2.5 Carrier diffusion

Carrier diffusion drives the movement of electrons and holes from regions of high concentration to regions of low concentration. This process is central to how p-n junctions, transistors, solar cells, and LEDs work, so understanding it well gives you a strong foundation for analyzing nearly any semiconductor device.

Carrier diffusion fundamentals

Diffusion vs drift

These are the two main mechanisms of carrier transport in semiconductors, and they arise from completely different causes.

Diffusion is the movement of carriers down a concentration gradient. If you have a lot of electrons bunched up in one region and fewer in an adjacent region, electrons will naturally spread out toward the lower-concentration side. No electric field is needed.

Drift is the movement of carriers in response to an electric field. The field exerts a force on charged particles, pushing electrons opposite to the field direction and holes along the field direction.

In most real devices, both mechanisms operate simultaneously. The total current density for electrons, for example, is the sum of drift and diffusion contributions:

$J_n = qn\mu_n E + qD_n \frac{dn}{dx}$

where the first term is drift and the second is diffusion.

Fick's laws of diffusion

Fick's first law states that the diffusion flux is proportional to the concentration gradient and flows from high to low concentration:

$J = -D \frac{dn}{dx}$

• $J$ is the particle flux (carriers per unit area per unit time)
• $D$ is the diffusion coefficient (units: $\text{cm}^2/\text{s}$)
• $\frac{dn}{dx}$ is the concentration gradient

The negative sign ensures that flux flows in the direction of decreasing concentration.

Fick's second law describes how the concentration profile evolves over time:

$\frac{\partial n}{\partial t} = D \frac{\partial^2 n}{\partial x^2}$

This is the equation you solve (with appropriate boundary conditions) to find how a carrier distribution spreads out as time progresses.

Einstein relation for diffusivity

The Einstein relation links the diffusion coefficient $D$ to the carrier mobility $\mu$:

$D = \frac{k_B T}{q} \mu$

• $k_B$ is Boltzmann's constant ($1.38 \times 10^{-23}$ J/K)
• $T$ is the absolute temperature in Kelvin
• $q$ is the elementary charge ($1.6 \times 10^{-19}$ C)

The quantity $\frac{k_B T}{q}$ is called the thermal voltage ($V_T$), which equals about 26 mV at room temperature (300 K). So at 300 K, $D = 0.026 \times \mu$ (with $\mu$ in $\text{cm}^2/\text{V·s}$ and $D$ in $\text{cm}^2/\text{s}$).

The physical insight here: carriers that are more mobile under an electric field also diffuse faster, and higher temperatures give carriers more thermal energy to spread out.

Diffusion coefficient temperature dependence

From the Einstein relation alone, $D$ increases with temperature because $V_T$ increases. But there's a second effect: mobility itself is temperature-dependent (it generally decreases at high temperatures due to increased phonon scattering).

For dopant or impurity diffusion in a lattice (as opposed to free carrier diffusion), the temperature dependence often follows an Arrhenius form:

$D = D_0 \exp\left(-\frac{E_a}{k_B T}\right)$

• $D_0$ is a pre-exponential factor
• $E_a$ is the activation energy for the diffusion process

This exponential dependence means that even modest temperature increases can significantly boost impurity diffusion rates, which matters a great deal during device fabrication steps like thermal annealing.

Carrier diffusion in semiconductors

Electron and hole diffusion

Both electrons in the conduction band and holes in the valence band undergo diffusion. Each carrier type has its own diffusion coefficient:

• Electrons: $D_n = \frac{k_B T}{q} \mu_n$
• Holes: $D_p = \frac{k_B T}{q} \mu_p$

Since electron mobility is typically higher than hole mobility in most semiconductors (for example, in silicon $\mu_n \approx 1350 \text{ cm}^2/\text{V·s}$ vs. $\mu_p \approx 480 \text{ cm}^2/\text{V·s}$), electrons generally have a larger diffusion coefficient than holes.

The corresponding diffusion current densities are:

$J_{n,\text{diff}} = qD_n \frac{dn}{dx} \quad \text{and} \quad J_{p,\text{diff}} = -qD_p \frac{dp}{dx}$

Note the sign difference: both expressions give current in the direction that positive charge flows.

