The PN Junction: From Cartoon to Poisson and Back

Put p-type silicon against n-type and electrons pour across the boundary into the holes, leaving behind a wall of stripped, immobile dopant ions — a region swept clean of carriers whose fixed charge builds a field that stops the flow. That is the depletion region, and the textbook picture of it is a cartoon so useful it survived seventy years: a sharp box of charge, empty inside, quasineutral outside. This page holds the cartoon up against the real thing — a full nonlinear Poisson solve with Boltzmann carriers — and finds it 0.8% right on the number it exists to predict. Then it does the move that makes device physics a measurement science rather than a story: put a known doping into a simulated capacitance–voltage experiment and read that same doping back out of the slope. The loop closes to 1%.

The cartoon, and why it works

The depletion approximation makes one brutal assumption — the transition from full depletion to full neutrality is a step — and buys closed forms for everything. The built-in potential is the difference in Fermi levels the two dopings arrived with,

and setting the integrated charge to zero across a box of width $W$ gives $W=\sqrt{2\epsilon V_{bi}(N_a+N_d)/q{N}_a{N}_d}$. For this junction — $N_a=10^{18}$ against $N_d=10^{16}$, a hundred-to-one one-sided junction — that is 330 nm, and it is nearly all on the lightly doped n side: 327 nm of the 330, versus 3.3 nm on the p side. Charge neutrality forces it — $N_a{x}_p=N_d{x}_n$ — so the side with less doping must deplete wider to expose the same total charge. This is why the lightly doped side of a junction is where all the action (and all the breakdown voltage) lives.

The junction, actually solved

def solve_poisson(Vr=0.0, psi0=None):
    """Damped Newton on 1D nonlinear Poisson: psi'' = -rho(psi)/eps,
    Boltzmann majority carriers, reverse bias Vr on the n side."""
    psi = psi0.copy() if psi0 is not None else initial_guess()
    psi[0], psi[-1] = psi_p, psi_n                 # Dirichlet: bulk potentials
    for it in range(400):
        n, p = carriers(psi, Vr)                   # ni*exp(+-psi/Vt), by side
        rho  = q * (p - n + Nnet)                  # Nnet = Nd - Na, doping
        F        = laplacian(psi) + rho / eps      # residual, zero at solution
        # Jacobian is tridiagonal: d rho / d psi = -q (p + n) / Vt
        diag = -2 / h**2 - q * (p + n) / (eps * Vt)
        dpsi = solve_tridiagonal(diag, 1/h**2, -F)
        psi[1:-1] += np.clip(dpsi, -2*Vt, 2*Vt)    # damp the Newton step
        if np.abs(dpsi).max() < 1e-10: break
    return psi, it

The real electrostatics is one nonlinear ODE: ${\psi }^{\prime \prime }=-\rho (\psi )/\epsilon$, where the charge density $\rho$ depends on the potential through the Boltzmann carrier statistics $n,p\propto e^{\pm \psi /V_t}$. Nonlinear because the carriers respond to the field they create; solved by Newton's method, which for this equation has a tridiagonal Jacobian and converges — with a little damping on the step — in 16 iterations.

PN junction  [Na = 1e18, Nd = 1e16, ni = 9.65e9 cm^-3 (Sze & Ng 3rd ed.), 300 K]:
  Vbi = 0.835 V
  W(0) = 330 nm   (n-side 327 nm, p-side 3.3 nm)
  Poisson solver: converged in 16 Newton steps
  numeric W (half-max charge) = 333 nm vs depletion approx 330 nm (+0.8%)
  numeric W (5% threshold)    = 453 nm — the Debye tail the sharp-edge picture ignores
  numeric built-in potential = 0.835 V vs formula 0.835 V

The three panels are the depletion region as it really is: charge, its integral the triangular field, and its integral the S-shaped potential rising by exactly $V_{bi}$. Two numbers score the cartoon. Measured at half-maximum charge, the depletion width is 333 nm against the approximation's 330 — 0.8% high, a genuinely good cartoon. But measured where the charge falls to 5% of its peak, it is 453 nm: the box has soft edges. Those edges are Debye tails — the carrier density cannot switch on in zero distance, it relaxes over a Debye length — and they are precisely what the sharp-box picture throws away. The built-in potential the solver develops, with no $V_{bi}$ put in anywhere, is 0.835 V to three digits: the same number the Fermi-level formula gives, now emerging from the electrostatics that formula summarizes.

