6.4 p-n junctions

P-n junctions form when p-type and n-type semiconductors meet, creating a boundary with unique electrical properties. They're the building blocks of diodes, transistors, solar cells, and LEDs, so understanding how they work is essential for nearly everything in semiconductor device physics.

This topic covers junction formation, energy band diagrams, biasing, capacitance, breakdown mechanisms, and applications.

Formation of p-n junctions

A p-n junction forms when a p-type and an n-type semiconductor region are brought into contact. The boundary between them develops special electrical properties that neither material has on its own. To understand why, you need to be clear on doping and what distinguishes the two semiconductor types.

Doping of semiconductors

Doping means intentionally adding impurity atoms to an intrinsic (pure) semiconductor to change its electrical conductivity.

• Donor dopants (e.g., phosphorus, arsenic in silicon) have one more valence electron than the host lattice. That extra electron is loosely bound and easily promoted to the conduction band, making the material n-type.
• Acceptor dopants (e.g., boron, gallium in silicon) have one fewer valence electron. They create empty states (holes) in the valence band, making the material p-type.

The dopant concentration directly controls how many majority carriers are available and therefore the conductivity. Typical doping levels range from about $10^{15}$ to $10^{19} \text{ cm}^{-3}$, compared to silicon's intrinsic carrier concentration of roughly $1.5 \times 10^{10} \text{ cm}^{-3}$ at room temperature.

p-type vs n-type semiconductors

• P-type: Holes are the majority carriers, electrons are the minority carriers. The Fermi level sits closer to the valence band edge.
• N-type: Electrons are the majority carriers, holes are the minority carriers. The Fermi level sits closer to the conduction band edge.

The position of the Fermi level relative to the band edges is what drives the interesting physics when you bring the two types into contact.

Contact between p-type and n-type regions

When the two regions meet, a concentration gradient exists: lots of electrons on the n-side, lots of holes on the p-side. Here's what happens step by step:

1. Diffusion begins. Electrons diffuse from the n-side into the p-side; holes diffuse from the p-side into the n-side.
2. Ionized dopants are exposed. As majority carriers leave, they uncover fixed charges: negative acceptor ions ($N_A^-$) on the p-side and positive donor ions ($N_D^+$) on the n-side.
3. An electric field builds up. These fixed charges create a built-in electric field directed from the n-side toward the p-side.
4. Equilibrium is reached. The electric field opposes further diffusion. When the drift current (driven by the field) exactly balances the diffusion current, the system reaches thermal equilibrium.

The region depleted of free carriers near the junction is called the depletion region (or space charge region).

Energy band diagram of p-n junctions

The energy band diagram is the primary tool for visualizing what's happening inside a p-n junction. It shows how the conduction band, valence band, and Fermi level vary with position across the device.

Built-in potential barrier

At equilibrium, the Fermi level must be constant throughout the entire structure (no net current flows). Because the Fermi level sits near the conduction band on the n-side and near the valence band on the p-side, the bands must bend across the junction. This band bending corresponds to the built-in potential $V_{bi}$.

The built-in potential is given by:

$V_{bi} = \frac{kT}{q} \ln\left(\frac{N_A N_D}{n_i^2}\right)$

where $N_A$ and $N_D$ are the acceptor and donor concentrations, $n_i$ is the intrinsic carrier concentration, $k$ is Boltzmann's constant, $T$ is absolute temperature, and $q$ is the electron charge. For a typical silicon junction, $V_{bi}$ is around 0.6–0.8 V.

Depletion region

The depletion region extends on both sides of the metallurgical junction, but not equally. It penetrates further into the more lightly doped side. The total depletion width $W$ depends on the doping concentrations and any applied voltage:

$W = \sqrt{\frac{2\epsilon_s}{q}\left(\frac{N_A + N_D}{N_A N_D}\right)(V_{bi} - V)}$

where $\epsilon_s$ is the semiconductor permittivity and $V$ is the applied voltage (positive for forward bias, negative for reverse bias).

Space charge distribution

Within the depletion region, the charge distribution follows the doping profile:

• On the p-side of the junction: fixed negative charge from ionized acceptors ($-qN_A$)
• On the n-side of the junction: fixed positive charge from ionized donors ($+qN_D$)

Under the abrupt junction approximation, the charge density is constant on each side and drops to zero at the depletion region edges. Overall charge neutrality requires $N_A x_p = N_D x_n$, where $x_p$ and $x_n$ are the depletion widths on the p-side and n-side respectively.

Electric field in depletion region

Solving Poisson's equation with the space charge distribution gives a triangular electric field profile:

• The field is zero outside the depletion region.
• It increases linearly from the p-side edge, reaches a maximum magnitude at the metallurgical junction, then decreases linearly to zero at the n-side edge.
• The field points from the n-side to the p-side (from positive to negative charge).

