PN Junction

When P-type and N-type regions form a junction, an electric field appears near the interface even without an external power source; however, this does not turn the diode into a battery capable of supplying continuous power. When forward-biased, current does not suddenly appear only after "exceeding 0.7 V." Understanding the PN junction requires linking charge distribution, potential barriers, and carrier transport.

Prerequisites include electroneutrality, drift, and diffusion from Energy Bands and Doping. The following discussion uses a PN homojunction made of the same material; Schottky junctions, heterojunctions, and insulated gate structures have their own physical boundaries, so one cannot claim that all semiconductor devices are merely PN junctions.

From Diffusion to Built-in Field

Before the junction forms, the N region has many electrons and the P region has many holes. The concentration difference at the interface causes electrons to diffuse into the P region and holes to diffuse into the N region, where they recombine with local majority carriers. Recombination reduces mobile electron-hole pairs, but this does not mean charge conservation is violated.

The impurity ions originally compensated by mobile carriers are exposed: positive ionized donors remain on the N side, and negative ionized acceptors remain on the P side. They are fixed in the lattice and cannot diffuse like carriers. This forms a space charge and a built-in electric field pointing from N to P.

The drift driven by the electric field and the diffusion driven by the concentration gradient are in opposite directions. At thermal equilibrium, their current contributions exactly cancel, resulting in zero net current; diffusion, random thermal motion, and generation-recombination do not stop. The depletion approximation ignores mobile majority carriers relative to fixed ions in the space charge region, but this does not claim that absolutely no free carriers exist there.

Assuming P is on the left and N is on the right, the electron diffusion particle flow is to the left, corresponding to an electron diffusion current to the right; the hole diffusion particle flow and diffusion current are both to the right. The built-in field points to the left, so the drift currents for both carriers point to the left. This accounting of directions explains why equilibrium can exist with no net current, rather than the electric field locking all carriers in place.

Equilibrium Energy Bands and Built-in Potential

At thermal equilibrium, the Fermi level is the same throughout the entire junction, but the conduction band bottom and valence band top bend with spatial potential. Since electron potential energy is −qφ, when the potential increases from P to N, the electron band edges move downward. The difference in band edges on both sides corresponds to qV_bi.

For a homojunction with uniform doping, non-degenerate, approximately fully ionized, and satisfying relevant approximations, the built-in potential is V_bi = V_T ln(N_A N_D/n_i²), where V_T = k_BT/q. It varies with temperature, material, and doping, and is not a fixed constant like 0.7 V for silicon or 0.3 V for germanium. MIT: PN Junction Electrostatics Course

For example, at 300 K, with N_A = 10¹⁷, N_D = 10¹⁶, and n_i = 10¹⁰ cm⁻³, the calculated V_bi is approximately 0.774 V. This internal equilibrium barrier cannot be directly treated as a power supply voltage that a standard voltmeter can read across an unpowered diode; the measurement loop also includes electrochemical potential relationships at contacts, and an equilibrium junction cannot continuously provide net work to the outside.

No Universal Turn-on Threshold

Define the external junction voltage as V_A = V_P−V_N, with the positive current direction from P to N. V_A > 0 is forward bias, which lowers the barrier for majority carriers crossing the junction; V_A < 0 is reverse bias, which raises the barrier. The so-called barrier approximation V_bi−V_A applies to models of the junction region and should not be used to assume that the entire packaged terminal voltage under high current constitutes a change in the barrier.

Forward Bias Increases Minority Carrier Injection

Forward bias makes it easier for electrons on the N side to enter the P side, becoming minority carriers in the P region; the same applies to holes entering the N region. After injection, minority carriers diffuse and recombine in the neutral regions, while external power supplies and contacts replenish carriers, maintaining a continuous current. Describing this merely as "forward bias has low resistance" only describes the phenomenon; to explain the exponential relationship, one must look at how carrier concentrations at the boundaries increase with bias.

The ideal low-injection model yields I = I_S[exp(V_A/V_T)−1]. In engineering, the ideality factor η is often used to write I = I_S[exp(V_A/(ηV_T))−1]; η is used to approximately describe the dominant transport/recombination mechanism and does not guarantee that a single η accurately fits a real curve from very small to very large currents.

Consider a hypothetical device with I_S = 1 pA, η = 1, and T = 300 K:

Current increases smoothly with voltage, and is already non-zero at 0.3 V. The "forward voltage drop" observed at a given current also depends on I_S, temperature, area, series resistance, and material; 0.7 V is merely a rough approximation for the operating point of some silicon diodes, not a requirement to exceed V_bi before conducting.

At high currents, series resistance voltage drop and high-injection effects may cause the exponential formula to become inaccurate; at low currents, depletion region recombination and leakage may also alter the curve. The model table here does not correspond to the rated capabilities of specific part numbers.

Reverse Current is Not Strictly Zero

When not in breakdown and the ideal model applies, reverse bias makes the exponential term very small, and the current approaches −I_S. However, real reverse current may also include depletion region generation, surface leakage, photocurrent, and defect-related mechanisms, varying with temperature and reverse voltage. When measuring reverse current, one must also distinguish between steady-state leakage and capacitive current at the moment the voltage is applied.

Depletion Region Numerical Derivation: Wider on the Lightly Doped Side

Consider a one-dimensional abrupt junction, with P-side depletion width x_p and N-side x_n, where W = x_p+x_n. The equality of the total fixed charge magnitude on both sides gives N_Ax_p = N_Dx_n, so the depletion region extends primarily toward the lighter doped side, rather than being half on each side.

