Thermal Oxidation: The Deal-Grove Model

Silicon did not win the semiconductor industry because it is the best semiconductor — germanium has higher mobility, and several III-V compounds beat it on almost every electronic figure of merit. Silicon won because of what happens when you heat it in oxygen: it grows its own insulator, a dense, adherent, electrically superb film of SiO₂, in a furnace, with no deposition step at all. Every gate oxide, every masking layer, every isolation region in seventy years of devices started as this reaction. This page is the first stop on the site's from sand to transistor arc: grow the oxide, then dope through its openings, then measure the junction you made, then draw the openings in the first place.

One ODE: diffuse through, react at the bottom

Deal and Grove (J. Appl. Phys. 36, 3770 (1965)) reduced oxidation to a supply chain with two bottlenecks in series. Oxidant arrives at the oxide surface, diffuses through the oxide already grown, and reacts at the buried Si/SiO₂ interface — the oxide grows from the bottom, not the top. In steady state the two fluxes match, and the growth rate is

whose integral is the famous linear-parabolic law $x^2+Ax=B(t+\tau )$. Two regimes fall out. While the film is thin ($x\ll A$), diffusion is easy and the interface reaction is the bottleneck: growth is linear at rate $B/A$. Once the film is thick, the oxidant's slog through the oxide dominates: growth is parabolic with constant $B$, and each nanometer costs more time than the last. The crossover sits exactly at $x=A$.

def dg_closed(t, A, B, tau):
    """Deal-Grove closed form: x^2 + A x = B (t + tau)."""
    return 0.5 * A * (np.sqrt(1 + 4 * B * (t + tau) / A**2) - 1)

def dg_ode(t_grid, A, B, tau):
    """Or integrate the physics directly: dx/dt = B / (2x + A)."""
    x = dg_closed(0.0, A, B, tau)
    out = [x]
    for t0, t1 in zip(t_grid[:-1], t_grid[1:]):
        h = t1 - t0
        f = lambda xx: B / (2 * xx + A)          # RK4
        k1 = f(x); k2 = f(x + 0.5*h*k1); k3 = f(x + 0.5*h*k2); k4 = f(x + h*k3)
        x = x + h / 6 * (k1 + 2*k2 + 2*k3 + k4)
        out.append(x)
    return np.array(out)

Using Deal and Grove's own published (111) rate constants at 1100°C (dry O₂: $A=0.090\mu \text{m}$, $B=0.027\mu {\text{m}}^2/\text{h}$, $\tau =0.076\text{h}$):

Deal-Grove, dry O2, 1100 C, (111) Si  [A = 0.090 um, B = 0.027 um^2/h]:
  closed form vs RK4 ODE: max |diff| = 1.05e-11 um
  x(1 h) = 131.3 nm   x(10 h) = 478.5 nm
  crossover x = A = 90 nm at t+tau = A^2/B = 0.30 h
  tau = 0.076 h encodes an 'initial' oxide of 18.9 nm — the thin-oxide fudge
  wet 1100 C, 1 h: 661 nm vs dry 131.3 nm (5x)
  Si consumed / oxide grown = (rho_ox/M_ox)/(rho_si/M_si) = 0.443

An hour buys 131 nm; ten hours buy only 479 — the parabolic regime punishing impatience with a square root. Note the closed form and the direct RK4 integration of $dx/dt$ agree to 10^-11 µm: the algebra and the physics are the same statement.

Wet versus dry: the speed-quality trade

Steam oxidizes five times faster at the same temperature — water is smaller than O₂ and far more soluble in the oxide, so the parabolic constant jumps twenty-fold. The price is a slightly looser, wetter film with more interface traps. Fabs use both accordingly: thick field and masking oxides grown wet, where microns matter and quality doesn't; thin gate oxides grown dry, where every interface state is a threshold-voltage error. The right panel is the other lever — both rate constants are Arrhenius in temperature, so a 100°C change moves an hour's growth by factors, which is why furnace temperature control is a tenth-of-a-degree business.

The model's own confession: τ and the thin-oxide anomaly

Deal and Grove needed the offset $\tau$ to fit their dry data — equivalent to pretending every wafer starts with 18.9 nm of oxide already grown. That is the model confessing where it fails: below roughly 25 nm, dry oxidation runs faster than the linear-parabolic law predicts, for reasons the model's serial-bottleneck picture cannot produce (Massoud's correction adds an exponential rate-enhancement term; the microscopic mechanism was argued about for decades). The fudge was benign in 1965. It stopped being benign when gate oxides became 2-5 nm thick — the entire film living inside the anomaly zone — and modern ultrathin-oxide recipes are calibrated empirically rather than trusted to Deal-Grove. A model plus the region where it lies is worth more than a model alone.

The oxide eats the wafer

The silicon in SiO₂ has to come from somewhere: the wafer. From densities and molar masses, $({\rho }_{ox}/M_{ox})/({\rho }_{Si}/M_{Si})=0.443$ — growing 100 nm of oxide consumes 44 nm of silicon, so the interface retreats into the wafer while the surface rises above it. Process engineers exploit this constantly: an oxide grown and then stripped leaves a recessed, atomically fresh silicon surface, and the consumed thickness is a built-in depth gauge.

Reproduce it

scripts/gen_semiconductors.py generates every number and figure on this page and its two siblings in seconds. Next on the arc: with oxide windows in place, push dopants through them — dopant diffusion.