p-n Junction Formation and Built-In Potential (V_bi)

Every diode, BJT, MOSFET junction, and photodiode relies on a single physical structure: a boundary between p-type and n-type semiconductor. Understanding how that boundary forms — and why it self-stabilizes at a specific built-in voltage — is the foundation for every I-V equation, capacitance model, and breakdown mechanism used in later device analysis.

Two Isolated Materials, Then Contact

Consider a p-type silicon bar (doped with acceptors, majority carriers = holes, concentration ≈ Nₐ) brought into intimate contact with an n-type bar (donors, majority carriers = electrons, concentration ≈ N_d). Before contact, each side is individually neutral and in thermal equilibrium, with its own Fermi level set by its doping.

At the instant of contact, the carrier concentration profile is discontinuous: high hole concentration on the p-side meets a region with almost no holes on the n-side, and symmetrically for electrons. This concentration gradient drives diffusion:

• Holes diffuse from p → n
• Electrons diffuse from n → p

As majority carriers diffuse across the junction, they leave behind their fixed, ionized dopant atoms — Nₐ⁻ acceptors on the p-side, N_d⁺ donors on the n-side — which are immobile in the crystal lattice. The departing carriers recombine with the majority carriers on the other side, so a region near the junction is stripped of free carriers. This is the depletion region, and the exposed ionized dopants form a net space charge: negative on the p-side, positive on the n-side.

The Built-In Electric Field

The space charge sets up an electric field pointing from n to p (from + charge to − charge), directly opposing further diffusion. This field:

• Pushes holes back toward p (drift opposing diffusion)
• Pushes electrons back toward n

Diffusion current and drift current are in opposite directions for each carrier type. As more carriers diffuse and more space charge accumulates, the field strengthens until the drift current exactly cancels the diffusion current for both electrons and holes independently. At that point, net current is zero everywhere and the junction reaches equilibrium — this happens in picoseconds to nanoseconds, far faster than any external circuit can respond.

The field is not uniform; it peaks at the metallurgical junction and falls to zero at the depletion region edges, integrating to a net potential difference across the region — the built-in potential, V_bi (also written φ₀ or ψ₀).

Deriving V_bi

Setting total electron current (drift + diffusion) to zero at equilibrium and integrating across the depletion width gives:

V_bi = (kT/q)·ln(Nₐ·N_d / ni²)

where:

• k = Boltzmann constant, T = absolute temperature
• q = elementary charge (kT/q ≈ 25.85 mV at 300 K, commonly rounded to 25–26 mV)
• Nₐ, N_d = acceptor/donor concentrations (cm⁻³)
• ni = intrinsic carrier concentration of silicon (≈ 1.0×10¹⁰ cm⁻³ at 300 K)

This is equivalent to expressing V_bi via the separation of the Fermi levels the two materials had before contact: V_bi = (1/q)·(Ei − EF,p) + (1/q)·(EF,n − Ei), i.e., the total band bending needed to align the Fermi level into a single flat level across the junction at equilibrium.

Key dependencies:

• V_bi increases (logarithmically) with heavier doping on either side.
• V_bi increases with lower ni, i.e., wider-bandgap materials (SiC, GaN) have significantly higher built-in potentials than silicon for equivalent doping, because ni drops exponentially with bandgap.
• V_bi decreases slightly with rising temperature for silicon, because ni grows faster with T than the ln() term compensates — this is the same effect behind a diode's negative temperature coefficient of forward voltage.

Worked Example

Silicon junction at T = 300 K, Nₐ = 1×10¹⁷ cm⁻³, N_d = 1×10¹⁵ cm⁻³, ni = 1.0×10¹⁰ cm⁻³.

V_bi = 0.02585 V · ln[(1×10¹⁷ · 1×10¹⁵) / (1.0×10¹⁰)²]
     = 0.02585 V · ln[1×10³² / 1×10²⁰]
     = 0.02585 V · ln(1×10¹²)
     = 0.02585 V · 27.63
     ≈ 0.714 V

Check: ln(10¹²) = 12·ln(10) = 12 · 2.3026 = 27.63 ✓. A typical silicon diode's built-in potential of ~0.6–0.8 V for moderate doping matches this result — consistent with the commonly quoted "0.7 V" silicon junction figure used throughout diode and BJT modeling.

If both regions were doped ten times heavier (Nₐ = 1×10¹⁸, N_d = 1×10¹⁶), the product Nₐ·N_d increases by 100×, adding 0.02585 · ln(100) = 0.02585 · 4.605 ≈ 0.119 V, giving V_bi ≈ 0.833 V — illustrating the logarithmic (not linear) sensitivity to doping.

Why This Matters in Real Devices

The built-in potential is not directly measurable with a voltmeter across the diode's terminals — it's exactly cancelled by the contact potentials at the metal probes in equilibrium, which is why an unbiased diode reads 0 V. V_bi only manifests indirectly, through its effect on depletion width, capacitance-voltage curves, and how much external forward bias is needed to collapse the barrier and enable significant net current flow (forward bias reduces the net barrier to V_bi − V_applied; reverse bias increases it to V_bi + |V_applied|).

Practical Design Implications

• Varactor/CV curve design: the CV curve of a varactor diode is centered on V_bi; SPICE models parametrize this directly as VJ (junction potential).
• Wide-bandgap devices: SiC and GaN junctions have V_bi in the 2–3 V range due to much lower ni, which shows up as higher forward knee voltages in SiC diodes compared to silicon equivalents.
• Temperature compensation: because V_bi (and hence VBE, VF) drifts with temperature, matched-pair biasing and bandgap references exploit this predictable drift rather than fighting it.
• Process/doping control: fabrication doping profiles directly set V_bi, which in turn sets depletion width at zero bias — a critical parameter for breakdown voltage ratings and parasitic capacitance in high-frequency parts.

Key Takeaways

• Contact between p-type and n-type semiconductor triggers carrier diffusion, leaving behind fixed ionized dopants that create a space-charge (depletion) region.
• The resulting electric field opposes further diffusion; equilibrium is reached when drift exactly balances diffusion for each carrier type, at zero net current.
• The built-in potential V_bi = (kT/q)·ln(Nₐ·N_d/ni²) quantifies this equilibrium barrier — typically 0.6–0.8 V for silicon, higher for wide-bandgap materials.
• V_bi scales logarithmically with doping product and inversely with ni², so it rises with heavier doping and falls slightly with temperature in silicon.
• V_bi cannot be measured directly across diode terminals, but governs depletion width, junction capacitance, forward knee voltage, and reverse breakdown — making it a hidden parameter behind nearly every semiconductor device equation.