Silicon: Melt to Junction (3D)

What is modelled

A silicon crystal is pulled from molten silicon (Czochralski process). The page follows four linked physical ideas: how fast the crystal grows relative to the temperature gradient decides which point defects freeze in; how dopant is rejected by the growing crystal decides how the doping (and therefore resistivity) changes along the boule; and the carrier statistics of silicon turn each dopant level into electrons, holes, a Fermi level and a resistivity.

Equations

• Solid fraction after pulling length L of a crystal of diameter D from a crucible of diameter 0.36 m with 0.25 m of melt: g = (ρs/ρl)(D/Dc)² L/H0, ρ_s = 2.33, ρ_l = 2.57 g/cm³.
• Scheil segregation: Cs(g) = k0 C0 (1 − g)k0 − 1. Boron k₀ ≈ 0.8, phosphorus 0.35, arsenic 0.3, antimony 0.023 (typical literature values): the lower k₀, the more the melt is enriched and the more strongly doping rises toward the tail.
• Voronkov criterion: V/G against a critical ratio ξc ≈ 0.13 mm²/(K min). Above it vacancies dominate (they condense into voids, COPs); below it self-interstitials dominate (dislocation loops); a narrow window (here ±10 %) near ξ_c gives a nearly defect-free crystal. The value and the window width are typical literature numbers and depend on the hot-zone design.
• Band gap (Varshni): Eg(T) = 1.170 − 4.73×10−4T²/(T + 636) eV. Intrinsic density ni = √(NcNv) exp(−Eg/2kT) with N_c, N_v ∝ T^3/2.
• Charge neutrality and mass action: n − p = ND − NA, np = ni², so n = ½(ND−NA) + √(¼(ND−NA)² + ni²); EF − Ei = kT ln(n/ni).
• Resistivity ρ = 1/[q(nμn + pμp)] with the Caughey-Thomas mobility μ = μmin + (μmax − μmin)/(1 + (N/Nref)α) (300 K parameter sets for electrons and holes).

The lattice cube and the p-n junction

The cube shows a block of the silicon diamond lattice (each atom bonded to four neighbours, 0.543 nm cell edge). Slice a wafer anywhere on the boule: its dopant concentration sets one side of a junction, and the slider "implanted opposite dopant" sets the other. Dopant atoms are drawn far more often than in reality (real doping replaces about one atom in 10⁶ to 10⁴), so the picture is schematic; donors are orange, acceptors blue, free electrons and holes are the glowing markers.

• Built-in voltage Vbi = kT ln(NAND/ni²).
• Depletion width (abrupt junction, full ionisation) W = √(2εs(Vbi − Va)/q · (1/NA + 1/ND)), with xn = W NA/(NA+ND) and xp = W ND/(NA+ND); peak field Emax = qNDxn/εs and capacitance per area εs/W. Reverse bias (negative V_a) widens W; forward bias shrinks it. The yellow slab in the cube is that region, scaled up so it is visible (the real width is read out in micrometres).
• The band diagram integrates Poisson's equation: the potential is parabolic on each side of the metallurgical junction, and the band edges are the bent levels E_c(x), E_v(x).

Additional simplifications for the junction

Abrupt (step) doping profile, depletion approximation (carriers are assumed absent inside W and neutral outside), room-temperature-like full ionisation, no current flow (only the equilibrium and the bias-dependent depletion, not the diode current), and no breakdown. The two extra checks compare a numerical double integration of the space charge with the closed form, and compare V_bi from the n_i formula with the band-edge difference computed from N_c, N_v and E_g.

Checks run in the page (the first two are about the junction)

• Poisson: numerical double integration of the depletion charge reproduces V_bi − V_a and the peak field.
• V_bi equals the band-edge difference from N_c, N_v, E_g; charge balance x_nN_D = x_pN_A.
• Scheil mass balance (numerical integral vs closed form) and the resistivity of 10¹⁶ cm⁻³ phosphorus against the tabulated 0.5 ohm cm.

Simplifications

Mobility uses 300 K values at every temperature; dopants are treated as fully ionised (a good approximation near room temperature for these levels, not at cryogenic temperature or for the heaviest doping); the axial gradient G falls linearly along the boule (a toy hot-zone, not a thermal simulation); crystal and crucible rotation change only the picture (the real effects are on the interface shape and oxygen content, not modelled); the neck, shoulder and the crucible dissolving are drawn but not simulated; the time lapse is a display speed-up of a real pull that takes many hours.

What to try

• Choose Pulled too fast: the defect strip turns red (voids); choose Pulled too slow: blue. Press Hold V/G on target to let the pull rate follow G and stay in the green window.
• Compare phosphorus with antimony: antimony's tiny k₀ makes the resistivity collapse toward the tail.
• Raise the wafer temperature to about 600 K and watch intrinsic carriers take over from a lightly doped slice.