Carrier concentration gradients

Diffusion only occurs when there's a non-uniform spatial distribution of carriers. Several common situations create these gradients:

• Non-uniform doping produces built-in concentration differences
• Carrier injection at a junction (e.g., forward-biased p-n junction) creates excess carriers near the junction boundary
• Optical generation from focused light creates localized excess carriers

The steeper the concentration gradient, the larger the diffusion flux. This is why the region right next to an injection point typically carries the highest diffusion current.

Minority carrier diffusion

Minority carriers are the less abundant carrier type in a doped region: electrons in p-type material, holes in n-type material. Their diffusion behavior dominates the operation of many devices.

Why does minority carrier diffusion matter so much? In a forward-biased p-n junction, electrons are injected into the p-side where they become minority carriers. These injected electrons must diffuse through the p-type material to be collected or to recombine. How far they get before recombining directly determines device performance.

Two key parameters govern this:

• Minority carrier lifetime ($\tau$): the average time a minority carrier survives before recombining
• Diffusion length ($L = \sqrt{D\tau}$): the characteristic distance a minority carrier travels before recombining

Ambipolar diffusion

When excess electrons and holes are generated together (as in optical excitation), they don't diffuse independently. Charge neutrality couples their motion: if electrons start to diffuse faster than holes, a small internal electric field builds up that slows the electrons and speeds up the holes. The result is that both carrier types effectively diffuse together.

The ambipolar diffusion coefficient captures this coupled behavior:

$D_a = \frac{n_0 D_p + p_0 D_n}{n_0 + p_0}$

where $n_0$ and $p_0$ are the equilibrium electron and hole concentrations. In strongly n-type material ($n_0 \gg p_0$), $D_a \approx D_p$, meaning the slower carrier (holes) limits the pair's diffusion rate. The opposite holds in p-type material.

Diffusion equations and solutions

One-dimensional diffusion equation

The basic 1D diffusion equation comes directly from Fick's second law:

$\frac{\partial n(x,t)}{\partial t} = D \frac{\partial^2 n(x,t)}{\partial x^2}$

This assumes $D$ is constant and independent of position. In real semiconductor problems, you often add a generation term $G$ and a recombination term $\frac{\Delta n}{\tau}$ to get the continuity equation:

$\frac{\partial (\Delta n)}{\partial t} = D \frac{\partial^2 (\Delta n)}{\partial x^2} - \frac{\Delta n}{\tau} + G$

where $\Delta n$ is the excess carrier concentration above equilibrium. This is the equation you'll solve most often in device problems.

Steady-state diffusion

In steady state, nothing changes with time, so $\frac{\partial n}{\partial t} = 0$. The continuity equation for excess minority carriers becomes:

$D \frac{d^2 (\Delta n)}{dx^2} - \frac{\Delta n}{\tau} = 0$

This can be rewritten as:

$\frac{d^2 (\Delta n)}{dx^2} = \frac{\Delta n}{L^2}$

where $L = \sqrt{D\tau}$ is the diffusion length. The general solution is:

$\Delta n(x) = A \exp\left(-\frac{x}{L}\right) + B \exp\left(\frac{x}{L}\right)$

The constants $A$ and $B$ are determined by boundary conditions. For a semi-infinite region (carrier concentration must stay finite as $x \to \infty$), the growing exponential drops out, leaving a simple decaying exponential.

Time-dependent diffusion

When the carrier distribution changes with time, you solve the full time-dependent equation. Two classic solutions appear frequently:

• Gaussian solution: If a pulse of carriers is deposited at $x = 0$ at $t = 0$, the concentration spreads as a Gaussian: $n(x,t) = \frac{N}{\sqrt{4\pi D t}} \exp\left(-\frac{x^2}{4Dt}\right)$, where $N$ is the total number of carriers per unit area.
• Complementary error function (erfc): If the concentration is held fixed at a boundary, the solution involves $\text{erfc}\left(\frac{x}{2\sqrt{Dt}}\right)$.

These solutions show up repeatedly in diffusion problems, so it's worth being comfortable with both functional forms.