Reading the doping back out

Here is where simulation earns its keep. A reverse-biased junction is a capacitor whose plates are the depletion edges; widen the depletion by pulling $V$ more negative and the capacitance drops. The depletion-approximation algebra predicts that $1/C^2$ is linear in bias, with a slope set by the doping and an intercept at $V_{bi}$:

  C-V closure: 1/C^2 fit -> Nd = 9.909e+15 cm^-3 (put in 1e16), Vbi = 0.792 V (formula 0.835)
  Shockley J0 = 5.256e-12 A/cm^2   J(0.6 V) = 6.313e-02 A/cm^2
  measured slope: 59.2 mV/decade (theory 2.303 kT/q = 59.5)

This is the whole closure. The capacitance came from differencing the depletion charge in the numeric Poisson solution across a bias sweep — a simulated C–V measurement, blind to the doping that was set — and the slope returns $N_d=9.9\times 10^{15}$ against the $10^{16}$ that went in. One percent. The input became a profile, the profile became a capacitance curve, and the curve gave the input back: exactly what a metrology engineer does to a real wafer, run in reverse to validate the physics. The intercept is the honest wrinkle worth keeping — it reads $V_{bi}=0.792$ V, about 0.04 V (roughly $2kT/q$) below the true 0.835. That gap is not solver error; it is the known majority-carrier correction the depletion-approximation C–V formula omits, the same Debye tail that softened the box edges now showing up in the extrapolated intercept. The doping — the thing you actually want — comes from the slope and is immune to it, which is why C–V profiling trusts slopes and treats the intercept as a rough guide.

Forward bias: sixty millivolts a decade

Forward-bias the junction and the current is Shockley's law, $J=J_0(e^{V/V_t}-1)$, with a saturation current $J_0=5.3\times 10^{-12}$ A/cm² set by minority-carrier diffusion. On a log axis it is a straight line whose slope is the number every circuit designer carries: $2.303kT/q=59.5$ mV per decade of current — measured here at 59.2. That 60 mV/decade is the thermodynamic speed limit on how sharply a diode (or a transistor's subthreshold) can turn on at room temperature, because it comes straight from the Boltzmann factor $e^{V/V_t}$. It is the reason low-power transistor scaling is hard and the reason "subthreshold swing" is a headline device metric — $kT/q$ does not care about your process node.

The junction the diffusion made

This junction did not come from nowhere. The p-side doping is exactly the boron profile driven in on the dopant diffusion page, crossing the wafer's $10^{16}$ background at the 1.75 µm junction depth computed there. Oxide grown, window opened, dopant driven, junction formed, junction measured — the four pages are one process flow, and every number on this page traces back to a published constant through code you can rerun. The device is where the whole arc was headed: this is the diode, and two of them back to back with a shared middle region is the bipolar transistor; a gate over an inverted channel between two of these is the MOSFET the computer architecture section is built out of.

Extensions

• Reverse breakdown — integrate the ionization rate across the triangular field until it hits unity: avalanche. The lightly doped side and its peak field set the breakdown voltage, the trade-off at the heart of every power device.
• Drift-diffusion — add current continuity for both carriers to the Poisson equation and you have the full semiconductor device model; the coupled system, solved by Gummel or Newton, is what TCAD tools integrate for a real transistor.
• The MOS capacitor — the same nonlinear-Poisson machinery with an oxide layer and a gate: accumulation, depletion, inversion, and the C–V curve whose shape reads out oxide thickness and flatband voltage. One boundary condition away from this page.
• Where it came from — the doping profile here is the output of dopant diffusion through a thermal-oxide mask patterned by photolithography. This page is where sand finishes becoming a device.

Reproduce it

scripts/gen_semiconductors.py runs the depletion algebra, the damped-Newton Poisson solve, and the simulated C–V sweep from $n_i$ and the doping alone (Sze & Ng's 300 K value). The C–V closure is the page's thesis in one number: doping in, doping out, to a percent — the sign that the model is not just plausible but calibrated against itself.