP -->|form| JUNCTION[Junction = device] DOPE -->|controlled by| DOSE[Dose Q atoms/cm2] DOPE -->|controlled by| PROFILE[Profile N of x] DOPE -->|method 1| DIFF[Diffusion thermal] DOPE -->|method 2| IMPLANT[Ion implantation] DIFF -->|net flux| FICK1[Fick 1st law J = -D dN/dx] FICK1 -->|D from| ARR[Arrhenius D0 exp -Ea/kT] FICK1 -->|conservation gives| FICK2[Fick 2nd law diffusion eqn] FICK2 -->|const surface Ns| ERFC[erfc profile predeposition] FICK2 -->|fixed dose Q| GAUSS[Gaussian profile drive-in] ERFC -->|defines| PROFILE GAUSS -->|defines| PROFILE ``` ## 🔊 Hinglish (regional understanding) > [!intuition]- Hinglish mein samjho > Dekho, transistor banane ke liye humein silicon wafer ke andar precise jagah par dopant atoms (jaise Boron, Phosphorus) daalne padte hain — kitne, kahan, aur kitni gehrai tak. Iske do main tarike hain: **diffusion** aur **ion implantation**. > > **Diffusion** matlab: dopant ko surface par rakho aur wafer ko garam karo (~1000 °C). Atoms garmi se hilte-dulte hain aur crowded region se empty region ki taraf apne aap phailte hain — bilkul paani mein ink phailne jaisa. Iska maths Fick ke laws se aata hai: flux gradient ke opposite (neeche ki taraf), aur continuity se diffusion equation. Yaad rakho: gehrai $\sqrt{Dt}$ ke hisaab se badhti hai — time double karne se depth sirf 1.4 guna hoti hai. Aur $D=D_0 e^{-E_a/kT}$, yani temperature exponential effect deता hai. > > **Ion implantation** mein hum dopant ko ionize karke high voltage se accelerate karte hain aur wafer par "goli" ki tarah maarte hain. Ions andar ghus kar ek certain depth $R_p$ (projected range) par ruk jaate hain, aur profile Gaussian hoti hai jiska peak surface par nahi, balki *andar* $R_p$ par hota hai. Dose ekdum precise hota hai kyunki $Q=It/(qA)$ — bas beam charge count karo. Peak concentration $N_p=Q/(\sqrt{2\pi}\Delta R_p)$. > > Ek important baat: implantation ke baad crystal damage ho jaata hai aur dopants electrically active nahi hote, isliye **annealing** (dobara heating) zaroori hai — isse lattice repair hota hai aur dopants activate hote hain. Exam aur real fabrication dono mein: implantation = precise depth + dose control + low temperature, diffusion = simple but slow aur lateral spread zyada. Yahi 80/20 concept hai. ![[audio/4.3.11-Ion-implantation-and-diffusion.mp3]]