3.2 Ballistic and diffusive transport

Electrons in materials can move in two ways: ballistic and diffusive transport. Ballistic transport happens when electrons zoom through without hitting anything, while diffusive transport involves lots of bumping and scattering. These different modes affect how well electricity flows.

Understanding electron movement is key to making tiny electronic devices work better. Ballistic transport leads to faster, more efficient circuits, while diffusive transport is more common in everyday electronics. Scientists are always trying to push the limits and get more ballistic behavior.

Ballistic and Diffusive Transport Regimes

Ballistic Transport and Mean Free Path

• Ballistic transport occurs when electrons move through a material without scattering
• Electrons maintain their momentum and energy during ballistic transport
• Typically observed in very small devices or at low temperatures
• Mean free path represents the average distance an electron travels before scattering
• In ballistic transport, device dimensions are smaller than the mean free path
• Ballistic transport leads to higher conductivity and reduced energy dissipation

Diffusive Transport and Scattering Events

• Diffusive transport involves frequent scattering of electrons as they move through a material
• Electrons change direction and lose energy due to collisions with lattice vibrations, impurities, or other electrons
• Occurs when device dimensions are larger than the mean free path
• Scattering events in diffusive transport reduce electron mobility and conductivity
• Diffusive transport is more common in conventional electronic devices
• Transition from diffusive to ballistic transport can be achieved by reducing device size or lowering temperature

Scattering and Carrier Mobility

Scattering Mechanisms in Semiconductor Materials

• Lattice scattering results from electron interactions with vibrating atoms in the crystal lattice
• Increases with temperature due to increased lattice vibrations
• Impurity scattering occurs when electrons collide with dopant atoms or crystal defects
• Dominates at low temperatures when lattice vibrations are reduced
• Electron-electron scattering involves collisions between electrons in the conduction band
• Becomes significant at high carrier concentrations

Mobility and Relaxation Time

• Mobility measures how easily charge carriers move through a material in response to an electric field
• Expressed as the ratio of drift velocity to applied electric field ($\mu = v_d / E$)
• Higher mobility leads to better conductivity and device performance
• Relaxation time represents the average time between scattering events
• Longer relaxation times correspond to higher mobility and more efficient electron transport
• Mobility and relaxation time are related through the equation $\mu = q\tau / m^*$, where q is the electron charge, τ is the relaxation time, and m* is the effective mass

Conductivity and Its Relationship to Mobility

• Conductivity quantifies a material's ability to conduct electric current
• Directly proportional to carrier concentration and mobility ($\sigma = nq\mu$)
• Influenced by temperature, doping levels, and material quality
• Higher mobility generally leads to higher conductivity
• Conductivity can be improved by increasing carrier concentration or reducing scattering mechanisms

Classical Transport Model

Drude Model Fundamentals

• Drude model provides a classical description of electron transport in metals
• Assumes electrons behave like a gas of non-interacting particles
• Electrons move freely between collisions with immobile ion cores
• Collisions occur randomly with a characteristic relaxation time
• Model successfully explains basic electrical and thermal properties of metals
• Limitations include neglecting quantum mechanical effects and electron-electron interactions

Conductivity in the Drude Model

• Drude model expresses conductivity as $\sigma = ne^2\tau / m$
• n represents the electron density, e is the electron charge, τ is the relaxation time, and m is the electron mass
• Predicts linear relationship between current density and electric field (Ohm's law)
• Explains temperature dependence of conductivity in metals
• Conductivity decreases with increasing temperature due to increased scattering
• Model accurately describes DC conductivity but fails for high-frequency AC conductivity

Mobility and the Drude Model

• Mobility in the Drude model is given by $\mu = e\tau / m$
• Relates the drift velocity of electrons to the applied electric field
• Assumes constant relaxation time, which is an oversimplification for many materials
• Predicts mobility is independent of electric field strength
• Fails to account for band structure effects and complex scattering mechanisms in semiconductors
• Despite limitations, provides a useful foundation for understanding electron transport in materials