Drift and diffusion are the two fundamental ways carriers move through a semiconductor. Every current you analyze in a metal-semiconductor junction is some combination of these two mechanisms, so getting comfortable with them is essential.

Drift current results from an applied electric field pushing carriers along. Diffusion current results from carriers spreading out from regions of high concentration to low concentration. The total current density in a semiconductor is always the sum of both components.

Drift Current in Semiconductors

When you apply an electric field across a semiconductor, it exerts a force on free carriers. Electrons drift opposite to the field direction (they're negative), while holes drift in the same direction as the field.

The drift current density is given by:

$J_{drift} = qn\mu E$

where $q$ is the elementary charge, $n$ is the carrier concentration, $\mu$ is the carrier mobility, and $E$ is the electric field strength. This tells you that drift current scales directly with both the field and how easily carriers can move (their mobility).

Mobility itself depends on how often carriers get scattered. Lattice vibrations (phonons), ionized impurities, and defects all scatter carriers and reduce mobility.

Diffusion Current in Semiconductors

Even without an electric field, carriers are in constant random thermal motion. If there's a concentration gradient, more carriers will randomly move away from the high-concentration region than toward it. The net result is a diffusion current.

The diffusion current density for electrons is:

$J_{diff} = qD\frac{dn}{dx}$

where $D$ is the diffusion coefficient and $\frac{dn}{dx}$ is the carrier concentration gradient. A steeper gradient produces a larger diffusion current.

Einstein Relation for Diffusion Coefficient

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

$D = \frac{kT}{q}\mu$

where $k$ is Boltzmann's constant and $T$ is temperature. The quantity $\frac{kT}{q}$ is the thermal voltage, roughly 26 mV at room temperature (300 K).

This relation makes physical sense: carriers with higher mobility also diffuse more readily, since both processes depend on how freely carriers can move. You'll use this relation frequently when analyzing bipolar transistors, solar cells, and Schottky diodes.

Carrier Mobility

Carrier mobility ( $\mu$ ) quantifies how easily a carrier moves through the lattice in response to an electric field. It's defined as:

$\mu = \frac{v_d}{E}$

where $v_d$ is the drift velocity. Higher mobility means carriers accelerate more effectively for a given field, which directly translates to better device performance.

Mobility vs. Conductivity

Electrical conductivity ties together carrier concentration and mobility:

$\sigma = qn\mu$

You can increase conductivity two ways: add more carriers through doping (increase $n$ ), or improve how freely those carriers move (increase $\mu$ ). In practice, there's a trade-off, since heavy doping introduces more impurity scattering, which reduces $\mu$ .

Factors Affecting Mobility

Lattice scattering (phonons): Thermal vibrations of the crystal lattice deflect carriers. This effect gets worse at higher temperatures.
Impurity scattering: Ionized dopant atoms create local electric fields that deflect passing carriers. Dominates in heavily doped material.
Surface and interface scattering: Roughness and defects at surfaces scatter carriers, particularly important in thin films and nanoscale devices.
Carrier-carrier scattering: At very high carrier densities, carriers interact with each other, further reducing mobility.

Temperature Dependence of Mobility

Mobility has a characteristic temperature behavior that depends on which scattering mechanism dominates:

At high temperatures, lattice scattering dominates and mobility decreases as temperature rises (more phonons means more scattering).
At low temperatures, impurity scattering dominates. Mobility can actually increase with temperature here because faster-moving carriers are deflected less by ionized impurities.

The crossover between these regimes depends on doping level. Empirical models like the Caughey-Thomas model capture this behavior for device simulation.

Velocity Saturation

At low electric fields, drift velocity increases linearly with field ( $v_d = \mu E$ ). But this can't continue forever. At high fields, the drift velocity levels off at a maximum value called the saturation velocity $v_{sat}$ .

Electric Field Effects on Velocity

The progression looks like this:

At low fields, $v_d = \mu E$ (linear regime).
As the field increases, carriers transfer more energy to the lattice through optical phonon emission, and velocity gains slow down.
Eventually the velocity plateaus at $v_{sat}$ , regardless of further field increases.

The field at which saturation begins depends on the semiconductor material, carrier type, and temperature.

Saturation Velocity in Semiconductors

Typical saturation velocities in silicon:

Electrons: approximately $1 \times 10^7$ cm/s
Holes: approximately $8 \times 10^6$ cm/s

The saturation velocity represents a balance point where the energy carriers gain from the field equals the energy they lose to scattering per unit time.

Drift current in semiconductors, Electric Current | Boundless Physics

Impact on Device Performance

Velocity saturation has major consequences for modern devices:

It caps the maximum current a transistor can deliver, since current depends on carrier velocity.
In short-channel MOSFETs, carriers can reach $v_{sat}$ across much of the channel, causing the drain current to saturate at a lower voltage than the long-channel model predicts.
Transconductance (how effectively gate voltage controls drain current) is reduced compared to the non-saturated case.
Any high-speed or high-power device design must account for velocity saturation effects.

