9.2 Minority carrier injection and transport

Minority Carrier Injection and Transport

Minority carrier injection and transport describe how electrons move through p-type regions and holes move through n-type regions. These processes are at the heart of how BJTs work: the transistor's ability to amplify current depends entirely on injecting minority carriers across the base and collecting them at the collector. This topic covers injection regimes, transport mechanisms, recombination, and the math that ties it all together.

Minority Carrier Injection

When you forward-bias a p-n junction, you push carriers across the junction into regions where they become the minority. Electrons get injected into the p-side (where holes dominate), and holes get injected into the n-side (where electrons dominate). The concentration of these injected minority carriers controls how much current flows through devices like BJTs and solar cells.

Electron and Hole Concentrations

In any semiconductor, the concentrations of electrons ($n$) and holes ($p$) determine its electrical behavior. A pure (intrinsic) semiconductor has equal numbers of both, set by the intrinsic carrier concentration $n_i$. At thermal equilibrium, the product $np = n_i^2$ always holds.

Doping changes the balance:

• n-type doping (adding donors like phosphorus to silicon) makes electrons the majority carriers, so $n \gg p$
• p-type doping (adding acceptors like boron) makes holes the majority carriers, so $p \gg n$

The minority carrier concentration in each region is what gets manipulated during device operation.

Quasi-Fermi Levels

At thermal equilibrium, a single Fermi level $E_F$ describes the occupation of all states. Under non-equilibrium conditions (applied bias, illumination), a single Fermi level no longer works. Instead, you use two quasi-Fermi levels:

• $E_{Fn}$ for electrons
• $E_{Fp}$ for holes

Each quasi-Fermi level marks the energy where the occupation probability is 0.5 for that carrier type. The splitting between $E_{Fn}$ and $E_{Fp}$ tells you how far the system is from equilibrium. Larger splitting means more excess carriers are present. In a forward-biased p-n junction, the quasi-Fermi level splitting across the junction equals $qV$, where $V$ is the applied voltage.

Low-Level Injection

Low-level injection is the regime where the injected minority carrier concentration stays much smaller than the equilibrium majority carrier concentration. For example, if you inject $10^{12}$ electrons/cm³ into a p-type region with $10^{17}$ holes/cm³, you're firmly in low-level injection.

In this regime:

• The majority carrier concentration barely changes
• The electric field distribution stays essentially the same as in equilibrium
• The math simplifies considerably, which is why most textbook BJT analysis assumes low-level injection

High-Level Injection

High-level injection occurs when the injected minority carrier concentration approaches or exceeds the majority carrier concentration. At this point, the assumptions of low-level injection break down:

• The majority carrier concentration shifts significantly to maintain charge neutrality
• The internal electric field is altered by the injected carriers
• Conductivity modulation occurs, changing the effective resistance of the region

High-level injection matters in power transistors, laser diodes, and any device operating at large current densities. In BJTs, high-level injection in the base causes the current gain to drop at high collector currents (the Kirk effect is a related phenomenon).

Minority Carrier Transport

Once minority carriers are injected, they need to move through the semiconductor. Two mechanisms govern this motion: drift (driven by electric fields) and diffusion (driven by concentration gradients). In the quasi-neutral regions of a BJT, diffusion typically dominates because the electric field is weak.

Drift and Diffusion Currents

Drift current arises when an electric field $E$ pushes carriers along:

• Electrons: $J_{n,drift} = qn\mu_n E$
• Holes: $J_{p,drift} = qp\mu_p E$

Here $\mu_n$ and $\mu_p$ are the electron and hole mobilities, and $q$ is the elementary charge.

Diffusion current arises when carriers move from regions of high concentration to low concentration:

• Electrons: $J_{n,diff} = qD_n \frac{dn}{dx}$
• Holes: $J_{p,diff} = -qD_p \frac{dp}{dx}$

The negative sign for holes reflects that holes diffusing down a concentration gradient produce current in the opposite direction to the gradient. $D_n$ and $D_p$ are the diffusion coefficients.

