The electronic properties of semiconductors are fundamentally governed by their band structures, which describe the allowed energy states for electrons. Crystalline semiconductors, such as silicon (c-Si) or gallium arsenide (GaAs), exhibit long-range atomic order, leading to well-defined band structures with distinct valence and conduction bands separated by a bandgap. In contrast, amorphous semiconductors, like hydrogenated amorphous silicon (a-Si:H), lack long-range order, resulting in significant modifications to their band structures, including the presence of localized states, mobility edges, and tail states. These differences have profound implications for carrier transport, optical properties, and device performance.

In crystalline semiconductors, the periodic arrangement of atoms creates a potential landscape that allows electrons to occupy delocalized states described by Bloch waves. The band structure consists of extended states forming continuous energy bands, with a clear distinction between the valence band (filled with electrons) and the conduction band (empty at absolute zero temperature). The energy difference between the top of the valence band and the bottom of the conduction band defines the bandgap, which is sharp and well-defined in crystals. For example, c-Si has an indirect bandgap of approximately 1.1 eV, while GaAs has a direct bandgap of about 1.4 eV. The absence of disorder in crystalline materials means that carriers (electrons and holes) can move freely through these extended states, leading to high carrier mobilities. For instance, electron mobility in c-Si can exceed 1000 cm²/Vs, while in GaAs, it can reach 8000 cm²/Vs.

Amorphous semiconductors, on the other hand, lack translational symmetry due to their disordered atomic structure. This disorder introduces localized states within the bandgap, which arise from defects, dangling bonds, and variations in bond lengths and angles. These localized states significantly alter the electronic properties compared to crystalline materials. The band structure of amorphous semiconductors is often described using the concept of mobility edges, proposed by Mott and Davis. Mobility edges separate extended states, where carriers can move freely, from localized states, where carriers are trapped and require thermal or tunneling assistance to contribute to conduction. In a-Si:H, the mobility edge for electrons lies near the conduction band, while for holes, it lies near the valence band. The energy region between these mobility edges is termed the mobility gap, which is broader than the bandgap of crystalline counterparts due to the presence of tail states.

Tail states are another critical feature of amorphous semiconductors. These states arise from the statistical fluctuations in bond lengths and angles, creating a distribution of energy levels that extend into the bandgap. The density of states (DOS) in amorphous materials shows exponential decays from the mobility edges into the gap, known as Urbach tails. The characteristic energy of these tails, called the Urbach energy, quantifies the disorder in the material. For a-Si:H, the Urbach energy typically ranges from 50 to 100 meV, depending on deposition conditions and hydrogen content. These tail states act as traps for charge carriers, reducing their effective mobility and contributing to phenomena like stretched-exponential recombination in photoconductive devices.

Localized states in amorphous semiconductors are primarily caused by defects such as dangling bonds, which introduce deep-level traps within the mobility gap. In a-Si:H, hydrogen passivation reduces the density of dangling bonds, but residual defects still dominate the electronic properties. The density of these mid-gap states can range from 10¹⁵ to 10¹⁷ cm⁻³eV⁻¹, significantly higher than in crystalline materials. These states act as recombination centers, limiting the lifetime of photoexcited carriers and reducing the efficiency of optoelectronic devices. For example, the minority carrier lifetime in a-Si:H is typically in the microsecond range, compared to milliseconds in c-Si.

The differences in band structure between crystalline and amorphous semiconductors directly impact carrier transport mechanisms. In crystalline materials, carriers move through extended states with high mobility, and scattering is primarily due to phonons or impurities. In amorphous materials, transport occurs via a combination of hopping between localized states and thermal excitation to mobility edges. At low temperatures, variable-range hopping dominates, where carriers tunnel between localized states with energies close to the Fermi level. At higher temperatures, carriers are thermally excited to the mobility edges, where they exhibit band-like transport with much lower mobility than in crystals. For instance, the room-temperature electron mobility in a-Si:H is typically around 1 cm²/Vs, several orders of magnitude lower than in c-Si.

Optical properties also differ significantly due to the modified band structure. Crystalline semiconductors exhibit sharp absorption edges corresponding to the bandgap energy, with absorption coefficients rising rapidly above the gap. In amorphous materials, the absorption edge is less sharp due to tail states, and absorption extends into the sub-gap region due to transitions involving localized states. The absorption coefficient of a-Si:H at 1.5 eV, for example, is about 10⁴ cm⁻¹, compared to nearly zero for c-Si at the same energy. This sub-gap absorption is crucial for applications like thin-film solar cells, where a-Si:H can absorb a broader spectrum of light despite its disordered structure.

The presence of localized states and tail states in amorphous semiconductors also affects doping behavior. In crystalline materials, dopants introduce shallow levels near the band edges, leading to predictable changes in conductivity. In amorphous materials, dopants can create additional defects or modify the existing density of states, making doping less efficient. For example, phosphorus doping in a-Si:H can increase the Fermi level toward the conduction band but also introduces additional tail states that may trap carriers. This compensation effect limits the maximum achievable conductivity in amorphous semiconductors compared to their crystalline counterparts.

Device applications of amorphous semiconductors often leverage their unique band structure properties. For instance, a-Si:H is widely used in thin-film transistors (TFTs) for display backplanes due to its uniform deposition over large areas and moderate carrier mobility. The localized states and tail states, however, lead to threshold voltage shifts under prolonged bias, a phenomenon known as the Staebler-Wronski effect. This instability is a direct consequence of defect creation and carrier trapping in the disordered network. In contrast, crystalline semiconductors like c-Si or GaAs are preferred for high-performance devices where high mobility and low defect densities are critical.

In summary, the band structures of crystalline and amorphous semiconductors differ fundamentally due to the presence or absence of long-range order. Crystalline materials exhibit sharp band edges and high carrier mobility, while amorphous materials feature localized states, mobility edges, and tail states that profoundly influence their electronic and optical properties. Understanding these differences is essential for designing materials and devices tailored to specific applications, from high-efficiency solar cells to flexible electronics. The interplay between disorder and electronic states continues to be a rich area of research, with implications for emerging materials like organic semiconductors and oxide glasses.