5.2 Depletion region and space charge

Formation of depletion region

When p-type and n-type semiconductors are joined, majority carriers on each side diffuse across the junction. Electrons from the n-side move into the p-side, and holes from the p-side move into the n-side. As these mobile carriers leave the region near the junction, they expose fixed ionized atoms that can't move. The result is a narrow zone stripped of mobile carriers: the depletion region (also called the space charge region).

This region is central to how p-n junctions, solar cells, transistors, and many other semiconductor devices operate.

Built-in potential barrier

The diffusion of carriers doesn't continue forever. As ionized donors and acceptors are exposed, they create an electric field pointing from the n-side (positive ions) toward the p-side (negative ions). This field opposes further diffusion.

The potential difference that develops across the depletion region is the built-in potential barrier, $V_{bi}$. It acts as an energy hill that majority carriers must overcome to cross the junction. At equilibrium, $V_{bi}$ is just large enough to stop net diffusion entirely.

Equilibrium condition

At thermal equilibrium, two opposing currents exist across the junction:

• Diffusion current driven by the concentration gradient of carriers
• Drift current driven by the built-in electric field

These two currents exactly cancel, so the net current is zero. This balance is reached when the Fermi level becomes constant (flat) throughout the entire device. A flat Fermi level is the defining signature of thermal equilibrium in any semiconductor system.

Fermi level alignment

The Fermi level represents the electrochemical potential of electrons in a material. In an isolated p-type semiconductor, the Fermi level sits near the valence band; in n-type, it sits near the conduction band.

When the two materials form a junction, their Fermi levels must align to a single, constant value at equilibrium. This alignment forces the conduction and valence bands to bend near the junction:

• On the p-side, bands bend upward
• On the n-side, bands bend downward

The total band bending equals $qV_{bi}$, where $q$ is the elementary charge. Reading a band diagram correctly is one of the most useful skills for analyzing any junction device.

Characteristics of depletion region

The depletion region's width, electric field profile, potential profile, and capacitance collectively determine how a junction device behaves under different conditions.

Width of depletion region

The total depletion width $W_D$ depends on doping levels and applied voltage. For an abrupt p-n junction:

$W_D = \sqrt{\frac{2\varepsilon_s(V_{bi}-V_A)}{q}\left(\frac{1}{N_A}+\frac{1}{N_D}\right)}$

where:

• $\varepsilon_s$ = permittivity of the semiconductor
• $V_{bi}$ = built-in potential
• $V_A$ = applied voltage (positive for forward bias)
• $N_A$, $N_D$ = acceptor and donor doping concentrations

A wider depletion region means lower junction capacitance and a higher breakdown voltage. Notice that the depletion region extends further into whichever side is more lightly doped.

Electric field distribution

The electric field inside the depletion region is not constant. For an abrupt junction:

• It peaks at the metallurgical junction ($x = 0$)
• It drops linearly to zero at each edge of the depletion region

You can find the field profile by solving Poisson's equation with the known charge density on each side. The maximum electric field at the junction is:

$E_{max} = \frac{qN_D x_n}{\varepsilon_s} = \frac{qN_A x_p}{\varepsilon_s}$

where $x_n$ and $x_p$ are the depletion widths on the n-side and p-side, respectively.

Potential distribution

The electrostatic potential is related to the electric field by:

$E(x) = -\frac{dV}{dx}$

Since the field varies linearly in an abrupt junction, the potential varies quadratically (parabolically) across the depletion region. The total potential drop across the region equals $V_{bi} - V_A$.

This potential profile determines how carriers move through the junction and is directly visible in band diagrams as the shape of the band bending.

Capacitance of depletion region

The depletion region behaves like a parallel-plate capacitor. The p-type and n-type neutral regions act as the "plates," and the depleted zone acts as the insulating dielectric between them.

The junction capacitance per unit area is:

$C_D = \frac{\varepsilon_s A}{W_D}$

where $A$ is the junction cross-sectional area. Because $W_D$ depends on voltage, the capacitance is voltage-dependent. This is not a parasitic effect; it's deliberately exploited in varactors and is critical for high-frequency circuit design.

Space charge in depletion region

The term "space charge" refers to the net charge density created by the exposed, immobile ionized impurities inside the depletion region. This charge distribution is what generates the electric field and built-in potential.