The total current for each carrier type is the sum of its drift and diffusion components. In the base of a BJT under normal operation, minority carriers (say, electrons in a p-type base) move primarily by diffusion from the emitter side toward the collector.

Einstein Relation

The diffusion coefficient and mobility aren't independent. They're linked by the Einstein relation:

$\frac{D_n}{\mu_n} = \frac{D_p}{\mu_p} = \frac{k_B T}{q}$

At room temperature ($T \approx 300$ K), $\frac{k_B T}{q} \approx 0.026$ V, which is the thermal voltage $V_T$. So if you know the mobility of a carrier, you can immediately find its diffusion coefficient: $D = \mu V_T$.

Ambipolar Transport

When both electrons and holes are present in comparable concentrations (as in high-level injection), their motions become coupled through the requirement of charge neutrality. You can't move a bunch of electrons without the holes responding to maintain local neutrality.

This coupled motion is described by the ambipolar diffusion coefficient and ambipolar mobility:

$D_a = \frac{nD_p + pD_n}{n + p}$

$\mu_a = \frac{n\mu_p - p\mu_n}{n + p}$

Note that the ambipolar diffusion coefficient is a weighted average that shifts toward the slower carrier's value, since the faster carrier is held back by the slower one.

Minority Carrier Recombination

Injected minority carriers don't last forever. They eventually recombine with majority carriers, releasing energy. The recombination rate determines the minority carrier lifetime, which directly affects how far carriers can travel before they're lost. In a BJT, recombination in the base is the main source of base current and limits the current gain.

Direct Recombination

In direct (band-to-band) recombination, an electron drops from the conduction band directly into a vacant state in the valence band, emitting a photon with energy approximately equal to the bandgap $E_g$.

This process dominates in direct bandgap semiconductors like GaAs and InP, where the conduction band minimum and valence band maximum occur at the same crystal momentum. The recombination rate is proportional to both the electron and hole concentrations: $R = B(np - n_i^2)$, where $B$ is the radiative recombination coefficient.

Indirect Recombination

In indirect bandgap semiconductors like silicon and germanium, the conduction band minimum and valence band maximum occur at different crystal momenta. A direct transition can't conserve both energy and momentum, so a phonon (lattice vibration) must participate.

This three-particle process is much less probable than direct recombination, which is why silicon is a poor light emitter but why minority carrier lifetimes in silicon can be relatively long (microseconds to milliseconds in high-quality material). In practice, recombination in silicon is usually dominated by trap-assisted (Shockley-Read-Hall) recombination through defect states in the bandgap, rather than by band-to-band indirect transitions.

Carrier Lifetime

The carrier lifetime $\tau$ is the average time a minority carrier survives before recombining. It connects to the recombination rate through:

$\tau = \frac{\Delta n}{R}$

where $\Delta n$ is the excess minority carrier concentration and $R$ is the recombination rate.

The lifetime is critical because it directly determines the diffusion length (covered below) and therefore how effectively a device can collect injected carriers. In a BJT, a longer minority carrier lifetime in the base means fewer carriers recombine before reaching the collector, which means higher current gain.

Surface Recombination

Semiconductor surfaces have broken (dangling) bonds that create energy states within the bandgap. These surface states act as efficient recombination centers, characterized by the surface recombination velocity $S$ (units of cm/s).

• A perfectly passivated surface has $S \approx 0$
• An unpassivated silicon surface can have $S > 10^5$ cm/s

Surface recombination is especially damaging in devices with large surface-to-volume ratios (thin films, nanowires, small-geometry BJTs). Passivation techniques like thermal oxide growth on silicon or chemical treatments reduce $S$ and improve device performance.

Continuity Equation

The continuity equation is the bookkeeping equation for carriers. It says: the rate of change of carrier concentration at any point equals what's generated minus what recombines, plus what flows in minus what flows out.

