Magnetoresistance refers to the change in electrical resistance of a material when subjected to an external magnetic field. In semiconductors, this phenomenon arises due to the interaction between charge carriers and the magnetic field, leading to modifications in carrier trajectories, scattering rates, and energy states. The effects can be broadly categorized into classical magnetoresistance, quantum magnetoresistance, and giant magnetoresistance, each governed by distinct physical mechanisms. Material-specific behaviors further influence these effects, with semiconductors like InSb and graphene exhibiting unique responses due to their electronic structures. Theoretical frameworks such as the Drude model and Boltzmann transport equation provide quantitative descriptions of these phenomena.

Classical magnetoresistance is primarily observed in bulk semiconductors and originates from the Lorentz force acting on moving charge carriers. Under an applied magnetic field, carriers undergo cyclotron motion, leading to curved trajectories that increase the effective path length between scattering events. This results in a higher resistance. The Drude model offers a simplified explanation by treating carriers as free particles with an average scattering time. The magnetoresistance in this regime is typically isotropic and quadratic in weak magnetic fields, saturating in strong fields when the cyclotron frequency exceeds the scattering rate. For example, in n-type InSb, the high electron mobility (exceeding 50,000 cm²/Vs at low temperatures) leads to pronounced classical magnetoresistance due to long mean free paths and strong Lorentz deflection.

Quantum magnetoresistance emerges at low temperatures and high magnetic fields, where quantum mechanical effects dominate. Landau quantization becomes significant, splitting the energy spectrum into discrete Landau levels. This leads to oscillatory resistance behavior known as Shubnikov-de Haas oscillations, periodic in the inverse magnetic field. The phenomenon is sensitive to carrier density and effective mass, providing a tool for probing Fermi surface properties. Graphene exhibits an anomalous quantum Hall effect and magnetoresistance due to its linear dispersion relation and Dirac fermion behavior. Unlike conventional semiconductors, graphene’s Landau levels are unequally spaced, leading to unique oscillation patterns. The Boltzmann transport equation, incorporating quantum corrections, is often employed to model these effects by accounting for Landau level occupation and scattering between states.

Giant magnetoresistance, while more commonly associated with metallic multilayers, can also manifest in certain semiconductor systems under specific conditions. It involves large resistance changes due to spin-dependent scattering or carrier confinement effects. In doped semiconductors with magnetic impurities, spin-split bands can lead to significant magnetoresistance if the magnetic field alters the spin polarization of carriers. For instance, Mn-doped GaAs shows enhanced magnetoresistance due to the exchange interaction between carriers and localized magnetic moments. However, this differs from the traditional giant magnetoresistance in metallic systems, as it does not rely on alternating ferromagnetic and non-magnetic layers.

Material-specific behaviors play a crucial role in determining magnetoresistance characteristics. InSb, a narrow-bandgap semiconductor with high electron mobility, exhibits large classical magnetoresistance due to its small effective mass and long scattering times. The non-parabolicity of its conduction band further complicates the response under high magnetic fields. Graphene, with its zero-bandgap and linear energy dispersion, shows unique magnetotransport properties. The absence of a bandgap leads to a non-zero minimum conductivity even at zero carrier density, while the magnetic field induces Landau levels that are detectable in magnetoresistance measurements. The quantum Hall effect in graphene is particularly notable for its half-integer quantization, a consequence of its Dirac cone structure.

Theoretical models provide frameworks for understanding and predicting magnetoresistance behavior. The Drude model, though simplistic, captures the essence of classical magnetoresistance by relating conductivity to carrier mobility and scattering time. However, it fails to account for quantum effects or anisotropic band structures. The Boltzmann transport equation offers a more comprehensive approach by incorporating carrier distribution functions and scattering integrals. It can be extended to include magnetic field effects through the Lorentz force term in the Boltzmann equation. For quantum magnetoresistance, Kubo formalism or Green’s function techniques are often employed to handle Landau level quantization and coherent transport effects.

Temperature and impurity concentration significantly influence magnetoresistance. At higher temperatures, increased phonon scattering reduces carrier mobility, diminishing classical magnetoresistance. Impurities introduce additional scattering centers, affecting both classical and quantum regimes. In heavily doped semiconductors, the interplay between ionized impurity scattering and magnetic field effects can lead to complex magnetoresistance behavior. For example, in degenerate InSb, the magnetoresistance may exhibit a crossover from quadratic to linear dependence on the magnetic field as impurity scattering dominates.

Experimental studies have validated these theoretical predictions across various semiconductors. InSb’s magnetoresistance has been extensively measured, showing agreement with classical models at moderate fields and quantum oscillations at low temperatures. Graphene’s magnetotransport has been characterized in high-quality samples, revealing Dirac fermion physics through quantum oscillations and anomalous quantum Hall plateaus. These measurements underscore the importance of material purity and defect control in observing intrinsic magnetoresistance effects.

In summary, magnetoresistance in semiconductors encompasses classical, quantum, and giant effects, each governed by distinct mechanisms and observable under specific conditions. Material properties such as band structure, carrier mobility, and impurity content critically influence the response. Theoretical models ranging from the Drude approximation to the Boltzmann equation provide tools for analysis, though quantum regimes often require more advanced treatments. Understanding these phenomena is essential for designing semiconductor devices where magnetic field sensitivity is desired, without delving into spintronic or memory-specific applications.