Ionized donors and acceptors

When mobile carriers are swept out of the depletion region by the electric field, they leave behind:

• Ionized donors ($N_D^+$) on the n-side, each carrying a charge of $+q$
• Ionized acceptors ($N_A^-$) on the p-side, each carrying a charge of $-q$

Assuming complete ionization (valid at room temperature for common dopants like B, P, As in Si), the concentration of ionized impurities equals the doping concentration in each region.

Charge density profile

For an abrupt junction under the depletion approximation, the charge density is a step function:

• N-side of depletion region: $\rho(x) = +qN_D$
• P-side of depletion region: $\rho(x) = -qN_A$
• Outside the depletion region: $\rho(x) = 0$

Overall charge neutrality requires that the total positive charge on the n-side equals the total negative charge on the p-side:

$N_D \cdot x_n = N_A \cdot x_p$

This constraint is why the depletion region extends further into the lightly doped side.

Poisson's equation in depletion region

Poisson's equation connects the charge density to the potential:

$\frac{d^2V}{dx^2} = -\frac{\rho(x)}{\varepsilon_s}$

Solving this equation is how you derive the electric field and potential profiles. The steps are:

1. Write $\rho(x)$ for each region using the depletion approximation
2. Integrate once to get $E(x)$, applying the boundary condition that $E = 0$ at the depletion edges
3. Integrate again to get $V(x)$, applying continuity of potential at the junction
4. Use the total potential drop ($V_{bi} - V_A$) to solve for the depletion width

Depletion approximation

The depletion approximation makes two key assumptions:

• The depletion region is completely free of mobile carriers
• The transition from depleted to neutral regions is abrupt (sharp boundaries)

These assumptions let you treat $\rho(x)$ as a simple step function, which makes Poisson's equation solvable analytically. The approximation works well for moderately doped junctions under normal operating conditions.

It breaks down in two situations:

• Heavily doped junctions, where quantum effects and incomplete ionization matter
• High injection conditions, where injected carrier densities become comparable to the doping level

Factors affecting depletion region

Doping concentrations

The doping levels $N_A$ and $N_D$ have a strong influence on the depletion region:

• Higher doping on both sides produces a narrower depletion region, higher peak electric field, and larger capacitance
• Asymmetric doping (e.g., $N_A \gg N_D$) causes the depletion region to extend mostly into the lightly doped side. In this case, the lightly doped side controls the depletion width

For a one-sided junction ($N_A \gg N_D$), the depletion width simplifies to approximately:

$W_D \approx \sqrt{\frac{2\varepsilon_s(V_{bi}-V_A)}{qN_D}}$

Applied bias voltage

The applied voltage $V_A$ directly modifies the barrier height and depletion width:

• Forward bias ($V_A > 0$, positive voltage on p-side): Reduces the barrier to $V_{bi} - V_A$, shrinks the depletion region, and increases capacitance
• Reverse bias ($V_A < 0$): Increases the barrier to $V_{bi} + |V_A|$, widens the depletion region, and decreases capacitance

This voltage dependence of $W_D$ is what makes the junction capacitance tunable and is the operating principle behind varactors.

Temperature dependence

Temperature affects the depletion region through several mechanisms:

• The intrinsic carrier concentration $n_i$ increases exponentially with temperature, which reduces $V_{bi}$ (since $V_{bi}$ depends on $\ln(N_A N_D / n_i^2)$)
• A lower $V_{bi}$ means a narrower depletion region at equilibrium
• Carrier mobilities also change with temperature, affecting current-voltage characteristics

For devices operating over a wide temperature range (e.g., automotive or space applications), these shifts must be accounted for in the design.

Depletion region in p-n junctions

The behavior of the depletion region depends significantly on the doping profile at the junction. Three common profiles are worth understanding.

Abrupt p-n junction

An abrupt junction has a sharp, step-like transition in doping at the metallurgical junction. This is the simplest model and the one most commonly analyzed in textbooks.

All the standard formulas apply directly. The depletion width is:

$W_D = \sqrt{\frac{2\varepsilon_s(V_{bi}-V_A)}{q}\left(\frac{1}{N_A}+\frac{1}{N_D}\right)}$

The electric field varies linearly, and the potential varies quadratically within the depletion region. Junctions formed by ion implantation with a steep profile are well-approximated by this model.

Linearly graded p-n junction

In a linearly graded junction, the net doping changes gradually across the junction rather than switching abruptly. The doping profile near the junction is described by:

$N_D - N_A = ax$

where $a$ is the doping gradient (in $\text{cm}^{-4}$) and $x$ is measured from the junction.