For electrons:

$\frac{\partial n}{\partial t} = G_n - R_n + \frac{1}{q}\frac{\partial J_n}{\partial x}$

For holes:

$\frac{\partial p}{\partial t} = G_p - R_p - \frac{1}{q}\frac{\partial J_p}{\partial x}$

The sign difference comes from the opposite charge of electrons and holes.

Time-Dependent Carrier Concentrations

The carrier concentrations $n(x,t)$ and $p(x,t)$ vary in both space and time. The continuity equation governs this variation by accounting for all the processes that create, destroy, or move carriers.

Time-dependent solutions matter for understanding transient behavior: how a BJT responds to a switching signal, how a photodetector responds to a light pulse, or how carriers build up and decay after injection.

Generation and Recombination Rates

• Generation rate $G$: the rate at which electron-hole pairs are created (by light absorption, impact ionization, or thermal excitation)
• Recombination rate $R$: the rate at which electron-hole pairs annihilate

The net recombination rate $U = R - G$ determines whether carriers are accumulating or depleting at a given point. Under illumination, $G$ can exceed $R$, creating excess carriers. In the dark with injection, $R$ exceeds the thermal generation rate, and excess carriers decay.

Steady-State Conditions

In steady state, nothing changes with time, so $\frac{\partial n}{\partial t} = 0$ and $\frac{\partial p}{\partial t} = 0$. The continuity equations simplify to:

• For electrons: $\frac{1}{q}\frac{dJ_n}{dx} = R_n - G_n$
• For holes: $-\frac{1}{q}\frac{dJ_p}{dx} = R_p - G_p$

Solving these alongside the drift-diffusion equations and Poisson's equation gives you the complete picture of carrier distributions and currents in a device. This is the foundation of BJT DC analysis.

Minority Carrier Diffusion Equation

By combining the continuity equation with the diffusion current expression (and neglecting drift in the quasi-neutral regions), you get the minority carrier diffusion equation. For excess electrons $\Delta n$ in a p-type region:

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

In steady state with no generation, this reduces to:

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

This is a second-order ODE with exponential solutions, and it's the equation you'll solve repeatedly when analyzing BJT base regions.

Diffusion Length

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

$L = \sqrt{D\tau}$

For electrons in a p-type region: $L_n = \sqrt{D_n \tau_n}$. For holes in an n-type region: $L_p = \sqrt{D_p \tau_p}$.

The diffusion length sets the spatial scale of the excess carrier distribution. In a BJT, the base width $W$ should be much smaller than the minority carrier diffusion length in the base ($W \ll L$) to ensure most injected carriers reach the collector without recombining. This is a key design requirement.

Boundary Conditions

To solve the diffusion equation, you need boundary conditions at the edges of each region. The most common ones:

• At the edge of the depletion region: The excess minority carrier concentration is set by the applied voltage via the law of the junction: $\Delta n(0) = n_{p0}(e^{qV/k_BT} - 1)$, where $n_{p0}$ is the equilibrium minority electron concentration in the p-region
• Surface recombination: $D\frac{d(\Delta n)}{dx}\bigg|_{surface} = S \cdot \Delta n_{surface}$
• Ohmic contact: The excess carrier concentration goes to zero ($\Delta n = 0$), since the contact provides infinite recombination
• Reflective/insulating boundary: The carrier current is zero, so $\frac{d(\Delta n)}{dx} = 0$

Choosing the right boundary conditions is half the work of solving any minority carrier problem.

Excess Carrier Distribution

Solving the diffusion equation with appropriate boundary conditions gives you the excess carrier profile $\Delta n(x)$. The general solution in a field-free region is:

$\Delta n(x) = A e^{-x/L} + B e^{x/L}$

where $A$ and $B$ are determined by boundary conditions.

In the base of a BJT (with base width $W \ll L$), the profile is approximately linear, which simplifies the current calculation. The slope of this profile determines the diffusion current, and therefore the collector current. This is why the excess carrier distribution is so central to BJT analysis.

Minority Carriers in p-n Junctions

The p-n junction is the building block of the BJT (which is essentially two p-n junctions back to back). Understanding minority carrier behavior in a single junction is the prerequisite for understanding transistor operation.

