In semiconductor physics, band bending is a fundamental phenomenon that occurs at surfaces and heterojunctions due to the redistribution of charge carriers. It plays a critical role in determining the electronic behavior of devices, influencing carrier transport, recombination, and the formation of potential barriers. The bending of energy bands arises from the equilibration of Fermi levels when two materials with different work functions or doping levels come into contact. This results in the formation of space charge regions, built-in potentials, and depletion layers, which are essential for understanding Schottky barriers and ohmic contacts.

At a semiconductor surface, band bending occurs due to the presence of surface states or the interaction with another material. Surface states can trap charges, leading to a net charge accumulation or depletion near the interface. For example, in an n-type semiconductor, if surface states trap electrons, a positive space charge region forms near the surface, causing the conduction and valence bands to bend upward. Conversely, in a p-type semiconductor, trapped holes induce a negative space charge region, bending the bands downward. The extent of band bending depends on the doping concentration, the density of surface states, and the external environment.

In heterojunctions, band bending arises from the alignment of two semiconductors with different bandgaps and electron affinities. When two semiconductors form a junction, their Fermi levels must align at equilibrium, leading to charge transfer across the interface. This creates a built-in potential and a depletion region where the electric field is non-zero. The band alignment can be classified into three types: straddling gap (Type I), staggered gap (Type II), or broken gap (Type III), each influencing carrier confinement and transport differently. For instance, in a Type I heterojunction like GaAs/AlGaAs, both electrons and holes are confined in the same layer, while in a Type II heterojunction like GaAs/InAs, carriers are spatially separated.

The depletion region is a key consequence of band bending, where mobile charge carriers are absent, leaving behind ionized dopants. The width of the depletion region depends on the doping concentrations and the built-in potential. For an abrupt p-n junction, the depletion width (W) can be expressed as:

W = sqrt((2ε_s (V_bi - V_a)) / (q (1/N_A + 1/N_D)))

where ε_s is the permittivity of the semiconductor, V_bi is the built-in potential, V_a is the applied bias, q is the electron charge, and N_A and N_D are the acceptor and donor concentrations, respectively. The built-in potential is determined by the difference in Fermi levels before contact and is typically in the range of 0.5 to 1.5 V for common semiconductors.

Schottky barriers form at metal-semiconductor junctions due to band bending. When a metal with a higher work function contacts an n-type semiconductor, electrons flow from the semiconductor to the metal, creating a depletion region and a potential barrier (Schottky barrier height) that opposes further electron transfer. The barrier height (φ_B) is given by:

φ_B = φ_M - χ_S

where φ_M is the metal work function and χ_S is the electron affinity of the semiconductor. For p-type semiconductors, the barrier forms when the metal work function is lower than that of the semiconductor. Schottky barriers are crucial for rectifying contacts in diodes and transistors.

Ohmic contacts, in contrast, are designed to minimize band bending and resistance at metal-semiconductor interfaces. This is achieved by heavily doping the semiconductor near the contact, reducing the depletion width to the point where carriers can tunnel through the barrier. For n-type semiconductors, a low-work-function metal or a highly doped n+ region ensures negligible barrier formation. Similarly, for p-type semiconductors, a high-work-function metal or p+ doping facilitates ohmic behavior. Ohmic contacts are essential for efficient current injection in devices like solar cells and LEDs.

The interplay between band bending and doping is critical in device operation. In MOSFETs, for example, gate-induced band bending controls the formation of inversion layers, enabling transistor action. In photodetectors, band bending separates photogenerated carriers, enhancing responsivity. The manipulation of band bending through surface passivation, doping gradients, or heterostructure engineering is a cornerstone of modern semiconductor technology.

Band bending also influences recombination dynamics. In depletion regions, the lack of free carriers reduces radiative recombination but can enhance non-radiative processes via trap states. Surface recombination velocity, a measure of how quickly carriers recombine at interfaces, is directly affected by the band bending magnitude. Techniques like surface passivation with dielectric layers or chemical treatments are employed to mitigate detrimental recombination effects.

Temperature and external biases further modulate band bending. At high temperatures, increased intrinsic carrier concentrations can narrow depletion regions, while reverse biasing a junction widens the depletion layer, increasing the electric field. These effects are exploited in devices like varactors and avalanche photodiodes.

In summary, band bending at semiconductor surfaces and heterojunctions is a cornerstone of device physics, governing the formation of depletion regions, built-in potentials, and contact properties. Understanding and controlling these phenomena enable the design of efficient Schottky barriers, ohmic contacts, and advanced electronic and optoelectronic devices. The quantitative relationships between doping, band alignment, and electric fields provide the foundation for optimizing performance across a wide range of applications.