Boundary conditions and initial conditions

To solve any diffusion equation, you need:

1. Initial condition: the carrier concentration profile at $t = 0$
2. Boundary conditions: constraints at the edges of your region

Common boundary condition types:

• Dirichlet (fixed concentration): $n = n_0$ at a boundary, used when a contact or junction fixes the carrier density
• Neumann (fixed flux): $\frac{dn}{dx} = \text{const}$ at a boundary, used when the current is specified (a zero-flux condition means no carriers cross that boundary)
• Mixed: a combination of concentration and flux conditions

Choosing the right boundary conditions is often the trickiest part of setting up a diffusion problem. Think carefully about the physics at each boundary before writing down equations.

Diffusion length and time

Diffusion length concept

The diffusion length $L$ is the average distance a carrier diffuses before recombining:

$L = \sqrt{D\tau}$

After traveling one diffusion length from the point of injection, the excess carrier concentration drops to $1/e$ (about 37%) of its initial value. Typical values in silicon:

• Minority electron diffusion length in p-type Si: 100–500 $\mu$m (high-quality material)
• Minority hole diffusion length in n-type Si: 50–200 $\mu$m

A longer diffusion length means carriers survive longer and travel farther, which directly improves the performance of devices that rely on minority carrier collection.

Minority carrier lifetime

The minority carrier lifetime $\tau$ is the average time a minority carrier exists before recombining. Three main recombination mechanisms determine it:

• Radiative recombination: an electron and hole recombine by emitting a photon (dominant in direct-bandgap semiconductors like GaAs)
• Shockley-Read-Hall (SRH) recombination: carriers recombine through defect or trap states in the bandgap (often dominant in indirect-bandgap semiconductors like Si)
• Auger recombination: the recombination energy is transferred to a third carrier instead of being emitted as a photon (important at high carrier concentrations)

The overall lifetime is determined by all mechanisms acting in parallel:

$\frac{1}{\tau} = \frac{1}{\tau_{\text{rad}}} + \frac{1}{\tau_{\text{SRH}}} + \frac{1}{\tau_{\text{Auger}}}$

The shortest individual lifetime dominates the total.

Diffusion length calculation

Here's a step-by-step approach to calculating diffusion length:

1. Find the diffusion coefficient from the Einstein relation: $D = \frac{k_B T}{q} \mu$. At 300 K, this simplifies to $D = 0.026 \times \mu$.
2. Determine the carrier lifetime $\tau$ (from measurement or given data).
3. Calculate: $L = \sqrt{D\tau}$

Example: For minority electrons in p-type silicon with $\mu_n = 1200 \text{ cm}^2/\text{V·s}$ and $\tau_n = 10 \text{ }\mu\text{s}$:

• $D_n = 0.026 \times 1200 = 31.2 \text{ cm}^2/\text{s}$
• $L_n = \sqrt{31.2 \times 10 \times 10^{-6}} = \sqrt{3.12 \times 10^{-4}} \approx 0.018 \text{ cm} = 180 \text{ }\mu\text{m}$

Transit time and dielectric relaxation time

Two additional time scales are important for understanding carrier dynamics:

Transit time ($\tau_{tr}$) is the average time for a carrier to diffuse across a distance $L$:

$\tau_{tr} = \frac{L^2}{2D}$

This tells you how quickly carriers can traverse a device region by diffusion alone. For the base of a bipolar transistor, a shorter transit time means faster switching.

Dielectric relaxation time ($\tau_{dr}$) is the time for a charge imbalance to be neutralized by the majority carriers:

$\tau_{dr} = \frac{\varepsilon}{q\mu n}$

where $\varepsilon$ is the semiconductor permittivity, $\mu$ is the majority carrier mobility, and $n$ is the majority carrier concentration. In doped silicon, $\tau_{dr}$ is extremely short (on the order of $10^{-12}$ s), which is why charge neutrality is restored almost instantaneously in the bulk.

Carrier diffusion applications

P-N junction carrier diffusion

At a p-n junction, there's a steep concentration gradient: many electrons on the n-side and many holes on the p-side. Electrons diffuse from n to p, and holes diffuse from p to n. As these carriers leave, they expose fixed ionized dopant atoms, creating a depletion region with a built-in electric field.