Depletion Region

When p-type and n-type materials are joined, majority carriers diffuse across the junction and recombine, leaving behind a depletion region of exposed, immobile dopant ions. This creates a built-in electric field that opposes further diffusion.

The depletion width depends on:

• Doping concentrations on each side (higher doping = narrower depletion on that side)
• Applied voltage (forward bias narrows it, reverse bias widens it)
• Semiconductor permittivity

Within the depletion region, the strong electric field sweeps any minority carriers across quickly. In BJT analysis, you typically assume carriers transit the depletion region without recombining.

Quasi-Neutral Regions

Outside the depletion region, the semiconductor is approximately charge-neutral (majority carriers balance the dopant ions). These are the quasi-neutral regions where minority carrier diffusion dominates transport.

The minority carrier diffusion equation applies here. The boundary condition at the depletion region edge is set by the junction voltage (law of the junction), and the boundary at the contact or far edge depends on the device geometry.

In a BJT, the base quasi-neutral region is where the critical minority carrier transport occurs. Its width and the minority carrier diffusion length within it determine the transistor's current gain.

Current-Voltage Characteristics

The I-V relationship of a p-n junction comes directly from solving the minority carrier diffusion equation in both quasi-neutral regions. The result is the Shockley diode equation:

$I = I_0 \left(e^{qV/k_BT} - 1\right)$

where $I_0$ is the reverse saturation current, determined by the minority carrier diffusion lengths, diffusion coefficients, and equilibrium minority carrier concentrations on each side.

• Forward bias ($V > 0$): Minority carriers are injected across the junction. The exponential term dominates, and current increases rapidly with voltage.
• Reverse bias ($V < 0$): The exponential term vanishes, and only a small current $I_0$ flows due to thermal generation and collection of minority carriers.

Minority Carrier Devices

All the physics above comes together in devices that depend on minority carrier behavior. Here's how each one uses injection, transport, and recombination.

Solar Cells

A solar cell is a large-area p-n junction optimized for light absorption. Photons generate electron-hole pairs, and the built-in field of the junction separates them. Minority carriers (electrons generated in the p-side, holes generated in the n-side) diffuse toward the junction, get swept across, and contribute to photocurrent.

The diffusion length must be long enough for photogenerated minority carriers to reach the junction before recombining. This is why high-purity silicon with long carrier lifetimes is essential for efficient solar cells.

Photodetectors

Photodetectors also convert light to electrical signals, but they're optimized for sensitivity and speed rather than power generation. Types include:

• p-n photodiodes: Operated in reverse bias to widen the depletion region and speed up carrier collection
• p-i-n photodiodes: An intrinsic layer between p and n regions creates a wide absorption/collection region
• Avalanche photodiodes: Use impact ionization to multiply photogenerated carriers for higher sensitivity

Performance metrics like responsivity, dark current, and bandwidth all trace back to minority carrier transport and recombination properties.

Light-Emitting Diodes (LEDs)

LEDs work in the opposite direction from solar cells: you inject minority carriers across a forward-biased junction, and they recombine radiatively to emit photons. The photon energy (and therefore the light color) equals the bandgap energy.

LEDs require direct bandgap materials (GaAs, GaN, InGaP) because radiative recombination is efficient in these materials. Silicon, with its indirect bandgap, makes a terrible LED.

Bipolar Junction Transistors (BJTs)

BJTs are the reason this topic is in Unit 9. A BJT has three regions (emitter, base, collector) forming two back-to-back p-n junctions. The operating principle:

1. The forward-biased emitter-base junction injects minority carriers into the base
2. These minority carriers diffuse across the narrow base ($W \ll L$)
3. The reverse-biased base-collector junction collects them

The current gain $\beta$ depends on what fraction of injected carriers make it across the base without recombining. This is directly controlled by the ratio $W/L$: a thinner base and longer diffusion length give higher gain. The base doping, emitter doping, and device geometry all feed into the minority carrier physics covered throughout this guide.