Oxidation (thermal SiO2 growth)

HOW growth works: the Deal–Grove model (derived)

We derive the growth law from three fluxes that must be equal in steady state (nothing piles up).

Let $x$ = oxide thickness, $C_g,C_s,C_i$ = oxidant concentrations in gas, at oxide surface, at Si interface.

Flux 1 — gas to surface (mass transport): $F_1 = h(C^* - C_s)$ where $C^*$ is equilibrium concentration in oxide, $h$ = gas-phase transfer coefficient.

Flux 2 — diffusion through existing oxide (Fick's law, linear gradient): $F_2 = D\frac{C_s - C_i}{x}$

Flux 3 — reaction at Si surface (first-order kinetics): $F_3 = k_s C_i$ where $k_s$ = surface reaction rate constant.

Steady state: $F_1 = F_2 = F_3 \equiv F$ . Solve for $C_i$ :

From $F=k_sC_i$ and $F = D(C_s-C_i)/x$ and $F=h(C^*-C_s)$ , eliminate $C_s,C_i$ : $F = \frac{k_s\,C^*}{1 + \dfrac{k_s}{h} + \dfrac{k_s x}{D}}$

The oxide grows because each unit of flux consumes $N_1$ oxidant molecules per unit volume of oxide: $\frac{dx}{dt} = \frac{F}{N_1} = \frac{k_s C^*/N_1}{1 + k_s/h + k_s x/D}$

WORKED EXAMPLES

Worked example 1 — Which regime am I in?

Dry O₂, 1000 °C: $B = 0.0117\,\mu m^2/hr$ , $B/A = 0.071\,\mu m/hr$ . Grow for $t=1$ hr, $x_i=0$ . Solve $x^2 + Ax = Bt$ with $A = B/(B/A) = 0.0117/0.071 = 0.165\,\mu m$ . $x^2 + 0.165x - 0.0117 = 0 \Rightarrow x = \frac{-0.165+\sqrt{0.165^2+4(0.0117)}}{2} \approx 0.062\,\mu m$ Why this step? We used the full equation (not a limit) because 1 hr in dry O₂ gives a thin oxide where neither regime dominates cleanly. Check: is $x \ll A$ ? $0.062$ vs $0.165$ — yes-ish → closer to linear.

COMMON MISTAKES (Steel-manned)

Flashcards

Dry oxidation reaction

\text{Si} + \text{O}_2 \to \text{SiO}_2

Wet oxidation reaction

\text{Si} + 2\text{H}_2\text{O} \to \text{SiO}_2 + 2\text{H}_2

Fraction of oxide below original Si surface

~46% (ratio 0.46)

Deal–Grove integrated growth law

x^2 + Ax = B(t+\tau)

Thin-oxide (early) regime, growth vs time

Linear,

x \approx (B/A)(t+\tau)

; reaction-limited

Thick-oxide (late) regime, growth vs time

Parabolic,

x \approx \sqrt{Bt}

; diffusion-limited

Physical meaning of parabolic constant

B

B=2DC^*/N_1

; controls diffusion through oxide

Physical meaning of linear constant

B/A

Surface reaction + gas transfer limited growth rate

Why growth slows with thickness

Oxidant must diffuse across thicker oxide → gentler gradient (

1/x

term)

Which oxidation for thin gate oxides & why

Dry — denser, higher quality

Which oxidation for thick field oxides & why

Wet — much faster growth

Temperature dependence of B and B/A

Arrhenius

\propto e^{-E_a/kT}

; law unchanged, constants rise

Three fluxes in Deal–Grove

Gas transport

h(C^*-C_s)

; oxide diffusion

D(C_s-C_i)/x

; interface reaction

k_sC_i

Recall Feynman: explain to a 12-year-old

Imagine you have a chocolate bar (silicon) and you leave it in the sun so a hard sugar crust (oxide) forms on top. At first the crust forms fast. But once there's a thick crust, the sun's heat has to get through the crust to melt more chocolate underneath — so it gets slower and slower. Also, the crust eats into the chocolate: part of it is where chocolate used to be. That's exactly how silicon grows glass on itself: fast at first, then slower, and it sinks in as it grows.

Connections

Fick's Law of Diffusion — source of the $D(C_s-C_i)/x$ flux term
Arrhenius Equation — temperature dependence of $B$ and $B/A$
Photolithography — SiO₂ used as mask / patterned after oxidation
Ion Implantation — oxide as screen/mask layer
MOSFET Gate Oxide — the critical application of thin dry oxide
LOCOS Isolation — thick wet field oxide for device isolation
Deal-Grove Model — the parent model derived here

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, thermal oxidation ka matlab hai silicon wafer ko furnace me 900–1200 °C par oxygen (dry) ya paani ke vapour (wet) ke saath react karana, jisse upar SiO₂ (glass) ki layer ban jaati hai. Sabse important intuition ye hai: jaise-jaise oxide mota hota jaata hai, oxygen ko us oxide ke through diffuse karke neeche fresh silicon tak pahunchna padta hai — isliye growth slow hoti jaati hai. Product khud hi rukawat ban jaata hai!

Deal–Grove model isi baat ko maths me daalta hai. Teen fluxes barabar hone chahiye (steady state): gas se surface tak, oxide ke andar diffusion (Fick's law, isiliye $1/x$ term aata hai), aur silicon interface par reaction. Inko solve karke milta hai $x^2 + Ax = B(t+\tau)$ . Shuru me, jab oxide patla hai, growth linear hoti hai — $x \approx (B/A)t$ — kyunki tab reaction rate limit karta hai. Baad me, jab oxide mota ho jaata hai, growth parabolic ho jaati hai — $x \approx \sqrt{Bt}$ — kyunki ab diffusion limit karta hai.

Do practical baatein yaad rakho: (1) SiO₂ silicon ko khaata hai — grown oxide ka ~46% original surface ke neeche hota hai. (2) Dry oxidation slow par high-quality hoti hai (gate oxide ke liye), aur wet fast par thodi kam quality (field/isolation oxide ke liye). Temperature badhaane se sirf $B$ aur $B/A$ badhte hain (Arrhenius), law wahi rehta hai.

Exam me common galti: log sochte hain growth constant rate se hoti hai — nahi! Thick oxide me time double karne se thickness sirf $\sqrt2 \approx 1.41$ guna hoti hai, double nahi. Ye samajh liya to poora chapter aasan.

WHAT is thermal oxidation?

HOW growth works: the Deal–Grove model (derived)

The two limiting regimes

WORKED EXAMPLES

COMMON MISTAKES (Steel-manned)

Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Test yourself — Semiconductor Fabrication

Connections