In semiconductor physics, the formation of minibands in superlattices is a critical phenomenon that arises from the periodic potential introduced by alternating layers of different materials. The Kronig-Penney model provides a simplified yet powerful framework to understand this behavior. By applying this model to superlattices, one can elucidate the electronic structure and transport properties that are pivotal for devices like quantum cascade lasers.
A superlattice consists of a periodic arrangement of two or more semiconductor materials with different bandgaps. The periodicity is typically on the order of nanometers, creating a potential landscape that modifies the electronic states of the system. The Kronig-Penney model approximates this potential as a series of rectangular barriers and wells, allowing analytical solutions to the Schrödinger equation. The model reveals that the original energy bands of the bulk materials split into subbands, or minibands, due to the superlattice periodicity.
The Kronig-Penney potential for a superlattice can be represented as a repeating sequence of barriers (height V0, width b) and wells (width a). The period of the superlattice is d = a + b. Solving the Schrödinger equation for this potential yields the dispersion relation, which determines the allowed energy states. The key result is the transcendental equation:
cos(kd) = cos(αa)cosh(βb) + (α² - β²)/(2αβ) sin(αa)sinh(βb),
where α = √(2m*E)/ħ, β = √(2m*(V0 - E))/ħ, and k is the Bloch wavevector. The solutions to this equation define the miniband structure, where energy ranges for which |cos(kd)| ≤ 1 are allowed (minibands) and others are forbidden (minigaps).
The width and spacing of minibands depend on the superlattice parameters. For instance, increasing the barrier height V0 or width b reduces the miniband width, leading to more localized states. Conversely, thinner barriers or smaller V0 enhance coupling between wells, broadening the minibands. The miniband width ΔE can be approximated for weak coupling as ΔE ≈ 4ħ²π²/(2m*d²) exp(-βb), showing exponential dependence on barrier thickness.
Miniband formation has profound implications for electron transport. In a superlattice, electrons propagate through the minibands via Bloch oscillations, provided the applied electric field is small enough to prevent Zener tunneling between minibands. The group velocity vg = (1/ħ) dE/dk and effective mass m* = ħ²/(d²E/dk²) are directly derived from the miniband dispersion, influencing the carrier mobility.
Quantum cascade lasers (QCLs) exploit miniband transport for their operation. These devices rely on electron transitions between quantized states in the conduction band, engineered through precise superlattice design. In a QCL, an electron injects into an upper state of a miniband, undergoes a radiative transition to a lower state, and then tunnels through the miniband to the next period. The periodic nature of the superlattice ensures population inversion across multiple stages, enabling high-power, mid-infrared to terahertz emission.
The design of QCLs involves optimizing the miniband structure for efficient carrier injection and extraction. For example, the miniband width must balance two competing effects: a narrow miniband reduces scattering and enhances upper-state lifetime, while a broad miniband improves carrier injection efficiency. Typical miniband widths in QCLs range from 10 to 50 meV, depending on the desired emission wavelength and operating temperature.
Another critical aspect is the alignment of minibands under an applied electric field. Stark tuning shifts the minibands relative to each other, and the design must ensure resonant tunneling conditions are maintained across the operating bias. The electric field at which alignment occurs is given by F = ΔE/(ed), where ΔE is the energy separation between minibands and d is the superlattice period. Fields in QCLs typically range from 10 to 100 kV/cm.
The performance of QCLs is also influenced by scattering mechanisms within minibands. Optical phonon scattering, interface roughness, and impurity scattering can all reduce the upper-state lifetime and degrade device efficiency. Engineering the superlattice to minimize these effects while maintaining the desired miniband structure is a key challenge. For instance, using materials with large optical phonon energies, such as GaN-based superlattices, can suppress phonon scattering at higher temperatures.
Beyond QCLs, miniband engineering has applications in other devices. Superlattice infrared photodetectors utilize miniband absorption for broadband detection, while resonant tunneling diodes exploit miniband alignment for negative differential resistance. The principles of miniband formation also extend to novel materials like graphene superlattices, where Dirac electrons exhibit unique miniband phenomena.
In summary, the Kronig-Penney model provides a foundational understanding of miniband formation in superlattices, enabling the design of advanced optoelectronic devices. Quantum cascade lasers, in particular, benefit from precise control over miniband widths and alignments, achieving high performance through tailored electronic states. The continued exploration of miniband physics promises further innovations in semiconductor technology.