Hot Carrier Effects

Hot carriers are electrons or holes that have acquired kinetic energy well above the thermal equilibrium value, typically from strong electric fields. They can damage devices and degrade performance over time, making them a major reliability concern.

Energy Balance in High Fields

Under normal conditions, carriers gain energy from the field and lose it through scattering at roughly equal rates, maintaining thermal equilibrium. In high-field regions (like near the drain of a MOSFET), carriers can gain energy faster than they lose it. This creates a non-equilibrium distribution where some carriers have very high energies.

The effective "temperature" of these carriers is much higher than the lattice temperature, which is why they're called "hot."

Hot Carrier Generation Mechanisms

Impact ionization: A hot carrier with enough energy collides with a lattice atom and generates a new electron-hole pair. This is the most fundamental hot carrier process.
Channel hot electrons (CHE): In MOSFETs, energetic electrons in the channel can gain enough energy to overcome the oxide barrier and get injected into the gate oxide. This causes threshold voltage shifts and transconductance degradation over time.
Drain avalanche hot carrier (DAHC) injection: Near the drain of high-voltage devices, impact ionization creates additional hot carriers that can also be injected into the gate oxide.

Impact Ionization and Avalanche Breakdown

Impact ionization becomes dangerous when it cascades:

A carrier gains enough energy from the field to ionize a lattice atom, creating an electron-hole pair.
The newly created carriers are also accelerated by the field.
If they gain enough energy, they cause further ionization events.
When this multiplication becomes self-sustaining, avalanche breakdown occurs and current increases rapidly.

Avalanche breakdown sets the maximum operating voltage for most power devices. It's also deliberately exploited in avalanche photodiodes (for signal amplification) and in Zener/avalanche diodes (for voltage regulation).

Tunneling Transport

Tunneling is a quantum mechanical process where carriers pass through a potential barrier they don't have enough classical energy to overcome. In metal-semiconductor junctions, tunneling becomes a significant or even dominant transport mechanism when barriers are thin or doping is very heavy.

Quantum Mechanical Tunneling

Because carriers behave as waves, their wavefunction doesn't abruptly drop to zero at a barrier. Instead, it decays exponentially inside the barrier. If the barrier is thin enough, there's a non-zero probability of the carrier appearing on the other side.

The tunneling probability decreases exponentially with:

Increasing barrier thickness
Increasing barrier height
Increasing carrier effective mass

Tunneling in Semiconductor Junctions

In heavily doped p-n junctions, the depletion region becomes extremely narrow (a few nanometers). Carriers can tunnel directly through this thin barrier, producing a tunneling current.

This is the operating principle behind tunnel diodes (Esaki diodes), which exhibit negative differential resistance: over a certain voltage range, increasing the voltage actually decreases the current. This property makes them useful for high-speed switching and oscillator circuits.

Tunneling also matters in heavily doped regions of bipolar transistors and in metal-semiconductor contacts where the semiconductor is degenerately doped (forming ohmic contacts).

Fowler-Nordheim Tunneling

Fowler-Nordheim (FN) tunneling occurs when a strong electric field bends the barrier into a triangular shape, allowing carriers to tunnel through the upper portion of the barrier rather than its full width.

The FN tunneling current density is:

$J_{FN} \propto E^2 \exp\left(-\frac{4\sqrt{2m^*}\phi^{3/2}}{3q\hbar E}\right)$

where $m^*$ is the carrier effective mass, $\phi$ is the barrier height, $\hbar$ is the reduced Planck constant, and $E$ is the electric field.

FN tunneling is the primary mechanism used to program and erase flash memory and EEPROM devices. Electrons are injected through the thin gate oxide into a floating gate, where they remain trapped and store data.

Ballistic Transport

Ballistic transport occurs when a carrier traverses a device without experiencing any scattering events. The carrier's motion is determined entirely by the potential profile and device geometry, not by random collisions.

Drift current in semiconductors, 6.6 The Hall Effect – Douglas College Physics 1207

Ballistic vs. Diffusive Transport

In diffusive transport, carriers scatter frequently, and their motion is well described by the drift-diffusion equations. This is the normal regime for most conventional devices.
In ballistic transport, the device length is shorter than the mean free path (the average distance between scattering events), so carriers travel from one contact to the other without scattering.

The key parameter is the ratio of device length $L$ to mean free path $\lambda$ . When $L \ll \lambda$ , transport is ballistic. When $L \gg \lambda$ , transport is diffusive.

Ballistic Transport in Nanostructures

Nanostructures like nanowires, carbon nanotubes, and quantum dots can exhibit ballistic transport because their dimensions are small enough to fall below the mean free path. In this regime, you can observe quantum effects such as:

Conductance quantization: Current flows in discrete steps as a function of the channel width or gate voltage, with each step equal to $\frac{2e^2}{h}$ .
Quantum Hall effect: In 2D systems under strong magnetic fields, Hall conductance is quantized.