The peak electric field at the junction is:

$\mathcal{E}_{max} = \frac{qN_D x_n}{\epsilon_s} = \frac{qN_A x_p}{\epsilon_s}$

This field is what sweeps minority carriers across the junction and opposes majority carrier diffusion.

Biasing of p-n junctions

Applying an external voltage to a p-n junction shifts it away from equilibrium and controls current flow. The direction of the applied voltage determines whether the junction conducts or blocks.

Forward vs reverse bias

Forward bias (positive terminal to p-side, negative to n-side):

• Reduces the potential barrier from $V_{bi}$ to $V_{bi} - V$
• Shrinks the depletion region
• Majority carriers can more easily cross the junction, producing a large current

Reverse bias (negative terminal to p-side, positive to n-side):

• Increases the potential barrier to $V_{bi} + |V|$
• Widens the depletion region
• Majority carriers are pushed away from the junction; only a tiny reverse saturation current flows due to thermally generated minority carriers

Current-voltage characteristics

The I-V curve of a p-n junction is strongly asymmetric, which is the basis of its rectifying behavior:

• Forward bias: Current rises exponentially once the applied voltage approaches $V_{bi}$. For silicon diodes, noticeable conduction begins around 0.6–0.7 V.
• Reverse bias: Current is very small and roughly constant (the reverse saturation current $I_s$, typically nanoamps to picoamps for silicon) until breakdown occurs.

Ideal diode equation

The relationship between current and voltage is described by the Shockley diode equation:

$I = I_s\left(e^{qV/nkT} - 1\right)$

• $I_s$ is the reverse saturation current
• $q$ is the electron charge ($1.6 \times 10^{-19}$ C)
• $V$ is the applied voltage
• $n$ is the ideality factor ($n = 1$ for an ideal junction; $n$ between 1 and 2 for real devices)
• $k$ is Boltzmann's constant ($1.38 \times 10^{-23}$ J/K)
• $T$ is absolute temperature

At room temperature (300 K), the thermal voltage $kT/q \approx 26$ mV. This sets the voltage scale for exponential turn-on.

The ideal form (with $n = 1$) assumes current is entirely due to minority carrier diffusion and neglects recombination/generation in the depletion region.

Deviations from ideal behavior

Real junctions deviate from the ideal equation in several ways:

• Series resistance: At high forward currents, the voltage drop across the bulk semiconductor and contacts becomes significant, causing the I-V curve to flatten.
• Recombination current: At low forward bias, recombination of carriers within the depletion region adds an extra current component (captured by $n \approx 2$ in the ideality factor).
• Generation current: Under reverse bias, thermal generation of electron-hole pairs in the depletion region produces a current that can exceed the ideal $I_s$.
• Surface effects: Leakage along the junction surface and surface recombination can alter the measured characteristics, especially in small-area devices.

Capacitance of p-n junctions

A p-n junction stores charge in the depletion region and in the injected minority carriers, so it behaves like a voltage-dependent capacitor. This capacitance matters for high-frequency and switching applications.

Junction capacitance

Junction (or depletion) capacitance arises because changing the applied voltage changes the depletion width, which changes the stored charge on the fixed ions. It behaves like a parallel-plate capacitor whose plate spacing varies with voltage:

$C_j = \frac{\epsilon_s A}{W} = C_{j0}\left(1 - \frac{V}{V_{bi}}\right)^{-1/2}$

where $A$ is the junction area, $W$ is the depletion width, and $C_{j0}$ is the zero-bias junction capacitance. The exponent is $-1/2$ for an abrupt junction and $-1/3$ for a linearly graded junction.

Under reverse bias, $C_j$ decreases as the depletion region widens. This voltage-dependent capacitance is the operating principle behind varactor diodes, used in voltage-controlled oscillators and tuning circuits.

Diffusion capacitance

Under forward bias, minority carriers are injected into the neutral regions and stored there for a time on the order of the minority carrier lifetime $\tau$. Changing the forward voltage changes this stored charge, giving rise to diffusion capacitance:

$C_d = \frac{\tau \, q \, I}{nkT} = \frac{\tau \, I}{nV_T}$

where $I$ is the forward current and $V_T = kT/q$ is the thermal voltage. Diffusion capacitance is proportional to the DC bias current, so it dominates at high forward bias and limits the switching speed of bipolar devices.

Variation with applied voltage

• Reverse bias: Junction capacitance dominates. It decreases with increasing reverse voltage.
• Forward bias: Diffusion capacitance dominates and increases with current.
• Total capacitance is the sum: $C_{total} = C_j + C_d$. In practice, one term usually dominates depending on the bias regime.

Breakdown mechanisms in p-n junctions

When the reverse voltage across a p-n junction exceeds a critical value, the reverse current increases sharply. This is breakdown. Two distinct physical mechanisms can cause it, and which one dominates depends on the doping level.