Integrating the Poisson equation under the depletion approximation yields:

W = √[(2ε_s/q)(1/N_A+1/N_D)(V_bi−V_A)].

If the reverse bias magnitude is written as V_R = −V_A ≥ 0, the term in parentheses becomes V_bi+V_R. The charge density near the origin is approximately constant on each side, the electric field is piecewise linear, and the potential is piecewise quadratic; this formula encapsulates the result of the constraint between charge quantity and potential change. MIT: Biased PN Junctions and Depletion Capacitance

Continuing with the previous doping levels and 0.774 V built-in potential, taking ε_s = 11.7ε₀ and using consistent cm units, we obtain:

Since N_D is only one-tenth of N_A, x_n is ten times x_p. Increasing reverse bias expands the space charge region and enhances the peak electric field; one cannot infer solely from the widening of W that the junction can withstand higher voltage, as doping, structure, and breakdown field conditions also matter.

This formula cannot be mechanically extrapolated to W = 0 when approaching strong forward bias where the depletion approximation fails, nor can the negative square root after V_A > V_bi be interpreted as a "general operating model after the depletion layer disappears." Finite device dimensions, graded doping, and fringe fields require other treatments.

Voltage Changes Stored Charge, So the Junction Has Capacitance

Depletion capacitance is the small-signal rate of change of terminal charge with respect to voltage. For an abrupt junction approximation, C_j = ε_s A/W; A is the junction area. Taking A = 10⁻⁴ cm², which is 100 μm×100 μm, the zero-bias capacitance is approximately 3.12 pF, and at 5 V reverse bias, it is approximately 1.14 pF. The greater the reverse bias and the wider the depletion region, the smaller the capacitance, which is the working basis for varactor devices.

The capacitance here does not equal the complete dynamic model for all biases. Minority carriers stored in the neutral region due to forward injection introduce diffusion capacitance, whose frequency and lifetime-related behaviors cannot be explained solely by the geometric εA/W.

from math import log, sqrt, expm1, isclose
q = 1.602176634e-19
vt = 8.617333262e-5*300
eps = 11.7*8.8541878128e-14  # F/cm
na, nd, ni, area = 1e17, 1e16, 1e10, 1e-4
vbi = vt*log(na*nd/ni**2)
values = []
for applied in [0, -5]:
    width = sqrt(2*eps/q*(1/na+1/nd)*(vbi-applied))
    xp, xn = width*nd/(na+nd), width*na/(na+nd)
    capacitance = eps*area/width
    assert isclose(na*xp, nd*xn)
    assert isclose(xn/xp, 10)
    values.append((width, capacitance))
    print(f"V_A={applied} V: W={width*1e4:.4f} μm, Cj={capacitance*1e12:.4f} pF")
assert values[1][0] > values[0][0]
assert values[1][1] < values[0][1]
assert 1e-12*expm1(.3/vt) > 0
print(f"V_bi={vbi:.4f} V; charge balance and reverse bias capacitance change verified")

Why is depletion wider on the lighter-doped side?

The two sides are not equal halves. A deeper blue charge rectangle must be narrower to preserve equal absolute areas. Reverse bias scales both widths at a fixed doping ratio, connecting the square-root width law with inverse capacitance.

$$N_A{x}_p=N_D{x}_n,W\propto \sqrt{V_{bi}+V_R},C_j=\frac{\epsilon A}{W}$$

Dynamic and Breakdown Boundaries

Reverse Recovery Involves Injected Charge

When switching from forward conduction to reverse bias, the previously injected minority carriers need to be swept out or recombined, potentially causing a period of reverse current. The reverse recovery charge, peak, and time depend on the prior current, switching slew rate, reverse voltage, temperature, and device structure; one cannot treat a t_rr as a fixed delay independent of test conditions.

Even without significant minority carrier storage, there may still be charging and discharging currents from the junction capacitance/other parasitic capacitances when the voltage changes. Therefore, "no reverse recovery" cannot be directly translated as "no reverse current at all during switching." Relevant selection criteria are elaborated on in the next article on specific diodes.

Tunneling and Avalanche are Different Current-Increasing Mechanisms

When reverse bias is strong, a strong electric field in the narrow depletion region can promote band-to-band tunneling; another mechanism is carriers gaining energy in the field and colliding to generate more electron-hole pairs, forming avalanche multiplication. The former is not classically "directly tearing covalent bonds," and the latter is not simply a sudden decrease in resistance.

Junctions with low breakdown voltages often have a strong tunneling component, while those with higher voltages are usually dominated by avalanche; both can coexist in the intermediate range, so 5 V should not be drawn as a strict physical boundary. The two mechanisms typically have different temperature coefficients, and the specific operating current can affect the observed results. Nexperia: Zener and Avalanche Mechanisms

The breakdown phenomenon itself does not necessarily mean permanent damage, but only if the device and circuit keep current, power, pulse energy, and temperature within applicable ranges can it operate repeatedly as designed. Ordinary junctions may not be rated for avalanche operation, so "reversible" cannot be taken as a guarantee that any device can directly enter breakdown.

Temperature Coefficients Must Specify Operating Conditions

Under constant forward current, the forward voltage drop of many silicon PN junctions decreases as temperature rises, but −2 mV/°C is a local approximation that varies with current and device; under constant voltage, it may manifest as a significant increase in current. There is also no universal "doubling every 10°C" rule for reverse leakage across devices and wide temperature ranges.

The next article, Diodes, maps these mechanisms to datasheet parameters for rectifiers, Schottky, voltage regulation, protection, and optoelectronic devices, retaining the operating conditions needed for reference.