Ballistic devices can potentially operate at higher speeds and lower power than their diffusive counterparts because carriers don't lose energy to scattering.

Mean Free Path and Scattering

The mean free path $\lambda$ is the average distance a carrier travels between consecutive scattering events. It depends on:

Material quality: Fewer defects means longer mean free path.
Temperature: Higher temperature increases phonon scattering, shortening $\lambda$ .
Carrier type: Electrons and holes have different effective masses and scattering cross-sections.

In silicon at room temperature, the electron mean free path is roughly 10-30 nm. As transistor gate lengths have shrunk below this range, ballistic and quasi-ballistic transport effects have become increasingly relevant to device modeling.

Thermionic Emission

Thermionic emission is the process by which carriers with sufficient thermal energy overcome a potential barrier and are emitted over it. This is one of the most important current transport mechanisms at metal-semiconductor (Schottky) junctions.

Thermionic Emission Theory

The theory, developed by Richardson and Dushman, describes how thermally energetic carriers in the high-energy tail of the Fermi-Dirac distribution can surmount a barrier. The resulting current density is:

$J = A^* T^2 \exp\left(-\frac{q\phi}{kT}\right)$

where $A^*$ is the effective Richardson constant (which depends on the carrier effective mass), $T$ is temperature, $\phi$ is the barrier height, and $k$ is Boltzmann's constant.

Richardson's Law

Richardson's law captures two competing effects:

Current increases strongly with temperature (the $T^2$ prefactor and the exponential both grow with $T$ ), because more carriers have enough energy to clear the barrier.
Current decreases exponentially with barrier height $\phi$ , because a taller barrier excludes more carriers.

By measuring thermionic emission current as a function of temperature, you can extract the barrier height $\phi$ from a Richardson plot (plotting $\ln(J/T^2)$ vs. $1/T$ , which yields a straight line with slope $-q\phi/k$ ).

Schottky Barrier and Contact Resistance

When a metal contacts a semiconductor, the difference in their work functions creates a Schottky barrier at the interface. The height of this barrier controls:

The thermionic emission current across the junction
The contact resistance, which adds parasitic resistance to the device

For device applications requiring low-resistance ohmic contacts, you want to minimize the Schottky barrier height. Strategies include:

Choosing metals with work functions that align favorably with the semiconductor band edges
Using very heavy doping at the semiconductor surface (so carriers can tunnel through the thin barrier)
Interface engineering techniques such as inserting thin interlayers to modify the barrier

Surface and Interface Effects

Surfaces and interfaces are where much of the action happens in metal-semiconductor junctions. The abrupt termination of the crystal lattice at a surface, or the transition between two different materials at an interface, introduces electronic states and scattering mechanisms that don't exist in the bulk.

Surface States and Fermi Level Pinning

Surface states are electronic energy levels that exist at the semiconductor surface due to dangling bonds (unsatisfied atomic bonds at the crystal termination), defects, or adsorbed atoms.

When the density of surface states is high enough, they can absorb enough charge to control the surface potential regardless of what metal is placed on top. This is Fermi level pinning: the Fermi level at the surface is "pinned" to a position within the bandgap determined by the surface states, not by the metal work function.

Fermi level pinning is a major practical problem because it means you can't freely tune the Schottky barrier height just by choosing different metals. Many III-V semiconductors (like GaAs) suffer from severe Fermi level pinning, making it difficult to form low-resistance ohmic contacts.

Carrier Scattering at Interfaces

Interfaces introduce additional scattering that reduces carrier mobility near the junction:

Interface roughness scattering: Atomic-scale variations in the interface position create a fluctuating potential that deflects carriers. This is particularly important at semiconductor-oxide interfaces in MOSFETs.
Coulomb scattering: Charged defects or trapped charges at the interface create local electric fields that scatter passing carriers.

These effects are most significant in thin-film devices and nanostructures where carriers spend a large fraction of their transit time near interfaces.

Surface Recombination and Leakage Current

Surface recombination occurs when electrons and holes recombine at surface states, which act as recombination centers (similar to Shockley-Read-Hall recombination in the bulk, but at the surface). This reduces carrier lifetime and degrades device efficiency, which is especially harmful in solar cells and photodetectors.

Surface passivation techniques reduce surface recombination by minimizing the density of active surface states:

Dielectric coatings (e.g., $SiO_2$ , $Si_3N_4$ , or $Al_2O_3$ on silicon)
Chemical treatments that terminate dangling bonds (e.g., hydrogen passivation)

Leakage current at surfaces and interfaces arises from carrier generation and recombination at defect states. It increases power consumption and degrades noise performance. Device-level mitigation strategies include guard rings and field plates, which shape the electric field to reduce surface leakage paths.

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