Zener breakdown

Zener breakdown occurs in heavily doped junctions where the depletion region is very narrow (on the order of tens of nanometers).

• The strong electric field across this thin barrier allows electrons to tunnel directly from the valence band on the p-side to the conduction band on the n-side. This is quantum mechanical tunneling (also called band-to-band tunneling).
• Zener breakdown produces a sharp, well-defined breakdown voltage.
• The Zener voltage typically decreases slightly with increasing temperature (negative temperature coefficient) because the bandgap narrows at higher temperatures, making tunneling easier.
• Zener breakdown dominates for breakdown voltages below about 5–6 V in silicon.

Avalanche breakdown

Avalanche breakdown occurs in lightly doped junctions with wide depletion regions.

1. Minority carriers entering the depletion region are accelerated by the strong electric field.
2. If a carrier gains enough kinetic energy (greater than the bandgap energy), it can knock a valence electron into the conduction band through impact ionization, creating a new electron-hole pair.
3. The newly created carriers are also accelerated and can cause further impact ionization.
4. This chain reaction (avalanche multiplication) produces a rapid increase in current.

• Avalanche breakdown has a positive temperature coefficient: at higher temperatures, more frequent lattice scattering reduces the mean free path of carriers, so a higher voltage is needed to reach the critical ionization energy.
• Avalanche breakdown dominates for breakdown voltages above about 5–6 V in silicon.

Breakdown voltage

The breakdown voltage $V_{BR}$ depends on:

• Doping concentration: Higher doping produces a narrower depletion region and stronger field at lower voltages, so $V_{BR}$ decreases with increasing doping.
• Semiconductor material: Wider bandgap materials (like SiC or GaN) have higher breakdown fields and therefore higher $V_{BR}$.
• Junction geometry: Sharp edges and corners concentrate the electric field and reduce $V_{BR}$. Edge termination techniques (guard rings, field plates) are used to mitigate this.

Controlled breakdown is exploited in Zener diodes (voltage regulation), avalanche photodiodes (high-sensitivity light detection), and TVS diodes (transient voltage suppression).

Applications of p-n junctions

The rectifying, light-emitting, and photosensitive properties of p-n junctions make them central to a huge range of devices.

Rectification and power conversion

The asymmetric I-V characteristic of a p-n junction makes it a natural rectifier: it conducts in forward bias and blocks in reverse bias.

• A half-wave rectifier uses a single diode to pass only one polarity of an AC waveform.
• A full-wave rectifier (typically four diodes in a bridge configuration) rectifies both halves of the AC cycle, producing a smoother DC output.
• Rectifier diodes are found in virtually every AC-to-DC power supply, from phone chargers to industrial converters.

Light-emitting diodes (LEDs)

When a p-n junction made from a direct bandgap semiconductor (like GaAs, GaN, or InGaP) is forward biased, electrons and holes recombine radiatively, emitting photons.

• The photon energy (and therefore the color of light) is approximately equal to the bandgap: $E_{photon} \approx E_g$.
• GaN-based LEDs emit blue/UV light ($E_g \approx 3.4$ eV); GaAs-based LEDs emit infrared ($E_g \approx 1.4$ eV). Visible colors across the spectrum are achieved by alloying different III-V compounds.
• Silicon, being an indirect bandgap semiconductor, is a poor light emitter because radiative recombination is inefficient without phonon assistance.

Solar cells and photovoltaics

Solar cells work in reverse compared to LEDs: they absorb photons and generate electrical power.

1. Photons with energy $\geq E_g$ are absorbed, creating electron-hole pairs.
2. Carriers generated within or near the depletion region are separated by the built-in electric field.
3. Electrons are swept to the n-side and holes to the p-side, producing a photocurrent.
4. Under open-circuit conditions, charge accumulation produces a photovoltage (the open-circuit voltage $V_{oc}$).

The efficiency of a solar cell depends on the bandgap (the Shockley-Queisser limit gives a theoretical maximum of about 33% for a single-junction cell at $E_g \approx 1.34$ eV), surface passivation, anti-reflection coatings, and light trapping.

Photodetectors and optical sensors

Reverse-biased p-n junctions act as photodiodes: incident photons generate electron-hole pairs in or near the depletion region, and the electric field sweeps them out as a measurable photocurrent.

• The photocurrent is proportional to the incident light intensity, making photodiodes useful as linear light sensors.
• Key performance metrics include responsivity (A/W), quantum efficiency, dark current (the reverse current with no light), and bandwidth (how fast the detector responds).
• Avalanche photodiodes (APDs) operate just below avalanche breakdown, using internal gain from impact ionization to amplify weak optical signals.
• Photodiodes are used in fiber-optic receivers, camera sensors, LiDAR systems, and scientific instruments.