This built-in field opposes further diffusion, and at equilibrium, the drift current exactly balances the diffusion current. Under forward bias, the barrier is reduced and diffusion current increases; under reverse bias, the barrier grows and diffusion current is suppressed.

Bipolar transistor base transport

The base region of a bipolar transistor is where minority carrier diffusion determines device performance. In an NPN transistor:

1. Electrons are injected from the emitter into the p-type base
2. These minority electrons diffuse across the base toward the collector
3. Electrons that reach the base-collector junction are swept into the collector by the junction field

The base transport factor ($\alpha_T$) measures the fraction of injected carriers that make it across without recombining. For a base of width $W$:

$\alpha_T \approx 1 - \frac{W^2}{2L_n^2}$

This is why you want $W \ll L_n$: a thin base relative to the diffusion length means nearly all injected carriers reach the collector, giving high current gain.

Solar cell carrier collection

When photons are absorbed in a solar cell, they generate electron-hole pairs throughout the material. Minority carriers must then diffuse to the p-n junction to be collected.

• Carriers generated within one diffusion length of the junction have a high probability of being collected
• Carriers generated farther away are likely to recombine before reaching the junction

This is why the diffusion length sets an effective "collection volume" for the solar cell. High-quality silicon with long minority carrier lifetimes (and therefore long diffusion lengths) produces more efficient cells. For a typical crystalline silicon solar cell, you want $L$ to be at least comparable to the cell thickness (~200 $\mu$m).

Light-emitting diode carrier injection

In an LED, forward bias injects electrons into the p-side and holes into the n-side. These injected minority carriers diffuse into the bulk and eventually recombine. If recombination is radiative (emitting a photon), you get light output.

The diffusion length determines the spatial extent of the recombination zone. A shorter diffusion length concentrates recombination near the junction, while a longer one spreads it out. LED designs often use heterostructures or quantum wells to confine carriers to a narrow active region, maximizing the chance of radiative recombination rather than relying on diffusion length alone.

Advanced diffusion topics

Diffusion in heterostructures

Heterostructures combine different semiconductor materials (e.g., GaAs/AlGaAs) with different band gaps. The band offsets at the interface create energy barriers or wells that strongly influence carrier diffusion.

• A conduction band offset can block electron diffusion across the interface
• A quantum well (thin low-bandgap layer sandwiched between higher-bandgap layers) traps carriers, confining them for efficient recombination

These effects are exploited in heterojunction bipolar transistors (HBTs), quantum well lasers, and high electron mobility transistors (HEMTs), where controlling carrier diffusion at interfaces is central to device design.

Surface recombination effects

Semiconductor surfaces have dangling bonds and defects that act as efficient recombination centers. This surface recombination is characterized by a surface recombination velocity $S$ (units: cm/s), where higher $S$ means faster carrier loss at the surface.

Surface recombination effectively reduces the minority carrier lifetime and diffusion length near the surface. This is especially problematic for devices with large surface-to-volume ratios, such as nanowires and thin films.

Passivation techniques reduce surface recombination by satisfying dangling bonds. Common approaches include thermal oxide growth on silicon, silicon nitride deposition, and atomic layer deposition of $\text{Al}_2\text{O}_3$. Good passivation can improve solar cell efficiency by several percentage points.

Grain boundary diffusion

In polycrystalline semiconductors (used in thin-film solar cells and display transistors), grain boundaries separate individual crystallites. These boundaries contain many defects and can:

• Act as fast diffusion paths for impurities, since the disordered structure at grain boundaries allows atoms to move more easily than through the bulk crystal
• Serve as recombination centers that reduce minority carrier lifetime

Both effects degrade device performance. Grain boundary passivation (e.g., hydrogen passivation) and optimizing grain size during deposition are common strategies to mitigate these issues.

Diffusion-induced stress and strain

When impurity atoms diffuse into a semiconductor lattice, they can distort the local crystal structure if their atomic radius differs from the host atoms. This creates stress and strain fields in the material.

The relationship goes both ways: stress and strain modify the local band structure and can alter carrier mobilities and diffusion rates. At high dopant concentrations or during high-temperature processing, these effects become significant and can impact device reliability.

Modeling diffusion-induced stress is particularly important in modern nanoscale devices, where even small lattice distortions can meaningfully affect electrical characteristics.