Compared to an abrupt junction:

• The electric field profile is parabolic rather than triangular
• The depletion width scales as $(V_{bi} - V_A)^{1/3}$ instead of $(V_{bi} - V_A)^{1/2}$
• The capacitance varies as $(V_{bi} - V_A)^{-1/3}$

Junctions formed by deep diffusion processes often approximate this graded profile.

Asymmetrical p-n junction

When one side is doped much more heavily than the other (e.g., a $p^+$-n junction where $N_A \gg N_D$), the depletion region extends almost entirely into the lightly doped side.

This has practical consequences:

• The lightly doped side determines the breakdown voltage
• Carrier collection in solar cells and photodetectors is optimized by controlling the width of the lightly doped region
• The junction capacitance is dominated by the lighter doping concentration

Most real devices use asymmetric doping intentionally to control where the depletion region sits and how wide it extends.

Depletion region in metal-semiconductor junctions

When a metal contacts a semiconductor, a junction forms whose properties depend on the relative work functions of the two materials. Unlike p-n junctions, only one side (the semiconductor) contributes a depletion region.

Schottky barrier

A Schottky barrier forms when the contact creates a rectifying junction. For an n-type semiconductor, this occurs when the metal work function $\phi_m$ exceeds the semiconductor electron affinity $\chi$.

The barrier height is:

$\phi_B = \phi_m - \chi$

This barrier controls current flow across the junction. In practice, the actual barrier height often deviates from this ideal relation due to Fermi-level pinning caused by interface states. Schottky diodes switch faster than p-n diodes because they are majority-carrier devices with no minority-carrier storage delay.

Ohmic contact

An ohmic contact shows a linear, low-resistance current-voltage characteristic. Ideally, it forms when:

• The metal work function is lower than the semiconductor's for n-type material
• The metal work function is higher than the semiconductor's for p-type material

In practice, ohmic contacts are usually made by heavily doping the semiconductor surface (creating a very thin barrier that carriers can tunnel through), rather than relying on work function matching alone. Every semiconductor device needs ohmic contacts to connect to external circuits.

Rectifying vs non-rectifying contacts

The type of contact depends on the metal-semiconductor work function difference and, in real devices, on surface states and interfacial layers that can pin the Fermi level. Choosing the right contact type for each terminal is a basic but critical part of device design.

Applications of depletion region

Semiconductor devices

The depletion region is at the heart of the most important semiconductor devices:

• P-N junction diodes: The depletion region enables rectification. Forward bias shrinks it and allows current; reverse bias widens it and blocks current.
• Bipolar junction transistors (BJTs): Two back-to-back junctions with depletion regions control current amplification. The base-emitter junction is forward biased while the base-collector junction is reverse biased.
• MOSFETs: A depletion region forms at the semiconductor-oxide interface and modulates the channel conductivity based on gate voltage.

Solar cells

Solar cells use the depletion region's built-in electric field to separate photogenerated electron-hole pairs. When a photon is absorbed near the junction, the field sweeps electrons toward the n-side and holes toward the p-side, generating a photocurrent.

Designing an efficient solar cell involves balancing the depletion width: wide enough to absorb a significant fraction of incoming light, but not so wide that carriers recombine before being collected. Typical silicon solar cells have depletion widths on the order of a few micrometers.

Photodetectors

Photodetectors also rely on carrier generation and collection in or near the depletion region. Different designs optimize for different performance metrics:

• P-N photodiodes: Simple structure, moderate speed and sensitivity
• P-I-N photodiodes: An intrinsic (undoped) layer widens the absorption region, improving quantum efficiency and speed
• Avalanche photodiodes (APDs): Operate under high reverse bias so that photogenerated carriers trigger impact ionization, providing internal gain

In all cases, the depletion region width and electric field strength are the key design parameters that determine quantum efficiency and response time.

Capacitors and varactors

The voltage-dependent capacitance of the depletion region is exploited in several device types:

• MOS capacitors: Used in integrated circuits for charge storage, filtering, and as the basis of DRAM memory cells
• Varactors (variable-capacitance diodes): Operated under reverse bias, where changing $V_A$ tunes the capacitance. The capacitance-voltage relationship follows $C \propto (V_{bi} - V_A)^{-1/2}$ for an abrupt junction.

Varactors are used in voltage-controlled oscillators (VCOs), tunable filters, and phase-locked loops, where precise electronic control of capacitance is needed for frequency tuning or impedance matching.