The growth of two-dimensional materials under intentional strain is a critical area of research in semiconductor science, offering unique opportunities to engineer electronic, optical, and mechanical properties. Strain can be introduced during growth by depositing 2D materials on curved substrates or lattice-mismatched templates, leading to controlled deformation of the atomic lattice. This approach differs from post-growth strain application, as the strain is inherently built into the material during synthesis, influencing nucleation, domain formation, and final film quality.

One common method for inducing intentional strain is the use of substrates with controlled curvature. When a 2D material such as graphene or a transition metal dichalcogenide (TMDC) is grown on a curved surface, the lattice must conform to the substrate’s topography. This results in non-uniform strain distribution, where regions of compression and tension arise depending on the local curvature. For example, graphene grown on a hemispherical substrate exhibits varying strain along its surface, altering its electronic band structure. The strain modifies the Dirac cone in graphene, creating pseudo-magnetic fields that can exceed 100 tesla in highly strained regions. Similarly, TMDCs like MoS2 experience shifts in their direct bandgap under strain, with tensile strain reducing the gap and compressive strain increasing it.

Another approach involves lattice-mismatched epitaxy, where 2D materials are grown on substrates with deliberate lattice constant differences. For instance, growing graphene on hexagonal boron nitride (hBN) introduces a moiré pattern due to the slight lattice mismatch of approximately 1.8%. The resulting periodic potential modulates the electronic states, leading to superlattice effects such as Hofstadter’s butterfly spectrum in graphene-hBN heterostructures. The moiré periodicity depends on the twist angle and strain, with smaller angles producing larger superlattices. In TMDCs, strain-induced moiré patterns can localize excitons, enhancing photoluminescence or creating correlated electronic states.

Strain also affects the growth kinetics of 2D materials. On strained substrates, adatom migration and nucleation are influenced by the local strain energy landscape. Compressive strain typically increases the energy barrier for adatom attachment, leading to slower growth rates and smaller domains. Conversely, tensile strain can enhance adatom mobility, promoting larger single-crystalline films. For example, WS2 grown on a stretched polymer substrate shows improved domain size due to reduced nucleation density under tensile strain. The strain distribution can be further controlled by using substrates with engineered step edges or patterned surfaces, which guide the growth of aligned 2D crystals.

The electronic properties of strained 2D materials are profoundly modified by the altered bond lengths and angles. In graphene, strain breaks the sublattice symmetry, inducing a bandgap that is tunable with strain magnitude. A uniaxial strain of 10% can open a bandgap of up to 0.2 eV, making strained graphene suitable for semiconductor applications. In TMDCs, strain shifts the valence and conduction band edges, affecting carrier effective masses and mobilities. Biaxial tensile strain of 2% in monolayer MoS2 reduces its bandgap by approximately 0.1 eV while increasing the electron mobility by over 30%. These changes are critical for designing optoelectronic devices with tailored performance.

Moiré patterns in strained 2D heterostructures create periodic potential landscapes that host exotic quantum phenomena. In twisted bilayer graphene, strain can modify the magic angle where flat bands emerge, leading to correlated insulator and superconducting states. The interplay between strain and twist angle determines the moiré wavelength and the resulting electronic phases. For instance, a combination of 0.5% heterostrain and a 1.1-degree twist angle in bilayer graphene can significantly alter the superconducting transition temperature. Similarly, in TMD heterostructures, strain-engineered moiré patterns enable the observation of moiré excitons and quantum emitters with spatially localized emission.

The growth of 2D materials on vicinal substrates—surfaces with intentional miscut angles—provides another route to strain engineering. Step edges on these substrates introduce periodic strain fields that template the growth of aligned nanostructures. For example, graphene grown on a vicinal SiC surface forms nanoribbons with width-dependent electronic properties due to edge states and quantum confinement. The step-induced strain also affects the doping level, with compressed regions showing p-type behavior and tensile regions exhibiting n-type characteristics.

Thermal mismatch between the 2D material and substrate further contributes to strain during growth cooling. For instance, the difference in thermal expansion coefficients between graphene and copper leads to compressive strain in graphene upon cooling from growth temperatures. This strain can reach 0.3% and influences the density of wrinkles and defects in the film. Similar effects occur in TMDCs grown on sapphire, where thermal strain modifies the phonon frequencies and exciton binding energies.

The controlled introduction of strain during growth enables the fabrication of 2D materials with properties unattainable in their unstrained counterparts. By selecting appropriate substrates and growth conditions, researchers can tailor the strain profile to achieve desired electronic, optical, and mechanical behaviors. Future advancements in strain engineering will likely focus on achieving atomic-level precision in strain distribution, enabling the realization of novel quantum phases and device functionalities. The continued exploration of strained 2D materials promises to unlock new possibilities in semiconductor technology, from high-performance electronics to quantum information science.